new lax pairs and darboux transformation and its application to a

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 548690 8 pageshttpdxdoiorg1011552013548690

Research ArticleNew Lax Pairs and Darboux Transformation and Its Applicationto a Shallow Water Wave Model of Generalized KdV Type

Huizhang Yang1 and Weiguo Rui2

1 College of Mathematics of Honghe University Mengzi Yunnan 661100 China2 College of Mathematics of Chongqing Normal University Chongqing 401331 China

Correspondence should be addressed to Weiguo Rui weiguorhhualiyuncom

Received 30 March 2013 Accepted 5 September 2013

Academic Editor Sarp Adali

Copyright copy 2013 H Yang and W RuiThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

New Lax pairs of a shallow water wave model of generalized KdV equation type are presented According to this Lax pair weconstructed a new spectral problem By using this spectral problem we constructed Darboux transformation with the help of agauge transformation Applying this Darboux transformation some new exact solutions including double-soliton solution of theshallow water wave model of generalized KdV equation type are obtained In order to visually show dynamical behaviors of thesedouble soliton solutions we plot graphs of profiles of them and discuss their dynamical properties

1 Introduction

It is well known that the Lax pair and Darboux transforma-tion can be employed to obtain multisoliton solution of non-linear evolution equations Darboux transformations provideus with purely algebraic powerful method to constructsolutions for systems of nonlinear equations In recent yearsmore and more researchers used the Lax pair and Darbouxtransformation to investigate soliton solutions of classicalnonlinear wave equations and some new soliton equationswhich were generated by new spectral problems see [1ndash30] and references cited therein In general a systematicaltheory on such Darboux transformation even for 119899 times 119899

matrix spectral problem and the resulting zero curvatureequation has a beautiful algebraic structure for associatedevolution equations which tells symmetry algebras of theobtained evolution equations see [31 32] and references citedtherein Sometimes it is found that there are many infinitysymmetries from the adopted zero curvature equation

In this paper we will investigate the Lax pairs Darbouxtransformation and double soliton solutions of the followingfamous shallow water wave model of generalized KdV equa-tion type

119906119905minus

ℎ2

0

3

119906119909119909119905+ 1198880119906119909+

3120572

2

119906119906119909minus

1198880ℎ2

0

6

119906119909119909119909

=

120572ℎ2

0

3

119906119909119906119909119909+

120572ℎ2

0

6

119906119906119909119909119909

(1)

which appeared in [33] where 0 lt 120572 ≪ 1 is a small-amplitude parameter Only dropping the right-hand side of(1) gives BBM equation Dropping the right-hand side of(1) and replacing the term 119906

119909119909119905by the term minus119888

0119906119909119909119909

giveKdV equation [34] Thus (1) can be seen as a BBM equationextended by retaining higher order terms in an asymptoticexpansion in terms of the small-amplitude parameter 120572Dropping the term minus(119888

0ℎ2

06)119906119909119909119909

from (1) and letting ℎ0=

radic3 1198880= 2120581 120572 = 2 (1) becomes the celebrated Camassa-

Holm equation [33] as follows

119906119905+ 2120581 minus 119906

119909119909119905+ 3119906119906

119909= 2119906119909119906119909119909+ 119906119906119909119909119909 (2)

where 119906 is the fluid velocity in the 119909 direction (or equivalentto the height of the waterrsquos free surface above a flat bottom) 120581is a constant related to the critical shallow water wave speedand subscripts denote partial derivatives Letting 119888

0= 1 ] =

minus(13)ℎ2

0 120574 = (32)120572 120573 = minus(16)ℎ

2

01198880 (1) can be rewritten as

119906119905+ 119906119909+ ]119906119909119909119905+ 120573119906119909119909119909

+ 120574119906119906119909+

1

3

120574] (119906119906119909119909119909

+ 2119906119909119906119909119909)=0

(3)

2 Mathematical Problems in Engineering

which comes from physical considerations via the method-ology introduced by Fuchssteiner and Fokas in [35 36] TheLax pairs of (3) with 120574 = 1 are given by [37 38] as follows

Ψ119909119909= [1198962(1 minus V) +

V

4]]Ψ = 0 120590V = 119906 + ]119906

119909119909

120590 =

120573 minus ]

2]

(4)

Ψ119905+ (minus

1

2

+ 119888)Ψ + (119906 +

120573

]+

2120590

4]1198962 minus 1)Ψ119909= 0 (5)

where 119888 is an arbitrary constant It is a pity that the [37] is not aformal publication and it is only preprint paper so we cannotknow its reality contents whether the authors have obtainedsoliton solutions of (3) by the Lax pairs (4) and (5) In fact(3) has been studied by many authors in recent years see thefollowing brief introductions

In [39] by using the bifurcation theory of dynamicsystem some subsection-function and implicit function solu-tions such as compactons solitary waves smooth periodicwaves and nonsmooth periodic waves with peaks as well asthe existence conditions have been presented by Bi By usingthe same method Li and Zhang [40] studied a generalizationform of the modified KdV equation which is more complexthan (3) In [40] the existence of solitary wave kink andantikink wave solutions and uncountably many smooth andnonsmooth periodic wave solutions are discussed By usingthe improved method named integral bifurcation method[41] Rui et al [42] obtain all kinds of soliton-like or kink-likewave solutions periodic wave solutions with loop or withoutloop smooth compacton-like periodic wave solutions and

nonsmooth periodic cusp wave solutions for (3) In [43]Long and Chen discussed the existence of solitary wave cuspwave periodic wave periodic cusp wave and compactonswere for (3) From the above references (1) (ie (3)) is a veryimportant water wave model

The rest of this paper is organized as follows In Section 2we will derive new Lax pair and Darboux transformationof (1) In Section 3 by using this Darboux transformationwe will investigate soliton solutions of (1) and discuss thedynamic properties of these soliton solutions

2 Lax Pair and Darboux Transformation of (1)Through a series of tedious computation we obtain Lax pairsof (1) as follows

120601119909119909= (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)120601 (6)

120601119905= minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)120601119909+

120572

4

119906119909120601 (7)

Obviously the Lax pairs (6) and (7) are different from theLax pairs (4) and (5) under 119888

0= 1 ] = minus(13)ℎ

2

0 120574 =

(32)120572 120573 = minus(16)ℎ2

01198880 They are new Lax pairs which we

obtained By using the new Lax pairs (6) and (7) we willconstruct a Darboux transformation for obtaining solitonsolutions of (1)

First we consider the following spectral problems

120601119909= M120601 120601

119905= N120601 (8)

with

M = (

0 1

minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

0

)

N = (

120572

4

119906119909

minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)(minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

) minus

120572

4

119906119909

)

(9)

where 120572 is a constant 120582 is a spectral parameter and 119906 is apotential function From compatibility condition 120601

119909119909119905= 120601119905119909119909

yields a zero curvature equation M119905minus N119909+ [MN] = O

Substituting M N into the zero curvature equation by adirect calculation (1) is obtained successfully

Next we will construct a Darboux Transformation (DT)of the spectral problems (8) In fact the DT is actually a gaugetransformation

120601 = T120601 (10)

of the spectral problems (8) It is required that 120601 also satisfiesthe same form of spectral problems

120601119909= M120601 M = (T

119909+ TM)Tminus1 (11)

120601119905= N120601 N = (T

119905+ TN)Tminus1 (12)

It means that we have to find a matrix T such that the oldpotential 119906 is replaced by the new one 119906

Suppose

T = T (120582) = (119860 (120582) 119861 (120582)119862 (120582) 119863 (120582)

) (13)

Mathematical Problems in Engineering 3

where

119860 (120582) = 119860119873(120582119873+

119873minus1

sum

119896=0

119860119896120582119896)

119861 (120582) = 119860119873(

119873minus1

sum

119896=0

119861119896120582119896)

(14)

119862 (120582) =

1

119860119873

(

119873minus1

sum

119896=0

119862119896120582119896)

119863 (120582) =

1

119860119873

(120582119873+

119873minus1

sum

119896=0

119863119896120582119896)

(15)

and 119860119873 119860119896 119861119896 119862119896 and 119863

119896(0 le 119896 le 119873 minus 1) are functions

of 119909 and 119905Let 120601(120582

119895) = (120601

1(120582119895) 1206012(120582119895))119879 120595(120582119895) = (120595

1(120582119895) 1205952(120582119895))119879

be two basic solutions of (8) From (10) there exist constants119903119895(0 le 119895 le 119873 minus 1) which satisfy

(119860 (120582119895) 1206011(120582119895) + 119861 (120582

119895) 1206012(120582119895))

minus 119903119895(119860 (120582

119895) 1205951(120582119895) + 119861 (120582

119895) 1205952(120582119895)) = 0

(119862 (120582119895) 1206011(120582119895) + 119863 (120582

119895) 1206012(120582119895))

minus 119903119895(119862 (120582

119895) 1205951(120582119895) + 119863 (120582

119895) 1205952(120582119895)) = 0

(16)

Further (16) can be written as a linear algebraic system

119860(120582119895) + 120575119895119861 (120582119895) = 0 119862 (120582

119895) + 120575119895119863(120582119895) = 0 (17)

That is119873minus1

sum

119896=0

(119860119896+ 120575119895119861119896) 120582119896

119895=minus120582119873

119895

119873minus1

sum

119896=0

(119862119896+ 120575119895119863119896) 120582119896

119895=minus120575119895120582119873

119895

(18)

where

120575119895=

1206012(120582119895) minus 1199031198951205952(120582119895)

1206011(120582119895) minus 1199031198951205951(120582119895)

1 le 119895 le 2119873 (19)

and the constants 120582119895(120582119896= 120582119904as 119896 = 119904) 119903

119895are suitably chosen

such that determinant of coefficients for (18) is nonzeroTherefore 119860

119873 119860119896 119861119896 119862119896 and 119863

119896(0 le 119896 le 119873 minus 1) are

uniquely determined by (18)Equations (14) and (15) show that the detT(120582) is a 2119873th-

order polynomial in 120582 and

detT (120582119895) = 119860 (120582

119895)119863 (120582

119895) minus 119861 (120582

119895) 119862 (120582

119895) (20)

On the other hand from (17) we have 119860(120582119895) = minus120575

119895119861(120582119895)

119862(120582119895) = minus120575

119895119863(120582119895) Thus we have

detT (120582) = 1205732119873minus1

prod

119895=1

(120582 minus 120582119895) (21)

where 120573 is independent of 120582 Equation (21) implies that120582119895(1 le 119895 le 2119873) are 2119873 roots of detT(120582)Second we prove the following theory of Darboux trans-

formation for special variable

Theorem 1 Let 119860119873satisfy

1198602

119873= 1 (22)

Then the matrixM determined by (11) has the same form asMthat is

M = (

0 1

minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

0

) (23)

where the transformation from the old potential 119906 into new one119906 is given by

119906 = 119906 +

6

ℎ2

0

119860119873minus1

(24)

119860119873minus1119909

= 119862119873minus1

minus

3

4ℎ2

0

119861119873minus1

119861119873minus1119909

= 119863119873minus1

minus 119860119873minus1

119862119873minus1119909

=

3

4ℎ2

0

(119860119873minus1


) +

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

119863119873minus1119909

=

3

4ℎ2

0

119861119873minus1


(25)

Proof Let Tminus1 = Tlowast detT and

(T119909+ TM)Tlowast = (11989111 (120582) 11989112 (120582)

11989121(120582) 119891

22(120582)) (26)

It is easy to see that 11989111(120582) and 119891

22(120582) are 2119873th-order

polynomials in 120582 11989112(120582) and 119891

21(120582) are (2119873 minus 1)th-order

polynomials in 120582 From (19) and (8) we find

120575119895119909= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

minus 1205752

119895 (27)

Through direct calculation all 120582119895(0 le 119895 le 2119873) are roots of

119891119899119904(119899 119904 = 1 2) Together with (21) and (26) we get

(T119909+ TM)Tlowast = (detT) 119875 (120582) (28)

with

119875 (120582) = (

119901(0)

11119901(0)

12

119901(0)

21119901(0)

22

) (29)

where 119901(0)119899119904(119899 119904 = 1 2) are independent of spectral parameter

120582 Indeed (28) can be written as

T119909+ TM = 119875 (120582)T (30)


Comparing the coefficients of 120582119873 in (30) we find

119901(0)

11= minus119901(0)

22= 120597119909ln119860119873 119901

(0)

12= 1198602

119873

119901(0)

21=

1

1198602

119873

3

4ℎ2

0

(31)

Substituting (22) into (31) yields

119901(0)

11= minus119901(0)

22= 0 119901

(0)

12= 1198602

119873= 1 (32)

From (22) (24) (25) and (31) and noticing 119906 in (23) we get

119901(0)

21= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

(33)

Thus 119875(120582) = M The proof of Theorem 1 is completed

Finally by using same way toTheorem 1 we prove thatNin (12) has the same form asN under the transformation (10)and (24) see the following theory and its proof

Theorem 2 ThematrixN defined by (12) has the same type asN in which the old potential 119906 is mapped into 119906 via the sameDT (24)


(T119905+ TN)Tlowast = (11989211 (120582) 11989212 (120582)

11989221(120582) 119892

22(120582)) (34)


22(120582) are 2119873th-order


21(120582) are (2119873 + 1)th-order

polynomials in 120582 By using (19) and (8) we obtain

120575119895119905=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times(minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

minus

120572

2

119906119909120575119895+ (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)1205752

119895

(35)



(T119905+ TN)Tlowast = (detT) 119876 (120582) (36)

with

119876 (120582) = (

119902(0)

11119902(1)

12120582 + 119902(0)

12

119902(1)

21120582 + 119902(0)

21119902(0)

22

) (37)

where 119902(119897)119899119904(119899 119904 = 1 2 119897 = 0 1) are independent of spectral

parameter 120582 Equation (37) can be written as

T119905+ TN = 119876 (120582)T (38)

Comparing the coefficients of 120582119873+1 and 120582119873 in (38) leadsto

119902(1)

12= minus

3120572

2ℎ2

0

1198602

119873 119902

(1)

21= minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(39)

119902(0)

11= 120597119905ln119860119873+

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

(40)

119902(0)

12= minus(

1198880

2

+

120572

2

119906)1198602

119873minus

3120572

2ℎ2

0

1198602

119873119860119873minus1

+

3120572

2ℎ2

0

1198602

119873119863119873minus1

(41)

119902(0)

21=

1

1198602

119873

120572

4

119906119909119909minus

1

1198602

119873

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

1

1198602

119873

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

)

+

119860119873minus1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(42)

119902(0)

22= minus120597119905ln119860119873minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

(43)

Substituting (22) (24) and (25) into (40) to (43) we canget

119902(0)

11=

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

=

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

=

120572

4

119906119909

119902(1)

12120582 + 119902(0)

12

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

2ℎ2

0

119860119873minus1

+

3120572

2ℎ2

0

119863119873minus1

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

ℎ2

0

+

3120572

2ℎ2

0

(119860119873minus1

+ 119863119873minus1

)

= minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

119902(0)

22= minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

= minus

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

= minus

120572

4

119906119909


119902(1)

21120582 + 119902(0)

21

= minus

3120572

2ℎ2

0

3

4ℎ2

0

120582+

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

) +

3120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906) +

3120572

2ℎ2

0

119862119873minus1119909

(44)

On the other hand from (24) (25) and (33) we have

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

=

120572

4

119906119909119909+

3120572

2ℎ2

0

119860119873minus1119909119909

minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

minus

6120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

+

3120572

2ℎ2

0

119862119873minus1119909

(45)

From (44) and (45) obviously we have

119902(1)

21120582 + 119902(0)

21=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

(46)

Thus 119876(120582) = N and this completes the proof of Theorem 2

3 Exact Soliton Solution of (1) andIts Dynamic Properties

In this section we will construct the explicit solutions ofthe integrable shallow water wave (1) by using the Darbouxtransformation (24)

Choosing 119906 = 1199060(1199060is an arbitrary constant) as a seed

solution of (1) and substituting 119906 = 1199060into the spectral

problems (8) Then we get two basic solutions of (8) asfollows

120601 (120582119895) = (

cosh 120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

sinh120583119895

)

120595 (120582119895) = (

sinh120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

cosh 120583119895

)

(47)

with

120583119895= radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

[119909 minus (

1198880

2

+

3120582119895120572

2ℎ2

0

+

1199060120572

2

) 119905]

(48)

where 120582119895is a nonzero constant and 0 lt 119895 le 2119873

According to (19) we have


1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

tanh120583119895minus 119903119895

1 minus 119903119895tanh120583

119895

(49)

where 119903119895(0 lt 119895 le 2119873) are nonzero constants

Using the Cramer rule to solve the linear algebraic system(18) we obtain

119860119873minus1

=

Δ119860119873minus1

Δ

(50)

where

Δ =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821

sdot sdot sdot 120582119896

11205751120582119896

1sdot sdot sdot 120582119873minus1

11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119860119873minus1

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896

1sdot sdot sdot minus120582

119873

11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


119873

21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


119873

21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(51)


0

5

0

250

100

150

u

x

tminus5

minus2

(a) 03 lt ℎ0lt 05 120582

1lt 0 120582

2lt 0

0

5

00

05

10

0

1000

2000

u

x

t

minus1000

minus5

minus05

minus10

(b) 02 lt ℎ0lt 03 120582

1lt 0 120582

2lt 0

Figure 1 The 3-D graphs of profiles of the singular double-soliton solution (53) for fixed parameters 1198880= 15 120572 = 02 120582

1= minus1 120582

2= minus02

1199031= minus80 1199032 = 20 and 119886 = 01

0

5

00

05

100

5000

u

x

t

minus5000

minus5

minus10

minus05

(a) 0 lt ℎ0lt 02 120582

1lt 0 120582

2lt 0

0

5

00

05

10u

x

tminus5

minus10

minus05

minus110

minus111

minus112

(b) ℎ0gt 1 120582

1gt 0 120582

2gt 0

Figure 2 The 3-D graphs of profiles of the singular double-soliton solution (53) for parameters (a) ℎ0= 015 119888

0= 15 120572 = 02 120582

1= minus1

1205822= minus02 119903

1= minus80 119903

2= 20 and 119886 = 01 (b) ℎ

0= 3 1198880= minus15 120572 = 02 120582

1= 1 120582

2= 02 119903

1= 8 119903

2= 20 and 119886 = minus2

As examples we will investigate exact solutions of (1) intwo simple cases119873 = 1 and119873 = 2 When119873 = 1 solving thelinear algebraic system (18) leads to

1198600=

12057511205822minus 12057521205821

1205752minus 1205751

(52)

Substituting (49) and (52) into (24) a singular double-solitonsolution of (1) is obtained as follows

119906 [1] =

6

ℎ2

0119891

[radicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

1205822

minus radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

1205821]

(53)

where

119891 = radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

minusradicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

(54)

By using program of computer it is easy to verify that thesolution (53) satisfies (1) and this shows that the Darbouxtransformation (24) which we obtained is correct In orderto show the properties of the above singular double-solitonsolutions visually as an example we plot the 3-D graphs ofsolution (53) for some fixed parameters which are shown inFigures 1 and 2


0

10

20

30

40

50

60

0 1 2

u

x

minus1

Figure 3 The 2-D graph of profile of the exact soliton solution (57)for fixed parameters

When 119873 = 2 using the Cramer rule to solve the linearalgebraic system (18) we obtain

1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)

where 120575119895(119895 = 1 2 3 4) are given by (49) From (24) an

explicit solution of (1) is obtained by the following

119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)

where 1198601is given by (55) Equation (57) is a very complex

solution and it is not soliton solution In order to show theproperties of solution (57) under the fixed parameters 120582

1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01

we plot its 2-D profile which is shown in Figure 3

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (no 11361023) the Natural Science

Foundation of Chongqing Normal University (no 13XLR20)the Scientific Foundation of Education of Yunnan Province(no 2012C199) and the Program Foundation of ChongqingInnovation Team Project in University under Grant noKJTD201308

References

[1] J Li T Xu X Meng Y Zhang H Zhang and B Tian ldquoLaxpair Backlund transformation and N-soliton-like solution fora variable-coefficient Gardner equation from nonlinear latticeplasma physics and ocean dynamics with symbolic computa-tionrdquo Journal of Mathematical Analysis and Applications vol336 no 2 pp 1443ndash1455 2007

[2] A U Khawaja ldquoA comparative analysis of Painleve Lax pairand similarity transformationmethods in obtaining the integra-bility conditions of nonlinear Schrodinger equationsrdquo Journal ofMathematical Physics vol 51 no 5 Article ID 007005JMP 2010

[3] Q X Qu B Tian K Sun and Y Jiang ldquoBacklund transfor-mation Lax pair and solutions for the Caudrey-Dodd-Gibbonequationrdquo Journal of Mathematical Physics vol 52 no 1 ArticleID 013511 2011

[4] J Lin B Ren H Li and Y Li ldquoSoliton solutions for twononlinear partial differential equations using a Darboux trans-formation of the Lax pairsrdquo Physical Review E vol 77 no 3Article ID 036605 2008

[5] P Wang B Tian W Liu Q Qu M Li and K Sun ldquoLaxpair conservation laws andN-soliton solutions for the extendedKorteweg-de Vries equations in fluidsrdquo European Physical Jour-nal D vol 61 no 3 pp 701ndash708 2011

[6] Y Jiang B Tian W Liu M Li P Wang and K SunldquoSolitons Backlund transformation and Lax pair for the (2+1)-dimensional Boiti-Leon-Pempinelli equation for the waterwavesrdquo Journal of Mathematical Physics vol 51 no 9 ArticleID 093519 2010

[7] H Zhi ldquoSymmetry reductions of the Lax pair for the 2 +1-dimensional Konopelchenko-Dubrovsky equationrdquo AppliedMathematics and Computation vol 210 no 2 pp 530ndash5352009

[8] E Fan and K W Chow ldquoDarboux covariant Lax pairs andinfinite conservation laws of the (2+1)-dimensional breakingsoliton equationrdquo Journal of Mathematical Physics vol 52 no2 Article ID 023504 2011

[9] H Q Zhang B Tian T Xu H Li C Zhang and H ZhangldquoLax pair and Darboux transformation for multi-componentmodified Korteweg-de Vries equationsrdquo Journal of Physics Avol 41 no 35 Article ID 355210 2008

[10] X Lu B Tian K Sun and P Wang ldquoBell-polynomial manip-ulations on the Backlund transformations and Lax pairs forsome soliton equations with one Tau-functionrdquo Journal ofMathematical Physics vol 51 no 11 Article ID 113506 2010

[11] A S Fokas ldquoLax pairs a novel type of separabilityrdquo InverseProblems vol 25 no 12 Article ID 123007 2009

[12] E Fan ldquoNew bilinear Backlund transformation and Lax pairfor the supersymmetric two-Boson equationrdquo Studies in AppliedMathematics vol 127 no 3 pp 284ndash301 2011

[13] S F Tian and H Q Zhang ldquoLax pair binary darbouxtransformation and new grammian solutions of nonisospec-tral kadomtsevpetviashvili equation with the two-singular-manifold methodrdquo Journal of Nonlinear Mathematical Physicsvol 17 no 4 pp 491ndash502 2010


[14] X Lu B Tian H Q Zhang T Xu and H Li ldquoGener-alized (2+1)-dimensional Gardner model bilinear equationsBacklund transformation Lax representation and interactionmechanismsrdquoNonlinearDynamics vol 67 no 3 pp 2279ndash22902012

[15] X L Gai Y T Gao L Wang et al ldquoPainleve property Laxpair and Darboux transformation of the variable-coefficientmodified Kortweg-de Vries model in fluid-filled elastic tubesrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 16 no 4 pp 1776ndash1782 2011

[16] J Avan and A Doikou ldquoBoundary Lax pairs for the An(1) Todafield theoriesrdquo Nuclear Physics B vol 821 no 3 pp 481ndash5052009

[17] Y He and H W Tam ldquoBilinear Backlund transformationand Lax pair for a coupled Ramani equationrdquo Journal ofMathematical Analysis and Applications vol 357 no 1 pp 132ndash136 2009

[18] H Q Zhang B Tian X H Meng X Lu and W Liu ldquoConser-vation laws soliton solutions and modulational instability forthe higher-order dispersive nonlinear Schrodinger equationrdquoEuropean Physical Journal B vol 72 no 2 pp 233ndash239 2009

[19] X Yu Y T Gao Z Y Sun and Y Liu ldquoWronskian solutionsand integrability for a generalized variable-coefficient forcedKorteweg-de Vries equation in fluidsrdquoNonlinear Dynamics vol67 no 2 pp 1023ndash1030 2012

[20] HQ Zhang B Tian L L Li andY Xue ldquoDarboux transforma-tion and soliton solutions for the (2 + 1)-dimensional nonlinearSchrodinger hierarchy with symbolic computationrdquo Physica Avol 388 no 1 pp 9ndash20 2009

[21] D Levi ldquoOn a new Darboux transformation for the construc-tion of exact solutions of the Schrodinger equationrdquo InverseProblems vol 4 no 1 pp 165ndash172 1988

[22] V B Matveev and M A Salle ldquoScattering of solitons inthe formalism of the Darboux transformrdquo Journal of SovietMathematics vol 34 no 5 pp 1983ndash1987 1986

[23] X Geng and H W Tam ldquoDarboux transformation and solitonsolutions for generalized nonlinear Schrodinger equationsrdquoJournal of the Physical Society of Japan vol 68 no 5 pp 1508ndash1512 1999

[24] Y Li W X Ma and J E Zhang ldquoDarboux transformationsof classical Boussinesq system and its new solutionsrdquo PhysicsLetters A vol 275 no 1-2 pp 60ndash66 2000

[25] X Li and A Chen ldquoDarboux transformation and multi-solitonsolutions of Boussinesq-Burgers equationrdquo Physics Letters Avol 342 no 5-6 pp 413ndash420 2005

[26] A Chen and X Li ldquoDarboux transformation and solitonsolutions for Boussinesq-Burgers equationrdquoChaos Solitons andFractals vol 27 no 1 pp 43ndash49 2006


[28] Z B Zhaqilao and Z Li ldquoDarboux transformation and bidi-rectional soliton solutions of a new (2 + 1)-dimensional solitonequationrdquo Physics Letters A vol 372 no 9 pp 1422ndash1428 2008

[29] H X Yang ldquoSoliton solutions by Darboux transformation for aHamiltonian lattice systemrdquo Physics Letters A vol 373 no 7 pp741ndash748 2009

[30] X Wu W Rui and X Hong ldquoA new discrete integrable systemderived from a generalized Ablowitz-Ladik hierarchy and itsDarboux transformationrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652076 19 pages 2012

[31] W XMa ldquoDarboux transformations for a lax integrable systemin 2n dimensionsrdquo Letters in Mathematical Physics vol 39 no1 pp 33ndash49 1997

[32] W X Ma ldquoThe algebraic structure of zero curvature repre-sentations and application to coupled KdV systemsrdquo Journal ofPhysics A vol 26 no 11 pp 2573ndash2582 1993

[33] R Camassa and D D Holm ldquoAn integrable shallow waterequation with peaked solitonsrdquo Physical Review Letters vol 71no 11 pp 1661ndash1664 1993

[34] G B Whitham Linear and Nonlinear Waves Wiley New YorkNY USA 1974

[35] B Fuchssteiner and A S Fokas ldquoSymplectic structures theirBacklund transformations and hereditary symmetriesrdquo PhysicaD vol 4 no 1 pp 47ndash66 1981

[36] B Fuchssteiner ldquoThe Lie algebra structure of nonlinear evolu-tion equations admitting infinite dimensional abelian symme-try groupsrdquoProgress ofTheoretical Physics vol 65 no 3 pp 861ndash876 1981

[37] A S Fokas and P M Santini ldquoAn inverse acoustic problem andlinearization of moderate amplitude dispersive wavesrdquo 1994

[38] A S Fokas ldquoOn a class of physically important integrableequationsrdquo Physica D vol 87 no 1ndash4 pp 145ndash150 1995

[39] Q Bi ldquoWave patterns associated with a singular line for a bi-Hamiltonian systemrdquo Physics Letters A vol 369 no 5-6 pp407ndash417 2007

[40] J Li and J Zhang ldquoBifurcations of travelling wave solutions forthe generalization form of the modified KdV equationrdquo ChaosSolitons and Fractals vol 21 no 4 pp 899ndash913 2004

[41] W Rui B He Y Long and C Chen ldquoThe integral bifurcationmethod and its application for solving a family of third-orderdispersive PDEsrdquo Nonlinear Analysis Theory Methods andApplications vol 69 no 4 pp 1256ndash1267 2008

[42] W Rui C Chen X Yang and Y Long ldquoSome new soliton-like solutions and periodic wave solutions with loop or withoutloop to a generalized KdV equationrdquo Applied Mathematics andComputation vol 217 no 4 pp 1666ndash1677 2010

[43] Y Long and C Chen ldquoExistence analysis of traveling wavesolutions for a generalization of KdV equationrdquo MathematicalProblems in Engineering vol 2013 Article ID 462957 7 pages2013

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


which comes from physical considerations via the method-ology introduced by Fuchssteiner and Fokas in [35 36] TheLax pairs of (3) with 120574 = 1 are given by [37 38] as follows

Ψ119909119909= [1198962(1 minus V) +

V

4]]Ψ = 0 120590V = 119906 + ]119906

119909119909

120590 =

120573 minus ]

2]

(4)

Ψ119905+ (minus

1

2

+ 119888)Ψ + (119906 +

120573

]+

2120590

4]1198962 minus 1)Ψ119909= 0 (5)

where 119888 is an arbitrary constant It is a pity that the [37] is not aformal publication and it is only preprint paper so we cannotknow its reality contents whether the authors have obtainedsoliton solutions of (3) by the Lax pairs (4) and (5) In fact(3) has been studied by many authors in recent years see thefollowing brief introductions

In [39] by using the bifurcation theory of dynamicsystem some subsection-function and implicit function solu-tions such as compactons solitary waves smooth periodicwaves and nonsmooth periodic waves with peaks as well asthe existence conditions have been presented by Bi By usingthe same method Li and Zhang [40] studied a generalizationform of the modified KdV equation which is more complexthan (3) In [40] the existence of solitary wave kink andantikink wave solutions and uncountably many smooth andnonsmooth periodic wave solutions are discussed By usingthe improved method named integral bifurcation method[41] Rui et al [42] obtain all kinds of soliton-like or kink-likewave solutions periodic wave solutions with loop or withoutloop smooth compacton-like periodic wave solutions and

nonsmooth periodic cusp wave solutions for (3) In [43]Long and Chen discussed the existence of solitary wave cuspwave periodic wave periodic cusp wave and compactonswere for (3) From the above references (1) (ie (3)) is a veryimportant water wave model

The rest of this paper is organized as follows In Section 2we will derive new Lax pair and Darboux transformationof (1) In Section 3 by using this Darboux transformationwe will investigate soliton solutions of (1) and discuss thedynamic properties of these soliton solutions

2 Lax Pair and Darboux Transformation of (1)Through a series of tedious computation we obtain Lax pairsof (1) as follows

120601119909119909= (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)120601 (6)

120601119905= minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)120601119909+

120572

4

119906119909120601 (7)

Obviously the Lax pairs (6) and (7) are different from theLax pairs (4) and (5) under 119888

0= 1 ] = minus(13)ℎ

2

0 120574 =

(32)120572 120573 = minus(16)ℎ2

01198880 They are new Lax pairs which we

obtained By using the new Lax pairs (6) and (7) we willconstruct a Darboux transformation for obtaining solitonsolutions of (1)

First we consider the following spectral problems

120601119909= M120601 120601

119905= N120601 (8)

with

M = (

0 1

minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

0

)

N = (

120572

4

119906119909

minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)(minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

) minus

120572

4

119906119909

)

(9)

where 120572 is a constant 120582 is a spectral parameter and 119906 is apotential function From compatibility condition 120601

119909119909119905= 120601119905119909119909

yields a zero curvature equation M119905minus N119909+ [MN] = O

Substituting M N into the zero curvature equation by adirect calculation (1) is obtained successfully

Next we will construct a Darboux Transformation (DT)of the spectral problems (8) In fact the DT is actually a gaugetransformation

120601 = T120601 (10)

of the spectral problems (8) It is required that 120601 also satisfiesthe same form of spectral problems

120601119909= M120601 M = (T

119909+ TM)Tminus1 (11)

120601119905= N120601 N = (T

119905+ TN)Tminus1 (12)

It means that we have to find a matrix T such that the oldpotential 119906 is replaced by the new one 119906

Suppose

T = T (120582) = (119860 (120582) 119861 (120582)119862 (120582) 119863 (120582)

) (13)


where

119860 (120582) = 119860119873(120582119873+

119873minus1

sum

119896=0

119860119896120582119896)

119861 (120582) = 119860119873(

119873minus1

sum

119896=0

119861119896120582119896)

(14)

119862 (120582) =

1

119860119873

(

119873minus1

sum

119896=0

119862119896120582119896)

119863 (120582) =

1

119860119873

(120582119873+

119873minus1

sum

119896=0

119863119896120582119896)

(15)

and 119860119873 119860119896 119861119896 119862119896 and 119863


of 119909 and 119905Let 120601(120582

119895) = (120601

1(120582119895) 1206012(120582119895))119879 120595(120582119895) = (120595

1(120582119895) 1205952(120582119895))119879


(119860 (120582119895) 1206011(120582119895) + 119861 (120582

119895) 1206012(120582119895))

minus 119903119895(119860 (120582

119895) 1205951(120582119895) + 119861 (120582

119895) 1205952(120582119895)) = 0

(119862 (120582119895) 1206011(120582119895) + 119863 (120582

119895) 1206012(120582119895))

minus 119903119895(119862 (120582

119895) 1205951(120582119895) + 119863 (120582

119895) 1205952(120582119895)) = 0

(16)


119860(120582119895) + 120575119895119861 (120582119895) = 0 119862 (120582

119895) + 120575119895119863(120582119895) = 0 (17)

That is119873minus1

sum

119896=0

(119860119896+ 120575119895119861119896) 120582119896

119895=minus120582119873

119895

119873minus1

sum

119896=0

(119862119896+ 120575119895119863119896) 120582119896

119895=minus120575119895120582119873

119895

(18)

where

120575119895=

1206012(120582119895) minus 1199031198951205952(120582119895)

1206011(120582119895) minus 1199031198951205951(120582119895)

1 le 119895 le 2119873 (19)




119873 119860119896 119861119896 119862119896 and 119863

119896(0 le 119896 le 119873 minus 1) are



detT (120582119895) = 119860 (120582

119895)119863 (120582

119895) minus 119861 (120582

119895) 119862 (120582

119895) (20)


119895119861(120582119895)

119862(120582119895) = minus120575

119895119863(120582119895) Thus we have

detT (120582) = 1205732119873minus1

prod

119895=1

(120582 minus 120582119895) (21)




1198602

119873= 1 (22)


M = (

0 1

minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

0

) (23)


119906 = 119906 +

6

ℎ2

0

119860119873minus1

(24)

119860119873minus1119909

= 119862119873minus1

minus

3

4ℎ2

0

119861119873minus1

119861119873minus1119909

= 119863119873minus1


119862119873minus1119909

=

3

4ℎ2

0

(119860119873minus1


) +

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

119863119873minus1119909

=

3

4ℎ2

0

119861119873minus1


(25)


(T119909+ TM)Tlowast = (11989111 (120582) 11989112 (120582)

11989121(120582) 119891

22(120582)) (26)


22(120582) are 2119873th-order




120575119895119909= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

minus 1205752

119895 (27)



(T119909+ TM)Tlowast = (detT) 119875 (120582) (28)

with

119875 (120582) = (

119901(0)

11119901(0)

12

119901(0)

21119901(0)

22

) (29)



T119909+ TM = 119875 (120582)T (30)



119901(0)

11= minus119901(0)

22= 120597119909ln119860119873 119901

(0)

12= 1198602

119873

119901(0)

21=

1

1198602

119873

3

4ℎ2

0

(31)


119901(0)

11= minus119901(0)

22= 0 119901

(0)

12= 1198602

119873= 1 (32)


119901(0)

21= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

(33)





(T119905+ TN)Tlowast = (11989211 (120582) 11989212 (120582)

11989221(120582) 119892

22(120582)) (34)


22(120582) are 2119873th-order


21(120582) are (2119873 + 1)th-order


120575119895119905=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times(minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

minus

120572

2

119906119909120575119895+ (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)1205752

119895

(35)



(T119905+ TN)Tlowast = (detT) 119876 (120582) (36)

with

119876 (120582) = (

119902(0)

11119902(1)

12120582 + 119902(0)

12

119902(1)

21120582 + 119902(0)

21119902(0)

22

) (37)



T119905+ TN = 119876 (120582)T (38)


119902(1)

12= minus

3120572

2ℎ2

0

1198602

119873 119902

(1)

21= minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(39)

119902(0)

11= 120597119905ln119860119873+

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

(40)

119902(0)

12= minus(

1198880

2

+

120572

2

119906)1198602

119873minus

3120572

2ℎ2

0

1198602

119873119860119873minus1

+

3120572

2ℎ2

0

1198602

119873119863119873minus1

(41)

119902(0)

21=

1

1198602

119873

120572

4

119906119909119909minus

1

1198602

119873

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

1

1198602

119873

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

)

+

119860119873minus1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(42)

119902(0)

22= minus120597119905ln119860119873minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

(43)


119902(0)

11=

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

=

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

=

120572

4

119906119909

119902(1)

12120582 + 119902(0)

12

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

2ℎ2

0

119860119873minus1

+

3120572

2ℎ2

0

119863119873minus1

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

ℎ2

0

+

3120572

2ℎ2

0

(119860119873minus1

+ 119863119873minus1

)

= minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

119902(0)

22= minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

= minus

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

= minus

120572

4

119906119909


119902(1)

21120582 + 119902(0)

21

= minus

3120572

2ℎ2

0

3

4ℎ2

0

120582+

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

) +

3120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906) +

3120572

2ℎ2

0

119862119873minus1119909

(44)


120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

=

120572

4

119906119909119909+

3120572

2ℎ2

0

119860119873minus1119909119909

minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

minus

6120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

+

3120572

2ℎ2

0

119862119873minus1119909

(45)


119902(1)

21120582 + 119902(0)

21=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

(46)







120601 (120582119895) = (

cosh 120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

sinh120583119895

)

120595 (120582119895) = (

sinh120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

cosh 120583119895

)

(47)

with


1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

[119909 minus (

1198880

2

+

3120582119895120572

2ℎ2

0

+

1199060120572

2

) 119905]

(48)




1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

tanh120583119895minus 119903119895

1 minus 119903119895tanh120583

119895

(49)



119860119873minus1

=

Δ119860119873minus1

Δ

(50)

where

Δ =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119860119873minus1

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


119873

11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


119873

21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


119873

21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(51)


0

5

0

250

100

150

u

x

tminus5

minus2

(a) 03 lt ℎ0lt 05 120582

1lt 0 120582

2lt 0

0

5

00

05

10

0

1000

2000

u

x

t

minus1000

minus5

minus05

minus10

(b) 02 lt ℎ0lt 03 120582

1lt 0 120582

2lt 0


1= minus1 120582

2= minus02

1199031= minus80 1199032 = 20 and 119886 = 01

0

5

00

05

100

5000

u

x

t

minus5000

minus5

minus10

minus05

(a) 0 lt ℎ0lt 02 120582

1lt 0 120582

2lt 0

0

5

00

05

10u

x

tminus5

minus10

minus05

minus110

minus111

minus112

(b) ℎ0gt 1 120582

1gt 0 120582

2gt 0


0= 15 120572 = 02 120582

1= minus1

1205822= minus02 119903

1= minus80 119903

2= 20 and 119886 = 01 (b) ℎ

0= 3 1198880= minus15 120572 = 02 120582

1= 1 120582

2= 02 119903

1= 8 119903

2= 20 and 119886 = minus2


1198600=

12057511205822minus 12057521205821

1205752minus 1205751

(52)


119906 [1] =

6

ℎ2

0119891

[radicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

1205822

minus radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

1205821]

(53)

where

119891 = radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

minusradicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

(54)



0

10

20

30

40

50

60

0 1 2

u

x

minus1



1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)



1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01


Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119860 (120582) = 119860119873(120582119873+

119873minus1

sum

119896=0

119860119896120582119896)

119861 (120582) = 119860119873(

119873minus1

sum

119896=0

119861119896120582119896)

(14)

119862 (120582) =

1

119860119873

(

119873minus1

sum

119896=0

119862119896120582119896)

119863 (120582) =

1

119860119873

(120582119873+

119873minus1

sum

119896=0

119863119896120582119896)

(15)

and 119860119873 119860119896 119861119896 119862119896 and 119863


of 119909 and 119905Let 120601(120582

119895) = (120601

1(120582119895) 1206012(120582119895))119879 120595(120582119895) = (120595

1(120582119895) 1205952(120582119895))119879


(119860 (120582119895) 1206011(120582119895) + 119861 (120582

119895) 1206012(120582119895))

minus 119903119895(119860 (120582

119895) 1205951(120582119895) + 119861 (120582

119895) 1205952(120582119895)) = 0

(119862 (120582119895) 1206011(120582119895) + 119863 (120582

119895) 1206012(120582119895))

minus 119903119895(119862 (120582

119895) 1205951(120582119895) + 119863 (120582

119895) 1205952(120582119895)) = 0

(16)


119860(120582119895) + 120575119895119861 (120582119895) = 0 119862 (120582

119895) + 120575119895119863(120582119895) = 0 (17)

That is119873minus1

sum

119896=0

(119860119896+ 120575119895119861119896) 120582119896

119895=minus120582119873

119895

119873minus1

sum

119896=0

(119862119896+ 120575119895119863119896) 120582119896

119895=minus120575119895120582119873

119895

(18)

where

120575119895=

1206012(120582119895) minus 1199031198951205952(120582119895)

1206011(120582119895) minus 1199031198951205951(120582119895)

1 le 119895 le 2119873 (19)




119873 119860119896 119861119896 119862119896 and 119863

119896(0 le 119896 le 119873 minus 1) are



detT (120582119895) = 119860 (120582

119895)119863 (120582

119895) minus 119861 (120582

119895) 119862 (120582

119895) (20)


119895119861(120582119895)

119862(120582119895) = minus120575

119895119863(120582119895) Thus we have

detT (120582) = 1205732119873minus1

prod

119895=1

(120582 minus 120582119895) (21)




1198602

119873= 1 (22)


M = (

0 1

minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

0

) (23)


119906 = 119906 +

6

ℎ2

0

119860119873minus1

(24)

119860119873minus1119909

= 119862119873minus1

minus

3

4ℎ2

0

119861119873minus1

119861119873minus1119909

= 119863119873minus1


119862119873minus1119909

=

3

4ℎ2

0

(119860119873minus1


) +

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

119863119873minus1119909

=

3

4ℎ2

0

119861119873minus1


(25)


(T119909+ TM)Tlowast = (11989111 (120582) 11989112 (120582)

11989121(120582) 119891

22(120582)) (26)


22(120582) are 2119873th-order




120575119895119909= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

minus 1205752

119895 (27)



(T119909+ TM)Tlowast = (detT) 119875 (120582) (28)

with

119875 (120582) = (

119901(0)

11119901(0)

12

119901(0)

21119901(0)

22

) (29)



T119909+ TM = 119875 (120582)T (30)



119901(0)

11= minus119901(0)

22= 120597119909ln119860119873 119901

(0)

12= 1198602

119873

119901(0)

21=

1

1198602

119873

3

4ℎ2

0

(31)


119901(0)

11= minus119901(0)

22= 0 119901

(0)

12= 1198602

119873= 1 (32)


119901(0)

21= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

(33)





(T119905+ TN)Tlowast = (11989211 (120582) 11989212 (120582)

11989221(120582) 119892

22(120582)) (34)


22(120582) are 2119873th-order


21(120582) are (2119873 + 1)th-order


120575119895119905=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times(minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

minus

120572

2

119906119909120575119895+ (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)1205752

119895

(35)



(T119905+ TN)Tlowast = (detT) 119876 (120582) (36)

with

119876 (120582) = (

119902(0)

11119902(1)

12120582 + 119902(0)

12

119902(1)

21120582 + 119902(0)

21119902(0)

22

) (37)



T119905+ TN = 119876 (120582)T (38)


119902(1)

12= minus

3120572

2ℎ2

0

1198602

119873 119902

(1)

21= minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(39)

119902(0)

11= 120597119905ln119860119873+

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

(40)

119902(0)

12= minus(

1198880

2

+

120572

2

119906)1198602

119873minus

3120572

2ℎ2

0

1198602

119873119860119873minus1

+

3120572

2ℎ2

0

1198602

119873119863119873minus1

(41)

119902(0)

21=

1

1198602

119873

120572

4

119906119909119909minus

1

1198602

119873

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

1

1198602

119873

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

)

+

119860119873minus1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(42)

119902(0)

22= minus120597119905ln119860119873minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

(43)


119902(0)

11=

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

=

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

=

120572

4

119906119909

119902(1)

12120582 + 119902(0)

12

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

2ℎ2

0

119860119873minus1

+

3120572

2ℎ2

0

119863119873minus1

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

ℎ2

0

+

3120572

2ℎ2

0

(119860119873minus1

+ 119863119873minus1

)

= minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

119902(0)

22= minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

= minus

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

= minus

120572

4

119906119909


119902(1)

21120582 + 119902(0)

21

= minus

3120572

2ℎ2

0

3

4ℎ2

0

120582+

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

) +

3120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906) +

3120572

2ℎ2

0

119862119873minus1119909

(44)


120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

=

120572

4

119906119909119909+

3120572

2ℎ2

0

119860119873minus1119909119909

minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

minus

6120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

+

3120572

2ℎ2

0

119862119873minus1119909

(45)


119902(1)

21120582 + 119902(0)

21=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

(46)







120601 (120582119895) = (

cosh 120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

sinh120583119895

)

120595 (120582119895) = (

sinh120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

cosh 120583119895

)

(47)

with


1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

[119909 minus (

1198880

2

+

3120582119895120572

2ℎ2

0

+

1199060120572

2

) 119905]

(48)




1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

tanh120583119895minus 119903119895

1 minus 119903119895tanh120583

119895

(49)



119860119873minus1

=

Δ119860119873minus1

Δ

(50)

where

Δ =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119860119873minus1

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


119873

11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


119873

21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


119873

21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(51)


0

5

0

250

100

150

u

x

tminus5

minus2

(a) 03 lt ℎ0lt 05 120582

1lt 0 120582

2lt 0

0

5

00

05

10

0

1000

2000

u

x

t

minus1000

minus5

minus05

minus10

(b) 02 lt ℎ0lt 03 120582

1lt 0 120582

2lt 0


1= minus1 120582

2= minus02

1199031= minus80 1199032 = 20 and 119886 = 01

0

5

00

05

100

5000

u

x

t

minus5000

minus5

minus10

minus05

(a) 0 lt ℎ0lt 02 120582

1lt 0 120582

2lt 0

0

5

00

05

10u

x

tminus5

minus10

minus05

minus110

minus111

minus112

(b) ℎ0gt 1 120582

1gt 0 120582

2gt 0


0= 15 120572 = 02 120582

1= minus1

1205822= minus02 119903

1= minus80 119903

2= 20 and 119886 = 01 (b) ℎ

0= 3 1198880= minus15 120572 = 02 120582

1= 1 120582

2= 02 119903

1= 8 119903

2= 20 and 119886 = minus2


1198600=

12057511205822minus 12057521205821

1205752minus 1205751

(52)


119906 [1] =

6

ℎ2

0119891

[radicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

1205822

minus radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

1205821]

(53)

where

119891 = radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

minusradicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

(54)



0

10

20

30

40

50

60

0 1 2

u

x

minus1



1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)



1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01


Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901(0)

11= minus119901(0)

22= 120597119909ln119860119873 119901

(0)

12= 1198602

119873

119901(0)

21=

1

1198602

119873

3

4ℎ2

0

(31)


119901(0)

11= minus119901(0)

22= 0 119901

(0)

12= 1198602

119873= 1 (32)


119901(0)

21= minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

(33)





(T119905+ TN)Tlowast = (11989211 (120582) 11989212 (120582)

11989221(120582) 119892

22(120582)) (34)


22(120582) are 2119873th-order


21(120582) are (2119873 + 1)th-order


120575119895119905=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times(minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

minus

120572

2

119906119909120575119895+ (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)1205752

119895

(35)



(T119905+ TN)Tlowast = (detT) 119876 (120582) (36)

with

119876 (120582) = (

119902(0)

11119902(1)

12120582 + 119902(0)

12

119902(1)

21120582 + 119902(0)

21119902(0)

22

) (37)



T119905+ TN = 119876 (120582)T (38)


119902(1)

12= minus

3120572

2ℎ2

0

1198602

119873 119902

(1)

21= minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(39)

119902(0)

11= 120597119905ln119860119873+

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

(40)

119902(0)

12= minus(

1198880

2

+

120572

2

119906)1198602

119873minus

3120572

2ℎ2

0

1198602

119873119860119873minus1

+

3120572

2ℎ2

0

1198602

119873119863119873minus1

(41)

119902(0)

21=

1

1198602

119873

120572

4

119906119909119909minus

1

1198602

119873

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

1

1198602

119873

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

)

+

119860119873minus1

1198602

119873

3120572

2ℎ2

0

3

4ℎ2

0

(42)

119902(0)

22= minus120597119905ln119860119873minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

(43)


119902(0)

11=

120572

4

119906119909minus

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

+

3120572

2ℎ2

0

119862119873minus1

=

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

=

120572

4

119906119909

119902(1)

12120582 + 119902(0)

12

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

2ℎ2

0

119860119873minus1

+

3120572

2ℎ2

0

119863119873minus1

= minus

3120572

2ℎ2

0

120582 minus (

1198880

2

+

120572

2

119906) minus

3120572

ℎ2

0

+

3120572

2ℎ2

0

(119860119873minus1

+ 119863119873minus1

)

= minus(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

119902(0)

22= minus

3120572

2ℎ2

0

119862119873minus1

minus

120572

4

119906119909+

3120572

2ℎ2

0

3

4ℎ2

0

119861119873minus1

= minus

120572

4

119906119909minus

3120572

2ℎ2

0

119860119873minus1119909

= minus

120572

4

119906119909


119902(1)

21120582 + 119902(0)

21

= minus

3120572

2ℎ2

0

3

4ℎ2

0

120582+

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

) +

3120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906) +

3120572

2ℎ2

0

119862119873minus1119909

(44)


120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

=

120572

4

119906119909119909+

3120572

2ℎ2

0

119860119873minus1119909119909

minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

minus

6120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

+

3120572

2ℎ2

0

119862119873minus1119909

(45)


119902(1)

21120582 + 119902(0)

21=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

(46)







120601 (120582119895) = (

cosh 120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

sinh120583119895

)

120595 (120582119895) = (

sinh120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

cosh 120583119895

)

(47)

with


1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

[119909 minus (

1198880

2

+

3120582119895120572

2ℎ2

0

+

1199060120572

2

) 119905]

(48)




1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

tanh120583119895minus 119903119895

1 minus 119903119895tanh120583

119895

(49)



119860119873minus1

=

Δ119860119873minus1

Δ

(50)

where

Δ =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119860119873minus1

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


119873

11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


119873

21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


119873

21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(51)


0

5

0

250

100

150

u

x

tminus5

minus2

(a) 03 lt ℎ0lt 05 120582

1lt 0 120582

2lt 0

0

5

00

05

10

0

1000

2000

u

x

t

minus1000

minus5

minus05

minus10

(b) 02 lt ℎ0lt 03 120582

1lt 0 120582

2lt 0


1= minus1 120582

2= minus02

1199031= minus80 1199032 = 20 and 119886 = 01

0

5

00

05

100

5000

u

x

t

minus5000

minus5

minus10

minus05

(a) 0 lt ℎ0lt 02 120582

1lt 0 120582

2lt 0

0

5

00

05

10u

x

tminus5

minus10

minus05

minus110

minus111

minus112

(b) ℎ0gt 1 120582

1gt 0 120582

2gt 0


0= 15 120572 = 02 120582

1= minus1

1205822= minus02 119903

1= minus80 119903

2= 20 and 119886 = 01 (b) ℎ

0= 3 1198880= minus15 120572 = 02 120582

1= 1 120582

2= 02 119903

1= 8 119903

2= 20 and 119886 = minus2


1198600=

12057511205822minus 12057521205821

1205752minus 1205751

(52)


119906 [1] =

6

ℎ2

0119891

[radicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

1205822

minus radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

1205821]

(53)

where

119891 = radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

minusradicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

(54)



0

10

20

30

40

50

60

0 1 2

u

x

minus1



1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)



1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01


Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119902(1)

21120582 + 119902(0)

21

= minus

3120572

2ℎ2

0

3

4ℎ2

0

120582+

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

120572

2

119906)minus

3120572

2ℎ2

0

3

4ℎ2

0

119863119873minus1

+

3120572

2ℎ2

0

(

1198880

4120572

+

119906 minus (13) ℎ2

0119906119909119909

2

) +

3120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus

3

4ℎ2

0

(

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906) +

3120572

2ℎ2

0

119862119873minus1119909

(44)


120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

=

120572

4

119906119909119909+

3120572

2ℎ2

0

119860119873minus1119909119909

minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

minus

6120572

2ℎ2

0

3

4ℎ2

0

119860119873minus1

=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

3

4ℎ2

0

+

3120572

2ℎ2

0

119862119873minus1119909

(45)


119902(1)

21120582 + 119902(0)

21=

120572

4

119906119909119909minus (

1198880

2

+

3120582120572

2ℎ2

0

+

120572

2

119906)

times (minus

1198880

4120582120572

+

3

4ℎ2

0

minus

119906 minus (13) ℎ2

0119906119909119909

2120582

)

(46)







120601 (120582119895) = (

cosh 120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

sinh120583119895

)

120595 (120582119895) = (

sinh120583119895

radicminus

1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

cosh 120583119895

)

(47)

with


1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

[119909 minus (

1198880

2

+

3120582119895120572

2ℎ2

0

+

1199060120572

2

) 119905]

(48)




1198880

4120582119895120572

+

3

4ℎ2

0

minus

1199060

2120582119895

tanh120583119895minus 119903119895

1 minus 119903119895tanh120583

119895

(49)



119860119873minus1

=

Δ119860119873minus1

Δ

(50)

where

Δ =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ119860119873minus1

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751

1205821

12057511205821


11205751120582119896


119873

11205751120582119873minus1

1

1 1205752

1205822

12057521205822


21205752120582119896


119873

21205752120582119873minus1

2

1 1205752119873

1205822119873

12057521198731205822119873


21198731205752119873120582119896


119873

21198731205752119873120582119873minus1

2119873

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(51)


0

5

0

250

100

150

u

x

tminus5

minus2

(a) 03 lt ℎ0lt 05 120582

1lt 0 120582

2lt 0

0

5

00

05

10

0

1000

2000

u

x

t

minus1000

minus5

minus05

minus10

(b) 02 lt ℎ0lt 03 120582

1lt 0 120582

2lt 0


1= minus1 120582

2= minus02

1199031= minus80 1199032 = 20 and 119886 = 01

0

5

00

05

100

5000

u

x

t

minus5000

minus5

minus10

minus05

(a) 0 lt ℎ0lt 02 120582

1lt 0 120582

2lt 0

0

5

00

05

10u

x

tminus5

minus10

minus05

minus110

minus111

minus112

(b) ℎ0gt 1 120582

1gt 0 120582

2gt 0


0= 15 120572 = 02 120582

1= minus1

1205822= minus02 119903

1= minus80 119903

2= 20 and 119886 = 01 (b) ℎ

0= 3 1198880= minus15 120572 = 02 120582

1= 1 120582

2= 02 119903

1= 8 119903

2= 20 and 119886 = minus2


1198600=

12057511205822minus 12057521205821

1205752minus 1205751

(52)


119906 [1] =

6

ℎ2

0119891

[radicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

1205822

minus radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

1205821]

(53)

where

119891 = radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

minusradicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

(54)



0

10

20

30

40

50

60

0 1 2

u

x

minus1



1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)



1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01


Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

0

250

100

150

u

x

tminus5

minus2

(a) 03 lt ℎ0lt 05 120582

1lt 0 120582

2lt 0

0

5

00

05

10

0

1000

2000

u

x

t

minus1000

minus5

minus05

minus10

(b) 02 lt ℎ0lt 03 120582

1lt 0 120582

2lt 0


1= minus1 120582

2= minus02

1199031= minus80 1199032 = 20 and 119886 = 01

0

5

00

05

100

5000

u

x

t

minus5000

minus5

minus10

minus05

(a) 0 lt ℎ0lt 02 120582

1lt 0 120582

2lt 0

0

5

00

05

10u

x

tminus5

minus10

minus05

minus110

minus111

minus112

(b) ℎ0gt 1 120582

1gt 0 120582

2gt 0


0= 15 120572 = 02 120582

1= minus1

1205822= minus02 119903

1= minus80 119903

2= 20 and 119886 = 01 (b) ℎ

0= 3 1198880= minus15 120572 = 02 120582

1= 1 120582

2= 02 119903

1= 8 119903

2= 20 and 119886 = minus2


1198600=

12057511205822minus 12057521205821

1205752minus 1205751

(52)


119906 [1] =

6

ℎ2

0119891

[radicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

1205822

minus radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

1205821]

(53)

where

119891 = radicminus

1198880

41205822120572

+

3

4ℎ2

0

minus

1199060

21205822

tanh1205832minus 1199032

1 minus 1199032tanh120583

2

minusradicminus

1198880

41205821120572

+

3

4ℎ2

0

minus

1199060

21205821

tanh1205831minus 1199031

1 minus 1199031tanh120583

1

(54)



0

10

20

30

40

50

60

0 1 2

u

x

minus1



1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)



1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01


Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40

50

60

0 1 2

u

x

minus1



1198601=

Δ1198601

Δ

(55)

with

Δ =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751120582112057511205821

1 1205752120582212057521205822

1 1205753120582312057531205823

1 1205754120582412057541205824

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

Δ1198601=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1 1205751minus1205822

112057511205821

1 1205752minus1205822

212057521205822

1 1205753minus1205822

312057531205823

1 1205754minus1205822

412057541205824

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(56)



119906 [2] = 119906 +

6

ℎ2

0

1198601 (57)



1=

minus021205822= minus03120582

3= minus04120582

4= minus01 119886 = 0 119888

0= 15120572 = 12

ℎ0= 08 119903

1= minus02 119903

2= minus03 119903

3= minus04 119903

4= minus05 119905 = 01


Acknowledgments



References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









new lax pairs and darboux transformation and its application to a

Documents