Research ArticleTravelling Wave Solutions for the Coupled IBq Equations byUsing the tanh-coth Method

Omer Faruk Gozukizil1 and Samil Akcagil2

1 Department of Mathematics Sakarya University 9054187 Sakarya Turkey2 Vocational School of Pazaryeri Bilecik Seyh Edebali University 9011800 Bilecik Turkey

Correspondence should be addressed to Omer Faruk Gozukizil farukgsakaryaedutr

Received 4 May 2013 Accepted 9 December 2013 Published 19 January 2014

Academic Editor Ch Tsitouras

Copyright copy 2014 O F Gozukizil and S Akcagil This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Based on the availability of symbolic computation the tanh-coth method is used to obtain a number of travelling wave solutionsfor several coupled improved Boussinesq equations The abundant new solutions can be seen as improvement of the previouslyknown data The obtained results in this work also demonstrate the efficiency of the method

1 Introduction

We consider the following two coupled improved Boussinesq(IMBq) equations of Sobolev type

119906119905119905minus 1205722119906119909119909119905119905

minus 119906119909119909= 119891(119906 119908)

119909119909 119909 isin R 119905 gt 0

119908119905119905minus 1205722119908119909119909119905119905

minus 119908119909119909= 119892(119906 119908)

119909119909 119909 isin R 119905 gt 0

(1)

where 119891 and 119892 are given nonlinear functions 119906(119909 119905) and119908(119909 119905) are unknown functions and 120572 is a constant that hasbeen derived to describe bidirectional wave propagation inseveral studies for instance in a Toda lattice model with atransversal degree of freedom in a two-layered lattice modeland in a diatomic lattice De Godefroy has studied (1) as theCauchy problem under certain conditions and showed thatthe solution for the Cauchy problem of this system blows upin finite time [1] Wang and Li have considered the Cauchyproblem for (1) proved the existence and uniqueness of theglobal solution and given sufficient conditions of blow-upof the solution in finite time by convex methods [2] TheCauchy problem for (1) has been studied and established theconditions for the global existence and finite-time blow-up ofsolutions in Sobolev spaces119867119904 times119867119904 for 119904 gt 12 [3] For moreinformation we refer the reader to [3] and references therein

Chen and Zhang have considered the initial boundaryvalue problem for the system of the generalized IMBq typeequations

119906119905119905minus 119860119906119909119909minus 119861119906119909119909119905119905

= 119898(119908)119909119909 0 lt 119909 lt 119897

1 119905 gt 0

119908119905119905minus 119861119908119909119909119905119905

= 119899(119908)119909119909 0 lt 119909 lt 119897

1 119905 gt 0

(2)

where 119906(119909 119905) and 119908(119909 119905) are unknown functions 119860 119861 gt 0are constants and119898 and 119899 are the given nonlinear functionsAs a result they have proved the existence and uniquenessof the global generalized solution and the global classicalsolution [4]

Rosenau is concerned with the problem of how todescribe the dynamics of a dense lattice via the system

119906119905119905= (119906 + 2120572119906119908)

119909119909+

1

12

119906119909119909119905119905

119908119905119905= (2119908 + 120573119908

2+ 1205721199062)119909119909+

1

12

119908119909119909119905119905

(3)

where 120572 120573 gt 0 are constants and pointed out that (3) was aconvenient vehicle to study the dynamics of a dense lattice [5]

In [6] a transversal degree of freedom was introducedin the Toda lattice For different order of magnitude of thelongitudinal and transversal strains coupled and uncoupled

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 486269 14 pageshttpdxdoiorg1011552014486269

2 Journal of Applied Mathematics

equations for these fields were derived in the discrete case aswell as the continuum limit The system

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(4)

has been obtained for the longitudinal and transversal strains119897 and 1119887 are characteristic lengths in the model 120573 is theirratio (ie120573 = 119897119887)120588denotes the linearmass density and120572 gt 0is a constant Besides travellingwave solutions andnumericalsolutions of the system have been found

Turitsyn has proved the existence of blow-up for thecontinuum limit model of the Toda lattice with a transversaldegree of freedom analytically and analysed the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(5)

Turitsyn has showed that the sufficient criterion for thecollapse in this model may be formulated as the requirementof nonpositively of the system Hamiltonian [7]

Pego et al have studied the stability of solitary waves oftwo coupled improved Boussinesq equations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(6)

which model weakly nonlinear vibrations in a cubic latticeThey have encountered this system for which the classicmethod fails [8]

Fan and Tian have expanded themodel in [6] and consid-ered the following general Toda dense lattice equation whichis governed by the following coupled improved Boussinesqequation

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(7)

They have studied compacton solutions andmulticompactonsolutions of transversal degree of freedom by direct sine andcosine method and found peakon solutions Moreover theyhave presented more solitary wave solutions for a specialcase of (7) using improved sin-cos and improved sinh-coshmethod [9]

The paper is arranged as followsWe described the outlineof the tanh-coth method in the following section and derivedvarious exact travelling wave solutions of the coupled IBqequations (3) (4) (5) (6) and (7) in the next sections Finallywe summarized our conclusions in the last section

2 Outline of the tanh-coth Method

Wazwaz has summarized the tanh-coth method [10] in thefollowing manner

(i) First consider a general form of nonlinear equation

119875 (119906 119906119905 119906119909 119906119909119909 ) = 0 (8)

(ii) To find the traveling wave solution of (8) the wavevariable 120585 = 119909 minus 119881119905 is introduced so that

119906 (119909 119905) = 119880 (120583120585) (9)

Based on this one may use the following changes

120597

120597119905

= minus119881

119889

119889120585

120597

120597119909

= 120583

119889

119889120585

1205972

1205971199092= 1205832 1198892

1198891205852

1205973

1205971199093= 1205833 1198893

1198891205853

(10)

and so on for other derivatives Using (10) changes thePDE (8) to an ODE

119876(1198801198801015840 11988010158401015840 ) = 0 (11)

(iii) If all terms of the resultingODE contain derivatives in120585 then by integrating this equation and by consider-ing the constant of integration to be zero one obtainsa simplified ODE

(iv) A new independent variable

119884 = tanh (120583120585) (12)

is introducedwhich leads to the change of derivatives

119889

119889120585

= 120583 (1 minus 1198842)

119889

119889119884

1198892

1198891205852= minus2120583

2119884 (1 minus 119884

2)

119889

119889119884

+ 1205832(1 minus 119884

2)

2 1198892

1198891198842

1198893

1198891205853= 21205833(1 minus 119884

2) (31198842minus 1)

119889

119889119884

minus 61205833119884(1 minus 119884

2)

2 1198892

1198891198842+ 1205833(1 minus 119884

2)

3 1198893

1198891198843

1198894

1198891205854= minus8120583

4119884 (1 minus 119884

2) (31198842minus 2)

119889

119889119884

+ 41205834(1 minus 119884

2)

2

(91198842minus 2)

1198892

1198891198842

minus 121205834119884(1 minus 119884

2)

3 1198893

1198891198843

+ 1205834(1 minus 119884

2)

4 1198894

1198891198844

(13)

where other derivatives can be derived in a similarmanner

Journal of Applied Mathematics 3

(v) The ansatz of the form

119880 (120583120585) = 119878 (119884) =

119872

sum

119896=0

119886119896119884119896+

119872

sum

119896=1

119887119896119884minus119896 (14)

is introduced where 119872 is a positive integer in mostcases that will be determined If119872 is not an integerthen a transformation formula is used to overcomethis difficulty Substituting (13) and (14) into the ODE(11) yields an equation in powers of 119884

(vi) To determine the parameter 119872 the linear terms ofthe highest order in the resulting equation with thehighest order nonlinear terms are balanced With119872determined one collects all coefficients of powers of119884 in the resulting equation where these coefficientshave to vanish This will give a system of algebraicequations involving the 119886

119896and 119887119896 (119896 = 0 119872) 119881

and 120583 Having determined these parameters knowingthat119872 is a positive integer in most cases and using(14) one obtains an analytic solution in a closed form

3 The Travelling Wave Solutions

31 The First Coupled We first consider the following twocoupled IBq equations

119906119905119905= (119906 + 2120572119906119908)

119909119909+

1

12

119906119909119909119905119905

119908119905119905= (2119908 + 120573119908

2+ 1205721199062)119909119909+

1

12

119908119909119909119905119905

(15)

Using the wave variable 120585 = 119909 minus 119881119905 then by integrating thisequation twice and considering the constants of integrationto be zero the system (15) is carried to a system of ODEs

12 (1198812minus 1)119880 minus 24120572119880119882 minus 119881

211988010158401015840= 0

12 (1198812minus 2)119882 minus 12120573119882

2minus 12120572119880

2minus 119881211988210158401015840= 0

(16)

Balancing 119881211988010158401015840 with 119880119882 and 119881211988210158401015840 with 1198802 in (16) gives

119872+ 2 = 119872 +119873

2119872 = 119873 + 2

(17)

so that119872 = 119873 = 2 We consider solutions in the form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(18)

Substituting (18) into the two components of (16) and col-lecting the coefficients of 119884 give two systems of algebraic

equations for 1198860 1198861 1198862 1198871 1198872 1198880 1198881 119888211988911198892119881 and120583 Solving

these systems together leads to the following sets

1198860= minus

1198862

3

1198862=

3radic120572120573 (1198814minus 31198812+ 2)

41205722

1198880=

120573 + 4120572

8120572 (120572 minus 120573)

1198882=

minus3120573

8120572 (120572 minus 120573)

119881 =

6

radicminus61205832+ 18

120583 =

radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= minus2119887

2 119886

2=

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2

1198872=

3radic120572120573 (1198814minus 31198812+ 2)

161205722

1198880=

minus3120573 + 8120572

16120572 (120572 minus 120573)

1198882=

3120573

32120572 (120572 minus 120573)

1198892= 1198882

119881 =

6

radic241205832+ 18

120583 =

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

1198861= 1198871= 1198881= 1198891= 0

1198860=

21198872

3

1198862=

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2

1198872=

3radic120572120573 (1198814minus 31198812+ 2)

161205722

1198880=

minus120573 + 8120572

16120572 (120572 minus 120573)

1198882=

minus3120573

32120572 (120572 minus 120573)

1198892= 1198882

119881 =

6

radicminus241205832+ 18

120583 =

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2 119886

2=

3radic120572120573 (1198814minus 31198812+ 2)

41205722

1198880=

minus3120573 + 4120572

8120572 (120572 minus 120573)

1198882=

3120573

8120572 (120572 minus 120573)

119881 =

6

radic61205832+ 18

120583 =

radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(19)

4 Journal of Applied Mathematics

Consequently we obtain the following travelling wave solu-tions

1199061(119909 119905) = minus

radic120572120573 (1198814minus 31198812+ 2)

41205722

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199081(119909 119905) =

120573 + 4120572

8120572 (120572 minus 120573)

minus

3120573

8120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199062(119909 119905) = minus

3radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199082(119909 119905) =

minus3120573 + 8120572

16120572 (120572 minus 120573)

+

3120573

32120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3120573

32120572 (120572 minus 120573)

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199063(119909 119905) =

radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199083(119909 119905) =

minus120573 + 8120572

16120572 (120572 minus 120573)

minus

3120573

32120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

minus

3120573

32120572 (120572 minus 120573)

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199064(119909 119905) =

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

Journal of Applied Mathematics 5

1199084(119909 119905) =

minus3120573 + 4120572

8120572 (120572 minus 120573)

+

3120573

8120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

(20)

32 The Second Coupled We next consider the system ofequations

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(21)

Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs

12 (120572120573 minus 1205881198812)119880 minus 6120572120573

21198802+ 6120572120573119882

2+ 1205881198972119881211988010158401015840= 0

12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881

2119882 = 0

(22)

Balancing minus612057212057321198802 with 120588119897

2119881211988010158401015840 and 12120572120573119880119882 with

1205881198972119881211988210158401015840 in (22) gives

2119872 = 119872 + 2

2119873 = 119872 +119873

(23)

so that119872 = 119873 = 2 The tanh-coth method admits the use of

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(24)

Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860=

1

120573 + 1

1198862=

minus1

2 (120573 + 1)

1198872= 1198862

1198892= plusmn

radic120573 + 2

2 (120573 + 1)

1198882= minus1198892

119881 =

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

120583 =

radic3

119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

1

120573 + 1

1198862=

3120573

61205732+ 14120573 + 8

1198872= 1198862

1198892= plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

1198882= minus1198892

119881 =

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

120583 =

radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

4 minus 120573

4 (120573 + 1)

1198862=

3120573

4120573 + 1

1198880=

minus1198882

3

1198882= plusmn

9120573

2radic611989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

120583 =

radicminus120573 (3120573 + 6)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

3120573 + 4

4 (120573 + 1)

1198862=

minus3120573

4120573 + 1

1198880= minus1198882

1198882= plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

120583 =

radic3120573 (120573 + 2)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 + 8

8 (120573 + 1)

1198862=

3120573

16 (120573 + 1)

1198872= 1198862 119888

0=

21198892

3

1198882= 1198892 119889

2= plusmn

9120573

8radic2411989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

120583 =

radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

1198860=

3 (120573 + 8)

8 (120573 + 1)

1198862=

minus3120573

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2

6 Journal of Applied Mathematics

1198882= 1198892 119889

2= plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

120583 =

radic120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

(25)

In accordance with these sets of solutions we have thefollowing solutions

1199061(119909 119905) =

1

120573 + 1

minus

1

2 (120573 + 1)

tanh2radic3

119897

times (119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

minus

1

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199081(119909 119905) = ∓

radic120573 + 2

2 (120573 + 1)

tanh2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

plusmn

radic120573 + 2

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199062(119909 119905) =

1

120573 + 1

+

3120573

61205732+ 14120573 + 8

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

+

3120573

61205732+ 14120573 + 8

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199082(119909 119905) = ∓

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199063(119909 119905) =

4 minus 120573

4 (120573 + 1)

+

3120573

4120573 + 1

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199083(119909 119905) = ∓

3120573

2radic611989721205832+ 36120573 + 18

plusmn

9120573

2radic611989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199064(119909 119905) =

3120573 + 4

4 (120573 + 1)

minus

3120573

4120573 + 1

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199084(119909 119905) = ∓

9120573

2radicminus611989721205832+ 36120573 + 18

plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199065(119909 119905) =

120573 + 8

8 (120573 + 1)

+

3120573

16 (120573 + 1)

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

equations for these fields were derived in the discrete case aswell as the continuum limit The system

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(4)

has been obtained for the longitudinal and transversal strains119897 and 1119887 are characteristic lengths in the model 120573 is theirratio (ie120573 = 119897119887)120588denotes the linearmass density and120572 gt 0is a constant Besides travellingwave solutions andnumericalsolutions of the system have been found

Turitsyn has proved the existence of blow-up for thecontinuum limit model of the Toda lattice with a transversaldegree of freedom analytically and analysed the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(5)

Turitsyn has showed that the sufficient criterion for thecollapse in this model may be formulated as the requirementof nonpositively of the system Hamiltonian [7]

Pego et al have studied the stability of solitary waves oftwo coupled improved Boussinesq equations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(6)

which model weakly nonlinear vibrations in a cubic latticeThey have encountered this system for which the classicmethod fails [8]

Fan and Tian have expanded themodel in [6] and consid-ered the following general Toda dense lattice equation whichis governed by the following coupled improved Boussinesqequation

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(7)

They have studied compacton solutions andmulticompactonsolutions of transversal degree of freedom by direct sine andcosine method and found peakon solutions Moreover theyhave presented more solitary wave solutions for a specialcase of (7) using improved sin-cos and improved sinh-coshmethod [9]

The paper is arranged as followsWe described the outlineof the tanh-coth method in the following section and derivedvarious exact travelling wave solutions of the coupled IBqequations (3) (4) (5) (6) and (7) in the next sections Finallywe summarized our conclusions in the last section

2 Outline of the tanh-coth Method

Wazwaz has summarized the tanh-coth method [10] in thefollowing manner

(i) First consider a general form of nonlinear equation

119875 (119906 119906119905 119906119909 119906119909119909 ) = 0 (8)

(ii) To find the traveling wave solution of (8) the wavevariable 120585 = 119909 minus 119881119905 is introduced so that

119906 (119909 119905) = 119880 (120583120585) (9)

Based on this one may use the following changes

120597

120597119905

= minus119881

119889

119889120585

120597

120597119909

= 120583

119889

119889120585

1205972

1205971199092= 1205832 1198892

1198891205852

1205973

1205971199093= 1205833 1198893

1198891205853

(10)

and so on for other derivatives Using (10) changes thePDE (8) to an ODE

119876(1198801198801015840 11988010158401015840 ) = 0 (11)

(iii) If all terms of the resultingODE contain derivatives in120585 then by integrating this equation and by consider-ing the constant of integration to be zero one obtainsa simplified ODE

(iv) A new independent variable

119884 = tanh (120583120585) (12)

is introducedwhich leads to the change of derivatives

119889

119889120585

= 120583 (1 minus 1198842)

119889

119889119884

1198892

1198891205852= minus2120583

2119884 (1 minus 119884

2)

119889

119889119884

+ 1205832(1 minus 119884

2)

2 1198892

1198891198842

1198893

1198891205853= 21205833(1 minus 119884

2) (31198842minus 1)

119889

119889119884

minus 61205833119884(1 minus 119884

2)

2 1198892

1198891198842+ 1205833(1 minus 119884

2)

3 1198893

1198891198843

1198894

1198891205854= minus8120583

4119884 (1 minus 119884

2) (31198842minus 2)

119889

119889119884

+ 41205834(1 minus 119884

2)

2

(91198842minus 2)

1198892

1198891198842

minus 121205834119884(1 minus 119884

2)

3 1198893

1198891198843

+ 1205834(1 minus 119884

2)

4 1198894

1198891198844

(13)

where other derivatives can be derived in a similarmanner

Journal of Applied Mathematics 3

(v) The ansatz of the form

119880 (120583120585) = 119878 (119884) =

119872

sum

119896=0

119886119896119884119896+

119872

sum

119896=1

119887119896119884minus119896 (14)

is introduced where 119872 is a positive integer in mostcases that will be determined If119872 is not an integerthen a transformation formula is used to overcomethis difficulty Substituting (13) and (14) into the ODE(11) yields an equation in powers of 119884

(vi) To determine the parameter 119872 the linear terms ofthe highest order in the resulting equation with thehighest order nonlinear terms are balanced With119872determined one collects all coefficients of powers of119884 in the resulting equation where these coefficientshave to vanish This will give a system of algebraicequations involving the 119886

119896and 119887119896 (119896 = 0 119872) 119881

and 120583 Having determined these parameters knowingthat119872 is a positive integer in most cases and using(14) one obtains an analytic solution in a closed form

3 The Travelling Wave Solutions

31 The First Coupled We first consider the following twocoupled IBq equations

119906119905119905= (119906 + 2120572119906119908)

119909119909+

1

12

119906119909119909119905119905

119908119905119905= (2119908 + 120573119908

2+ 1205721199062)119909119909+

1

12

119908119909119909119905119905

(15)

Using the wave variable 120585 = 119909 minus 119881119905 then by integrating thisequation twice and considering the constants of integrationto be zero the system (15) is carried to a system of ODEs

12 (1198812minus 1)119880 minus 24120572119880119882 minus 119881

211988010158401015840= 0

12 (1198812minus 2)119882 minus 12120573119882

2minus 12120572119880

2minus 119881211988210158401015840= 0

(16)

Balancing 119881211988010158401015840 with 119880119882 and 119881211988210158401015840 with 1198802 in (16) gives

119872+ 2 = 119872 +119873

2119872 = 119873 + 2

(17)

so that119872 = 119873 = 2 We consider solutions in the form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(18)

Substituting (18) into the two components of (16) and col-lecting the coefficients of 119884 give two systems of algebraic

equations for 1198860 1198861 1198862 1198871 1198872 1198880 1198881 119888211988911198892119881 and120583 Solving

these systems together leads to the following sets

1198860= minus

1198862

3

1198862=

3radic120572120573 (1198814minus 31198812+ 2)

41205722

1198880=

120573 + 4120572

8120572 (120572 minus 120573)

1198882=

minus3120573

8120572 (120572 minus 120573)

119881 =

6

radicminus61205832+ 18

120583 =

radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= minus2119887

2 119886

2=

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2

1198872=

3radic120572120573 (1198814minus 31198812+ 2)

161205722

1198880=

minus3120573 + 8120572

16120572 (120572 minus 120573)

1198882=

3120573

32120572 (120572 minus 120573)

1198892= 1198882

119881 =

6

radic241205832+ 18

120583 =

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

1198861= 1198871= 1198881= 1198891= 0

1198860=

21198872

3

1198862=

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2

1198872=

3radic120572120573 (1198814minus 31198812+ 2)

161205722

1198880=

minus120573 + 8120572

16120572 (120572 minus 120573)

1198882=

minus3120573

32120572 (120572 minus 120573)

1198892= 1198882

119881 =

6

radicminus241205832+ 18

120583 =

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2 119886

2=

3radic120572120573 (1198814minus 31198812+ 2)

41205722

1198880=

minus3120573 + 4120572

8120572 (120572 minus 120573)

1198882=

3120573

8120572 (120572 minus 120573)

119881 =

6

radic61205832+ 18

120583 =

radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(19)

4 Journal of Applied Mathematics

Consequently we obtain the following travelling wave solu-tions

1199061(119909 119905) = minus

radic120572120573 (1198814minus 31198812+ 2)

41205722

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199081(119909 119905) =

120573 + 4120572

8120572 (120572 minus 120573)

minus

3120573

8120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199062(119909 119905) = minus

3radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199082(119909 119905) =

minus3120573 + 8120572

16120572 (120572 minus 120573)

+

3120573

32120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3120573

32120572 (120572 minus 120573)

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199063(119909 119905) =

radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199083(119909 119905) =

minus120573 + 8120572

16120572 (120572 minus 120573)

minus

3120573

32120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

minus

3120573

32120572 (120572 minus 120573)

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199064(119909 119905) =

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

Journal of Applied Mathematics 5

1199084(119909 119905) =

minus3120573 + 4120572

8120572 (120572 minus 120573)

+

3120573

8120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

(20)

32 The Second Coupled We next consider the system ofequations

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(21)

Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs

12 (120572120573 minus 1205881198812)119880 minus 6120572120573

21198802+ 6120572120573119882

2+ 1205881198972119881211988010158401015840= 0

12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881

2119882 = 0

(22)

Balancing minus612057212057321198802 with 120588119897

2119881211988010158401015840 and 12120572120573119880119882 with

1205881198972119881211988210158401015840 in (22) gives

2119872 = 119872 + 2

2119873 = 119872 +119873

(23)

so that119872 = 119873 = 2 The tanh-coth method admits the use of

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(24)

Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860=

1

120573 + 1

1198862=

minus1

2 (120573 + 1)

1198872= 1198862

1198892= plusmn

radic120573 + 2

2 (120573 + 1)

1198882= minus1198892

119881 =

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

120583 =

radic3

119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

1

120573 + 1

1198862=

3120573

61205732+ 14120573 + 8

1198872= 1198862

1198892= plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

1198882= minus1198892

119881 =

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

120583 =

radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

4 minus 120573

4 (120573 + 1)

1198862=

3120573

4120573 + 1

1198880=

minus1198882

3

1198882= plusmn

9120573

2radic611989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

120583 =

radicminus120573 (3120573 + 6)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

3120573 + 4

4 (120573 + 1)

1198862=

minus3120573

4120573 + 1

1198880= minus1198882

1198882= plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

120583 =

radic3120573 (120573 + 2)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 + 8

8 (120573 + 1)

1198862=

3120573

16 (120573 + 1)

1198872= 1198862 119888

0=

21198892

3

1198882= 1198892 119889

2= plusmn

9120573

8radic2411989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

120583 =

radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

1198860=

3 (120573 + 8)

8 (120573 + 1)

1198862=

minus3120573

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2

6 Journal of Applied Mathematics

1198882= 1198892 119889

2= plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

120583 =

radic120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

(25)

In accordance with these sets of solutions we have thefollowing solutions

1199061(119909 119905) =

1

120573 + 1

minus

1

2 (120573 + 1)

tanh2radic3

119897

times (119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

minus

1

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199081(119909 119905) = ∓

radic120573 + 2

2 (120573 + 1)

tanh2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

plusmn

radic120573 + 2

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199062(119909 119905) =

1

120573 + 1

+

3120573

61205732+ 14120573 + 8

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

+

3120573

61205732+ 14120573 + 8

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199082(119909 119905) = ∓

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199063(119909 119905) =

4 minus 120573

4 (120573 + 1)

+

3120573

4120573 + 1

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199083(119909 119905) = ∓

3120573

2radic611989721205832+ 36120573 + 18

plusmn

9120573

2radic611989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199064(119909 119905) =

3120573 + 4

4 (120573 + 1)

minus

3120573

4120573 + 1

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199084(119909 119905) = ∓

9120573

2radicminus611989721205832+ 36120573 + 18

plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199065(119909 119905) =

120573 + 8

8 (120573 + 1)

+

3120573

16 (120573 + 1)

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

(v) The ansatz of the form

119880 (120583120585) = 119878 (119884) =

119872

sum

119896=0

119886119896119884119896+

119872

sum

119896=1

119887119896119884minus119896 (14)

is introduced where 119872 is a positive integer in mostcases that will be determined If119872 is not an integerthen a transformation formula is used to overcomethis difficulty Substituting (13) and (14) into the ODE(11) yields an equation in powers of 119884

(vi) To determine the parameter 119872 the linear terms ofthe highest order in the resulting equation with thehighest order nonlinear terms are balanced With119872determined one collects all coefficients of powers of119884 in the resulting equation where these coefficientshave to vanish This will give a system of algebraicequations involving the 119886

119896and 119887119896 (119896 = 0 119872) 119881

and 120583 Having determined these parameters knowingthat119872 is a positive integer in most cases and using(14) one obtains an analytic solution in a closed form

3 The Travelling Wave Solutions

31 The First Coupled We first consider the following twocoupled IBq equations

119906119905119905= (119906 + 2120572119906119908)

119909119909+

1

12

119906119909119909119905119905

119908119905119905= (2119908 + 120573119908

2+ 1205721199062)119909119909+

1

12

119908119909119909119905119905

(15)

Using the wave variable 120585 = 119909 minus 119881119905 then by integrating thisequation twice and considering the constants of integrationto be zero the system (15) is carried to a system of ODEs

12 (1198812minus 1)119880 minus 24120572119880119882 minus 119881

211988010158401015840= 0

12 (1198812minus 2)119882 minus 12120573119882

2minus 12120572119880

2minus 119881211988210158401015840= 0

(16)

Balancing 119881211988010158401015840 with 119880119882 and 119881211988210158401015840 with 1198802 in (16) gives

119872+ 2 = 119872 +119873

2119872 = 119873 + 2

(17)

so that119872 = 119873 = 2 We consider solutions in the form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(18)

Substituting (18) into the two components of (16) and col-lecting the coefficients of 119884 give two systems of algebraic

equations for 1198860 1198861 1198862 1198871 1198872 1198880 1198881 119888211988911198892119881 and120583 Solving

these systems together leads to the following sets

1198860= minus

1198862

3

1198862=

3radic120572120573 (1198814minus 31198812+ 2)

41205722

1198880=

120573 + 4120572

8120572 (120572 minus 120573)

1198882=

minus3120573

8120572 (120572 minus 120573)

119881 =

6

radicminus61205832+ 18

120583 =

radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= minus2119887

2 119886

2=

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2

1198872=

3radic120572120573 (1198814minus 31198812+ 2)

161205722

1198880=

minus3120573 + 8120572

16120572 (120572 minus 120573)

1198882=

3120573

32120572 (120572 minus 120573)

1198892= 1198882

119881 =

6

radic241205832+ 18

120583 =

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

1198861= 1198871= 1198881= 1198891= 0

1198860=

21198872

3

1198862=

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2

1198872=

3radic120572120573 (1198814minus 31198812+ 2)

161205722

1198880=

minus120573 + 8120572

16120572 (120572 minus 120573)

1198882=

minus3120573

32120572 (120572 minus 120573)

1198892= 1198882

119881 =

6

radicminus241205832+ 18

120583 =

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2 119886

2=

3radic120572120573 (1198814minus 31198812+ 2)

41205722

1198880=

minus3120573 + 4120572

8120572 (120572 minus 120573)

1198882=

3120573

8120572 (120572 minus 120573)

119881 =

6

radic61205832+ 18

120583 =

radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(19)

4 Journal of Applied Mathematics

Consequently we obtain the following travelling wave solu-tions

1199061(119909 119905) = minus

radic120572120573 (1198814minus 31198812+ 2)

41205722

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199081(119909 119905) =

120573 + 4120572

8120572 (120572 minus 120573)

minus

3120573

8120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199062(119909 119905) = minus

3radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199082(119909 119905) =

minus3120573 + 8120572

16120572 (120572 minus 120573)

+

3120573

32120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3120573

32120572 (120572 minus 120573)

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199063(119909 119905) =

radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199083(119909 119905) =

minus120573 + 8120572

16120572 (120572 minus 120573)

minus

3120573

32120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

minus

3120573

32120572 (120572 minus 120573)

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199064(119909 119905) =

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

Journal of Applied Mathematics 5

1199084(119909 119905) =

minus3120573 + 4120572

8120572 (120572 minus 120573)

+

3120573

8120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

(20)

32 The Second Coupled We next consider the system ofequations

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(21)

Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs

12 (120572120573 minus 1205881198812)119880 minus 6120572120573

21198802+ 6120572120573119882

2+ 1205881198972119881211988010158401015840= 0

12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881

2119882 = 0

(22)

Balancing minus612057212057321198802 with 120588119897

2119881211988010158401015840 and 12120572120573119880119882 with

1205881198972119881211988210158401015840 in (22) gives

2119872 = 119872 + 2

2119873 = 119872 +119873

(23)

so that119872 = 119873 = 2 The tanh-coth method admits the use of

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(24)

Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860=

1

120573 + 1

1198862=

minus1

2 (120573 + 1)

1198872= 1198862

1198892= plusmn

radic120573 + 2

2 (120573 + 1)

1198882= minus1198892

119881 =

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

120583 =

radic3

119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

1

120573 + 1

1198862=

3120573

61205732+ 14120573 + 8

1198872= 1198862

1198892= plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

1198882= minus1198892

119881 =

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

120583 =

radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

4 minus 120573

4 (120573 + 1)

1198862=

3120573

4120573 + 1

1198880=

minus1198882

3

1198882= plusmn

9120573

2radic611989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

120583 =

radicminus120573 (3120573 + 6)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

3120573 + 4

4 (120573 + 1)

1198862=

minus3120573

4120573 + 1

1198880= minus1198882

1198882= plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

120583 =

radic3120573 (120573 + 2)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 + 8

8 (120573 + 1)

1198862=

3120573

16 (120573 + 1)

1198872= 1198862 119888

0=

21198892

3

1198882= 1198892 119889

2= plusmn

9120573

8radic2411989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

120583 =

radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

1198860=

3 (120573 + 8)

8 (120573 + 1)

1198862=

minus3120573

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2

6 Journal of Applied Mathematics

1198882= 1198892 119889

2= plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

120583 =

radic120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

(25)

In accordance with these sets of solutions we have thefollowing solutions

1199061(119909 119905) =

1

120573 + 1

minus

1

2 (120573 + 1)

tanh2radic3

119897

times (119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

minus

1

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199081(119909 119905) = ∓

radic120573 + 2

2 (120573 + 1)

tanh2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

plusmn

radic120573 + 2

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199062(119909 119905) =

1

120573 + 1

+

3120573

61205732+ 14120573 + 8

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

+

3120573

61205732+ 14120573 + 8

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199082(119909 119905) = ∓

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199063(119909 119905) =

4 minus 120573

4 (120573 + 1)

+

3120573

4120573 + 1

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199083(119909 119905) = ∓

3120573

2radic611989721205832+ 36120573 + 18

plusmn

9120573

2radic611989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199064(119909 119905) =

3120573 + 4

4 (120573 + 1)

minus

3120573

4120573 + 1

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199084(119909 119905) = ∓

9120573

2radicminus611989721205832+ 36120573 + 18

plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199065(119909 119905) =

120573 + 8

8 (120573 + 1)

+

3120573

16 (120573 + 1)

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Consequently we obtain the following travelling wave solu-tions

1199061(119909 119905) = minus

radic120572120573 (1198814minus 31198812+ 2)

41205722

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199081(119909 119905) =

120573 + 4120572

8120572 (120572 minus 120573)

minus

3120573

8120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus3120573 + 4120572

times(119909 minus

6

radicminus61205832+ 18

119905)

1199062(119909 119905) = minus

3radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199082(119909 119905) =

minus3120573 + 8120572

16120572 (120572 minus 120573)

+

3120573

32120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

+

3120573

32120572 (120572 minus 120573)

coth2radic120573 (9120573 minus 12120572)

minus6120573 + 8120572

times(119909 minus

6

radic241205832+ 18

119905)

1199063(119909 119905) =

radic120572120573 (1198814minus 31198812+ 2)

81205722

+

91205732(2120572 minus 120573)

102412057231198872(120572 minus 120573)

2tanh2

radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

+

3radic120572120573 (1198814minus 31198812+ 2)

161205722

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199083(119909 119905) =

minus120573 + 8120572

16120572 (120572 minus 120573)

minus

3120573

32120572 (120572 minus 120573)

tanh2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

minus

3120573

32120572 (120572 minus 120573)

coth2radic3120573 (minus3120573 + 4120572)

minus6120573 + 8120572

times(119909 minus

6

radicminus241205832+ 18

119905)

1199064(119909 119905) =

minus91205732(2120572 minus 120573)

6412057231198862(120572 minus 120573)

2

+

3radic120572120573 (1198814minus 31198812+ 2)

41205722

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

Journal of Applied Mathematics 5

1199084(119909 119905) =

minus3120573 + 4120572

8120572 (120572 minus 120573)

+

3120573

8120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

(20)

32 The Second Coupled We next consider the system ofequations

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(21)

Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs

12 (120572120573 minus 1205881198812)119880 minus 6120572120573

21198802+ 6120572120573119882

2+ 1205881198972119881211988010158401015840= 0

12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881

2119882 = 0

(22)

Balancing minus612057212057321198802 with 120588119897

2119881211988010158401015840 and 12120572120573119880119882 with

1205881198972119881211988210158401015840 in (22) gives

2119872 = 119872 + 2

2119873 = 119872 +119873

(23)

so that119872 = 119873 = 2 The tanh-coth method admits the use of

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(24)

Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860=

1

120573 + 1

1198862=

minus1

2 (120573 + 1)

1198872= 1198862

1198892= plusmn

radic120573 + 2

2 (120573 + 1)

1198882= minus1198892

119881 =

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

120583 =

radic3

119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

1

120573 + 1

1198862=

3120573

61205732+ 14120573 + 8

1198872= 1198862

1198892= plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

1198882= minus1198892

119881 =

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

120583 =

radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

4 minus 120573

4 (120573 + 1)

1198862=

3120573

4120573 + 1

1198880=

minus1198882

3

1198882= plusmn

9120573

2radic611989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

120583 =

radicminus120573 (3120573 + 6)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

3120573 + 4

4 (120573 + 1)

1198862=

minus3120573

4120573 + 1

1198880= minus1198882

1198882= plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

120583 =

radic3120573 (120573 + 2)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 + 8

8 (120573 + 1)

1198862=

3120573

16 (120573 + 1)

1198872= 1198862 119888

0=

21198892

3

1198882= 1198892 119889

2= plusmn

9120573

8radic2411989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

120583 =

radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

1198860=

3 (120573 + 8)

8 (120573 + 1)

1198862=

minus3120573

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2

6 Journal of Applied Mathematics

1198882= 1198892 119889

2= plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

120583 =

radic120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

(25)

In accordance with these sets of solutions we have thefollowing solutions

1199061(119909 119905) =

1

120573 + 1

minus

1

2 (120573 + 1)

tanh2radic3

119897

times (119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

minus

1

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199081(119909 119905) = ∓

radic120573 + 2

2 (120573 + 1)

tanh2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

plusmn

radic120573 + 2

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199062(119909 119905) =

1

120573 + 1

+

3120573

61205732+ 14120573 + 8

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

+

3120573

61205732+ 14120573 + 8

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199082(119909 119905) = ∓

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199063(119909 119905) =

4 minus 120573

4 (120573 + 1)

+

3120573

4120573 + 1

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199083(119909 119905) = ∓

3120573

2radic611989721205832+ 36120573 + 18

plusmn

9120573

2radic611989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199064(119909 119905) =

3120573 + 4

4 (120573 + 1)

minus

3120573

4120573 + 1

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199084(119909 119905) = ∓

9120573

2radicminus611989721205832+ 36120573 + 18

plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199065(119909 119905) =

120573 + 8

8 (120573 + 1)

+

3120573

16 (120573 + 1)

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

1199084(119909 119905) =

minus3120573 + 4120572

8120572 (120572 minus 120573)

+

3120573

8120572 (120572 minus 120573)

tanh2radic120573 (9120573 minus 12120572)

minus3120573 + 4120572

times(119909 minus

6

radic61205832+ 18

119905)

(20)

32 The Second Coupled We next consider the system ofequations

120588

120572

119906119905119905= 120573119906119909119909minus

1205732

2

(1199062)119909119909+

120573

2

(1199082)119909119909+

120588

120572

1198972

12

119906119909119909119905119905

120588

120572

119908119905119905= 120573(119906119908)

119909119909+

120588

120572

1198972

12

119908119909119909119905119905

(21)

Using thewave variable 120585 = 119909minus119881119905 and then by integrating thisequation twice and considering the constants of integration tobe zero the system (21) is carried to a system of ODEs

12 (120572120573 minus 1205881198812)119880 minus 6120572120573

21198802+ 6120572120573119882

2+ 1205881198972119881211988010158401015840= 0

12120572120573119880119882 + 1205881198972119881211988210158401015840minus 12120588119881

2119882 = 0

(22)

Balancing minus612057212057321198802 with 120588119897

2119881211988010158401015840 and 12120572120573119880119882 with

1205881198972119881211988210158401015840 in (22) gives

2119872 = 119872 + 2

2119873 = 119872 +119873

(23)

so that119872 = 119873 = 2 The tanh-coth method admits the use of

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(24)

Substituting (24) into (22) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860=

1

120573 + 1

1198862=

minus1

2 (120573 + 1)

1198872= 1198862

1198892= plusmn

radic120573 + 2

2 (120573 + 1)

1198882= minus1198892

119881 =

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

120583 =

radic3

119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

1

120573 + 1

1198862=

3120573

61205732+ 14120573 + 8

1198872= 1198862

1198892= plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

1198882= minus1198892

119881 =

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

120583 =

radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

4 minus 120573

4 (120573 + 1)

1198862=

3120573

4120573 + 1

1198880=

minus1198882

3

1198882= plusmn

9120573

2radic611989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

120583 =

radicminus120573 (3120573 + 6)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

3120573 + 4

4 (120573 + 1)

1198862=

minus3120573

4120573 + 1

1198880= minus1198882

1198882= plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

120583 =

radic3120573 (120573 + 2)

(120573 + 2) 119897

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 + 8

8 (120573 + 1)

1198862=

3120573

16 (120573 + 1)

1198872= 1198862 119888

0=

21198892

3

1198882= 1198892 119889

2= plusmn

9120573

8radic2411989721205832+ 36120573 + 18

119881 =

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

120583 =

radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

1198860=

3 (120573 + 8)

8 (120573 + 1)

1198862=

minus3120573

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2

6 Journal of Applied Mathematics

1198882= 1198892 119889

2= plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

120583 =

radic120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

(25)

In accordance with these sets of solutions we have thefollowing solutions

1199061(119909 119905) =

1

120573 + 1

minus

1

2 (120573 + 1)

tanh2radic3

119897

times (119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

minus

1

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199081(119909 119905) = ∓

radic120573 + 2

2 (120573 + 1)

tanh2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

plusmn

radic120573 + 2

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199062(119909 119905) =

1

120573 + 1

+

3120573

61205732+ 14120573 + 8

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

+

3120573

61205732+ 14120573 + 8

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199082(119909 119905) = ∓

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199063(119909 119905) =

4 minus 120573

4 (120573 + 1)

+

3120573

4120573 + 1

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199083(119909 119905) = ∓

3120573

2radic611989721205832+ 36120573 + 18

plusmn

9120573

2radic611989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199064(119909 119905) =

3120573 + 4

4 (120573 + 1)

minus

3120573

4120573 + 1

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199084(119909 119905) = ∓

9120573

2radicminus611989721205832+ 36120573 + 18

plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199065(119909 119905) =

120573 + 8

8 (120573 + 1)

+

3120573

16 (120573 + 1)

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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6 Journal of Applied Mathematics

1198882= 1198892 119889

2= plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

119881 =

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

120583 =

radic120573 (3120573 + 6)

2 (120573 + 2) 119897

1198861= 1198871= 1198881= 1198891= 0

(25)

In accordance with these sets of solutions we have thefollowing solutions

1199061(119909 119905) =

1

120573 + 1

minus

1

2 (120573 + 1)

tanh2radic3

119897

times (119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

minus

1

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199081(119909 119905) = ∓

radic120573 + 2

2 (120573 + 1)

tanh2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

plusmn

radic120573 + 2

2 (120573 + 1)

coth2radic3

119897

(119909 minus

radic3120588120572120573 (120573 + 1)

3120588 (120573 + 1)

119905)

1199062(119909 119905) =

1

120573 + 1

+

3120573

61205732+ 14120573 + 8

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

+

3120573

61205732+ 14120573 + 8

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199082(119909 119905) = ∓

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

tanh2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

plusmn

3120573radic120573 + 2

2 (3120573 + 4) (120573 + 1)

coth2radicminus120573 (15120573 + 12)

(5120573 + 4) 119897

times (119909 minus

radic15120588120572120573 (211989721205832+ 9120573 + 15)

120588 (211989721205832+ 9120573 + 15)

119905)

1199063(119909 119905) =

4 minus 120573

4 (120573 + 1)

+

3120573

4120573 + 1

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199083(119909 119905) = ∓

3120573

2radic611989721205832+ 36120573 + 18

plusmn

9120573

2radic611989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

(120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 11989721205832)

120588 (minus3 + 11989721205832)

119905)

1199064(119909 119905) =

3120573 + 4

4 (120573 + 1)

minus

3120573

4120573 + 1

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199084(119909 119905) = ∓

9120573

2radicminus611989721205832+ 36120573 + 18

plusmn

9120573

2radicminus611989721205832+ 36120573 + 18

tanh2radic3120573 (120573 + 2)

(120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 11989721205832)

120588 (3 + 11989721205832)

119905)

1199065(119909 119905) =

120573 + 8

8 (120573 + 1)

+

3120573

16 (120573 + 1)

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

times(119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

+

3120573

16 (120573 + 1)

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199085(119909 119905) = plusmn

3120573

4radic2411989721205832+ 36120573 + 18

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

tanh2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

plusmn

9120573

8radic2411989721205832+ 36120573 + 18

coth2radicminus120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radicminus3120588120572120573 (minus3 + 411989721205832)

120588 (minus3 + 411989721205832)

119905)

1199066(119909 119905) =

3 (120573 + 8)

8 (120573 + 1)

minus

3120573

16 (120573 + 1)

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

minus

3120573

16 (120573 + 1)

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

1199086(119909 119905) = ∓

9120573

4radicminus2411989721205832+ 36120573 + 18

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

tanh2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

plusmn

9120573

8radicminus2411989721205832+ 36120573 + 18

coth2radic120573 (3120573 + 6)

2 (120573 + 2) 119897

times (119909 minus

radic3120588120572120573 (3 + 411989721205832)

120588 (3 + 411989721205832)

119905)

(26)

33 The Third Coupled We now examine the coupled equa-tions

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

1

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= (119906119908)

119909119909+ 1205722119908119909119909119905119905

(27)

Using the wave variable 120585 = 119909 minus119881119905 the system (27) is carriedto a system of ODEs Integrating these equations twice andneglecting constants of integration we find

2 (1198812minus 120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus119880119882 minus 120572

2119881211988210158401015840= 0

(28)

Balancing 1205731198822 with 21205722119881211988010158401015840 and 119881211988210158401015840 with 119880119882 in (28)gives

2119873 = 119872 + 2

2 + 119873 = 119872 +119873

(29)

so that

119872 = 119873 = 2 (30)

As a result we seek solutions to (27) in the form

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(31)

Substitution of (31) into (28) leads to an algebraic system inthe unknowns 119886

0 1198861 1198862 1198871 1198872 1198880 1198881 1198882 1198891 1198892 119881 and 120583

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Collecting the coefficients of 119884 and solving the resultingsystem we find the following sets of solutions

1198860=

120573 (3120573 + 8)

8 (120573 + 1)

1198862=

minus31205732

16 (120573 + 1)

1198872= 1198862 119888

0= minus2119889

2 119888

2= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

120583 =

radic120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

120573 (120573 + 8)

8 (120573 + 1)

1198862=

31205732

16 (120573 + 1)

1198872= 1198862

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

16

119881 =

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

4120572 (120573 + 2)

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus120573 (120573 minus 4)

4 (120573 + 1)

1198862=

31205732

4 (120573 + 1)

1198880= minus

1198882

3

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573 (3120573 + 4)

4 (120573 + 1)

1198862=

minus31205732

4 (120573 + 1)

1198880= minus2

1198882=

plusmn3radicminus41198814minus 41198812+ 21205732+ 4120573

4

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (120573 + 2)

2120572 (120573 + 2)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

120573

120573 + 1

1198862=

31205732

2 (31205732+ 7120573 + 4)

1198872= 1198862 119888

2= minus1198892

1198892=

plusmn3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

119881 =

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

120583 =

radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

1198861= 1198871= 1198880= 1198881= 1198891= 0

1198860=

120573

120573 + 1

1198862=

minus120573

2 (120573 + 1)

1198872= 1198862

1198882= minus1198892 119889

2=

plusmnradic91198812+ 6

2

119881 =

radic3120573 (120573 + 1)

3 (120573 + 1)

120583 =

minus1

2120572

1198861= 1198871= 1198880= 1198881= 1198891= 0

(32)

Consequently we obtain the following solutions

1199061(119909 119905) =

120573 (3120573 + 8)

8 (120573 + 1)

minus

31205732

16 (120573 + 1)

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

minus

31205732

16 (120573 + 1)

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

1199081(119909 119905) =

∓3radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radic120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radic120573 (1612057221205832+ 1)

1612057221205832+ 1

119905)

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

1199062(119909 119905) =

120573 (120573 + 8)

8 (120573 + 1)

+

31205732

16 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

+

31205732

16 (120573 + 1)

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199082(119909 119905) = plusmn

radicminus41198814minus 41198812+ 21205732+ 4120573

8

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

tanh2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

16

coth2radicminus120573 (120573 + 2)

4120572 (120573 + 2)

times (119909 minus

radicminus120573 (1612057221205832minus 1)

1612057221205832minus 1

119905)

1199063(119909 119905) =

minus120573 (120573 minus 4)

4 (120573 + 1)

+

31205732

4 (120573 + 1)

tanh2radicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199083(119909 119905) = ∓

radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanhradicminus120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199064(119909 119905) =

120573 (3120573 + 4)

4 (120573 + 1)

minus

31205732

4 (120573 + 1)

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199084(119909 119905) = ∓

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

plusmn

3radicminus41198814minus 41198812+ 21205732+ 4120573

4

tanh2radic120573 (120573 + 2)

2120572 (120573 + 2)

times (119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199065(119909 119905) =

120573

120573 + 1

+

31205732

2 (31205732+ 7120573 + 4)

tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

+

31205732

2 (31205732+ 7120573 + 4)

coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199085(119909 119905)

= ∓

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

times tanh2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

plusmn

3radicminus421198814120573 minus 58119881

4minus 21198812120573 minus 120119881

2+ 120120573

8

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

times coth2radicminus120573 (5120573 + 4)

2120572 (5120573 + 4)

times (119909 minus

radic5120573 (812057221205832+ 3120573 + 5)

812057221205832+ 3120573 + 5

119905)

1199066(119909 119905) =

120573

120573 + 1

minus

120573

2 (120573 + 1)

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

minus

120573

2 (120573 + 1)

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

1199086(119909 119905)

= ∓

radic91198812+ 6

2

tanh2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

plusmn

radic91198812+ 6

2

coth2minus12120572

(119909 minus

radic3120573 (120573 + 1)

3 (120573 + 1)

119905)

(33)

34 The Fourth Coupled We now consider the coupled IBqequations

119906119905119905= 119906119909119909minus 120573(119906

2)119909119909+ (1199082)119909119909+ 119906119909119909119905119905

119908119905119905= 119908119909119909+ 2(119906119908)

119909119909+ 119908119909119909119905119905

(34)

Using the wave variable 120585 = 119909 minus 119881119905 (34) can be converted toa system of ODEs

(1198812minus 1)119880 + 120573119880

2minus1198822minus 119881211988010158401015840= 0

(1198812minus 1)119882 minus 2119880119882 minus 119881

211988210158401015840= 0

(35)

Balancing1198822 with 11988010158401015840 and 119880119882 with11988210158401015840 in (35) gives

2119873 = 119872 + 2

119872 + 119873 = 119873 + 2

(36)

so that

119872 = 119873 = 2 (37)

The tanh-coth method substitutes the finite expansions

119880(120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(38)

In (35) collecting the coefficients of each power of 119884 andsolving the resulting system we obtain

1198860=

minus21205832

161205832+ 1

1198862=

minus31205832

161205832+ 1

1198862= 1198872

1198880=

21198892

3

1198882= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radic161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

minus61205832

161205832minus 1

1198862=

31205832

161205832minus 1

1198862= 1198872

1198880= minus2119889

2 119888

2= 1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

119881 =

1

radicminus161205832+ 1

1198861= 1198871= 1198881= 1198891= 0

1198860=

1205832

41205832+ 1

1198862=

minus31205832

41205832+ 1

1198880=

minus1198882

3

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radic41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198862=

31205832

41205832minus 1

1198880= minus1198882

1198882=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860=

minus31205832

41205832minus 1

1198872=

31205832

41205832minus 1

1198880= minus1198892

1198892=

plusmn3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

119881 =

1

radicminus41205832+ 1

1198861= 1198862= 1198871= 1198881= 1198882= 1198891= 0

(39)

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

where 120583 is left as a free parameter Consequently we obtainthe following travelling wave solutions

1199061(119909 119905) = minus

21205832

161205832+ 1

minus

31205832

161205832+ 1

tanh2120583(119909 minus 1

radic161205832+ 1

119905)

minus

31205832

161205832+ 1

coth2120583(119909 minus 1

radic161205832+ 1

119905)

1199081(119909 119905) = plusmn

(119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radic161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radic161205832+ 1

119905)

1199062(119909 119905) = minus

61205832

161205832minus 1

+

31205832

161205832minus 1

tanh2120583(119909 minus 1

radicminus161205832+ 1

119905)

+

31205832

161205832minus 1

coth2120583(119909 minus 1

radicminus161205832+ 1

119905)

1199082(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

8

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

tanh2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

16

coth2120583

times(119909 minus

1

radicminus161205832+ 1

119905)

1199063(119909 119905) =

1205832

41205832+ 1

minus

31205832

41205832+ 1

tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199083(119909 119905) = ∓

(119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radic41205832+ 1

119905)

1199064(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199084(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

times tanh2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199065(119909 119905) = minus

31205832

41205832minus 1

+

31205832

41205832minus 1

coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

1199085(119909 119905) = ∓

3 (119881 + 1) (119881 minus 1)radic120573 + 2

4

plusmn 3

(119881 + 1) (119881 minus 1)radic120573 + 2

4

times coth2120583(119909 minus 1

radicminus41205832+ 1

119905)

(40)

35 The Fifth Coupled We finally consider the coupled IBqequations

119906119905119905= 120573119906119909119909minus

120573

2

(1199062)119909119909+

120573

2

(1199082)119909119909+ 1205722119906119909119909119905119905

119908119905119905= 120574(119906119908)

119909119909+ 120588(119908

2)119909119909+ 1205722119908119909119909119905119905

(41)

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

This system of equations (41) can be converted to the follow-ing system of ODEs by using the wave variable 120585 = 119909 minus 119881119905

(21198812minus 2120573)119880 + 120573119880

2minus 120573119882

2minus 21205722119881211988010158401015840= 0

1198812119882minus 120574119880119882 minus 120588119882

2minus 1205722119881211988210158401015840= 0

(42)

Using the balancing procedure in (42) gives

2119872 = 119872 +119873

2119873 = 119872 +119873

(43)

so that

119872 = 119873 = 2 (44)

We consider solutions in the following form

119880 (120583120585) = 119878 (119884) =

2

sum

119896=0

119886119896119884119896+

2

sum

119896=1

119887119896119884minus119896

119882 (120583120585) = 119878 (119884) =

2

sum

119896=0

119888119896119884119896+

2

sum

119896=1

119889119896119884119896

(45)

Substituting (45) into (42) collecting the coefficients of 119884and solving the resulting system we find the following setsof solutions

1198860= (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

1198882= (plusmn3 (minus 120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742

minus41205882120574120573 + 4120588

21198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

120583 =

radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

1198860= (120573(6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198862=

minus (41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

1198880= (120573(minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882

minus 312058821205733minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

1198882= (plusmn3 (120573120588 + (minus120588

21205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

119881 =

radic120573 (412057221205832+ 1)

412057221205832+ 1

120583 =

radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

1198861= 1198871= 1198872= 1198881= 1198891= 1198892= 0

(46)

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 13

In view of this we obtain the following travelling wavesolutions

1199061(119909 119905) = (minus120573 (8120588119888

21205732120574 minus 8120573120574

21205881198882minus 18120588

21205732+ 24120573120588

31198882

+1212057321205742minus 16120574

31205881198882+ 24120588

31198882120574 minus 3120574120573

3))

times (4 (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732))

minus1

minus

(41205881198882120573 + 4119888

2120588120574 minus 3120573

2)

4120574 (120583 + 120573)

tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199081(119909 119905) = (120573 (120588

212057321198882+ 120574211988821205732+ 12057311988821205743+ 2120574212058821198882

+312057412058821205731198882minus

3

2

1205881205732120574 minus

3

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882+ 312058821205733

+ 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882minus 312057331205742minus 312057431205732)

minus1

plusmn (3 (minus 120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radicminus120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radicminus120573 (412057221205832minus 1)

412057221205832minus 1

119905)

1199062(119909 119905) = (120573 (6120588119888

21205732120574 + 10120573120574

21205881198882+

3

2

12058821205732+ 2120573120588

31198882

+312057321205742+ 412057431205881198882+ 212058831198882120574 +

9

4

1205741205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

minus

(41205881198882120573 + 4119888

2120588120574 + 3120573

2)

4120574 (120583 + 120573)

tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

times(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

1199082(119909 119905) = (120573 (minus 3120588

212057321198882minus 3120574211988821205732minus 312057311988821205743

minus6120574212058821198882minus 9120574120588

21205731198882minus

9

2

1205881205732120574 minus

9

4

1205881205733))

times (minus812057312058831198882120574 minus 4120588

312057421198882minus 4120573212058831198882minus 312058821205733

minus 312058821205741205732+ 4120588120574

41198882+ 4120573212057421205881198882

+812057312057431205881198882+ 312057331205742+ 312057431205732)

minus1

plusmn (3 (120573120588 + (minus12058821205732+ 41205743120573 minus 4120574

21198814

minus 412057431198812+ 212057321205742minus 41205882120574120573

+412058821198814+ 412058821198812120574)

12

))

times (4 (1205742minus 1205882))

minus1

times tanh2radic120573 (2120574 + 120573)

2120572 (2120574 + 120573)

(119909 minus

radic120573 (412057221205832+ 1)

412057221205832+ 1

119905)

(47)

4 Conclusion

In summary the tanh-coth method has been successfullyimplemented to find new travelling wave solutions for severalkinds of two coupled improved Boussinesq equations Thesolutions that have been found can be seen as an expansionand verification of the previously known data However thisstudy confirmed again the belief that the tanh-coth methodis a powerful and reliable technique to handle nonlineardispersive-dissipative equations and system of equationsThroughout the work Maple has been used to overcome thetedious algebraic calculations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A De Godefroy ldquoBlow up of solutions of a generalized Boussi-nesq equationrdquo IMA Journal of AppliedMathematics vol 60 no2 pp 123ndash138 1998

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Journal of Applied Mathematics

[2] S Wang and M Li ldquoThe Cauchy problem for coupled IMBqequationsrdquo IMA Journal of Applied Mathematics vol 74 no 5pp 726ndash740 2009

[3] N Duruk H A Erbay and A Erkip ldquoBlow-up and globalexistence for a general class of nonlocal nonlinear coupled waveequationsrdquo Journal of Differential Equations vol 250 no 3 pp1448ndash1459 2011

[4] G Chen and H Zhang ldquoInitial boundary value problem for asystem of generalized IMBq equationsrdquoMathematical Methodsin the Applied Sciences vol 27 no 5 pp 497ndash518 2004

[5] P Rosenau ldquoDynamics of dense latticesrdquo Physical Review B vol36 no 11 pp 5868ndash5876 1987

[6] P L Christiansen P S Lomdahl andVMuto ldquoOn aToda latticemodel with a transversal degree of freedomrdquo Nonlinearity vol4 no 2 pp 477ndash501 1991

[7] S K Turitsyn ldquoOn a Toda lattice model with a transversaldegree of freedom Sufficient criterion of blow-up in thecontinuum limitrdquo Physics Letters A vol 173 no 3 pp 267ndash2691993

[8] R L Pego P Smereka and M I Weinstein ldquoOscillatoryinstability of solitary waves in a continuum model of latticevibrationsrdquo Nonlinearity vol 8 no 6 pp 921ndash941 1995

[9] X Fan and L Tian ldquoCompacton solutions and multiple com-pacton solutions for a continuum Toda lattice modelrdquo ChaosSolitons and Fractals vol 29 no 4 pp 882ndash894 2006

[10] A M Wazwaz ldquoThe Hirotarsquos bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomtsev-Petviashvili equationrdquoAppliedMathematicsand Computation vol 200 no 1 pp 160ndash166 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of