research article bright solitons in a pt...

Research ArticleBright Solitons in a PT-Symmetric Chain of Dimers

Omar B Kirikchi1 Alhaji A Bachtiar2 and Hadi Susanto1

1Department of Mathematical Sciences University of Essex Wivenhoe Park Colchester Essex CO4 3SQ UK2Department of Mathematics University of Indonesia Depok Indonesia

Correspondence should be addressed to Hadi Susanto hsusantoessexacuk

Received 17 July 2016 Revised 17 September 2016 Accepted 17 October 2016

Academic Editor Yao-Zhong Zhang

Copyright copy 2016 Omar B Kirikchi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study the existence and stability of fundamental bright discrete solitons in a parity-time- (PT-) symmetric coupler composedby a chain of dimers that is modelled by linearly coupled discrete nonlinear Schrodinger equations with gain and loss terms Weuse a perturbation theory for small coupling between the lattices to perform the analysis which is then confirmed by numericalcalculations Such analysis is based on the concept of the so-called anticontinuum limit approach We consider the fundamentalonsite and intersite bright solitons Each solution has symmetric and antisymmetric configurations between the armsThe stabilityof the solutions is then determined by solving the corresponding eigenvalue problem We obtain that both symmetric andantisymmetric onsite mode can be stable for small coupling in contrast to the reported continuum limit where the antisymmetricsolutions are always unstable The instability is either due to the internal modes crossing the origin or the appearance of a quartetof complex eigenvalues In general the gain-loss term can be considered parasitic as it reduces the stability region of the onsitesolitons Additionally we analyse the dynamic behaviour of the onsite and intersite solitons when unstable where typically it iseither in the form of travelling solitons or soliton blow-ups

1 Introduction

A system of equations is PT-symmetric if it is invariantwith respect to combined parity (P) and time-reversal (T)transformations The symmetry is interesting as it forms aparticular class of non-Hermitian Hamiltonians in quantummechanics [1] whichmay have a real spectrumup to a criticalvalue of the complex potential parameter above which thesystem is in the ldquobrokenPT symmetryrdquo phase [2ndash4]

The most basic configuration having PT symmetry isa dimer that is a system of two coupled oscillators whereone of the oscillators has damping losses and the other onegains energy from external sources Considerably dimersare also the most important PT systems as the concept ofPT symmetry was first realised experimentally on dimersconsisting of two coupled optical waveguides [5 6] (see alsothe review [7] for PT symmetry in optical applications)The experiments have been rapidly followed by many otherobservations of PT symmetry in different branches ofphysics from mechanical to electrical analogues (see thereview [8])

When nonlinearity is present in a PT system one mayhave nontrivial behaviours that do not exist in the linear casesuch as the presence of blow-up dynamics in the parameterregion of the unbroken phase in the linear counterpart [9ndash11]When nonlinear dimers are put in arrays where elementswith gain and loss are linearly coupled to the elements of thesame type belonging to adjacent dimers one can also obtaina distinctive feature in the form of the existence of solutionslocalized in space as continuous families of their energyparameter [12] The system therefore has two arms with eacharm described by a discrete nonlinear Schrodinger equationwith gain or loss Here we study the nonlinear localisedsolutions which loosely we also refer to as bright discretesolitons and their stability analytically and numerically

In the continuous limit the coupled equations withoutgain-loss have been studied in [13ndash16] where it has beenshown that the system admits symmetric antisymmetricand asymmetric solitons between the arms Unstable asym-metric solutions bifurcate from the symmetric ones througha subcritical symmetry breaking bifurcation which then

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 9514230 12 pageshttpdxdoiorg10115520169514230

2 Advances in Mathematical Physics

become stable after a tangent (saddle-center) bifurcationWhen one adds a gain and loss term in each arm one obtainsPT-symmetric couplers which have been considered in[17ndash22] In the presence of the linear-gain and loss termsasymmetric solitons cease to exist while antisymmetric soli-tons are always unstable [20] even though those with smallamplitudes can live long due to weak underlying instability[17] Symmetric solitons can be stable in a similar fashion tothose in the system without gain-loss [20]

The stability of bright discrete solitons inPT-symmetriccouplers was discussed in [12] using variational methodswhere it was shown that symmetric onsite solutions can bestable and there is a critical solution amplitude above whichthe PT symmetry is broken The case when the polarity ofthe PT-symmetric dimers is staggered along the chain isconsidered in [23]The same equations without gain and losswere considered in [24] where the symmetric soliton losesits stability through the symmetry-breaking bifurcation at afinite value of the energy similarly to that in the continuouscounterpart [13ndash16] Recently a similarPT chain of dimerswith a slightly different nonlinearity was derived [25] todescribe coupled chains of parametrically driven pendula asa mechanical analogue of PT-symmetric systems [26] Thestability of bright discrete solitons was established throughthe applications of the Hamiltonian energy and an indextheorem The nonlinear long-time stability of the discretesolitons was also established using the Lyapunov method inthe asymptotic limit of a weak coupling between the pendula[27]

In this work we determine the eigenvalues of dis-crete solitons in PT-symmetric couplers analytically usingasymptotic expansions The computation is based on the so-called method of weak coupling or anticontinuum limit Theapplication of the method in the study of discrete solitonswas formulated rigorously in [28] for conservative systemsIt was then applied to PT-symmetric networks in [29 30]However no explicit expression of the asymptotic series ofthe eigenvalues for the stability of discrete solitons has beenpresented before Here in addition to the asymptotic limitof weak coupling between the dimers we also propose toconsider expansions in the coefficient of the gain-loss termsIn this case explicit computations of the asymptotic series ofthe eigenvalues become possible

The manuscript is outlined as follows In Section 2 wepresent the mathematical model In Section 3 we use per-turbation theory for small coupling to analyse the existenceof fundamental localised solutions Such analysis is based onthe concept of the so-called anticontinuum limit approachThe stability of the solitons is then considered analyticallyin Section 4 by solving a corresponding eigenvalue problemIn this section in addition to small coupling the expansionis also performed under the assumption of small coefficientof the gain-loss term due to the nonsimple expression of theeigenvectors of the linearised operatorThe findings obtainedfrom the analytical calculations are then compared with thenumerical counterparts in Section 5 We also produce stabil-ity regions for all the fundamental solitons numerically Inthis section we present the typical dynamics of solitons in theunstable parameter ranges by direct numerical integrations

of the governing equation We present the conclusion inSection 6

2 Mathematical Model

The governing equations describing PT-symmetric chainsof dimers are of the form [12]

119899 = 119894120590 100381610038161003816100381611990611989910038161003816100381610038162 119906119899 + 119894120598Δ 2119906119899 + 120574119906119899 + 119894V119899V119899 = 119894120590 1003816100381610038161003816V11989910038161003816100381610038162 V119899 + 119894120598Δ 2V119899 minus 120574V119899 + 119894119906119899

(1)

The derivative with respect to the evolution variable (ie thepropagation distance if we consider their application in fiberoptics) is denoted by the overdot 119906119899 = 119906119899(119905) and V119899 = V119899(119905)are complex-valued wave function at site 119899 isin Z 120598 gt 0is the constant coefficient of the horizontal linear coupling(coupling constant between two adjacent sites) Δ 2119906119899 =(119906119899+1 minus 2119906119899 + 119906119899minus1) and Δ 2V119899 = (V119899+1 minus 2V119899 + V119899minus1) are thediscrete Laplacian term in one spatial dimension the gain andloss acting on complex variables 119906119899 and V119899 are representedby the positive coefficient 120574 that is 120574 gt 0 The nonlinearitycoefficient is denoted by 120590 which can be scaled to +1withoutloss of generality due to the case of focusing nonlinearitythat we consider Bright discrete soliton solutions satisfy thelocalisation conditions 119906119899 V119899 rarr 0 as 119899 rarr plusmninfin

The focusing system has static localised solutions that canbe obtained from substituting

119906119899 = 119860119899119890119894120596119905V119899 = 119861119899119890119894120596119905 (2)

into (1) to yield the equations

120596119860119899 = 100381610038161003816100381611986011989910038161003816100381610038162 119860119899 + 120598 (119860119899+1 minus 2119860119899 + 119860119899minus1) minus 119894120574119860119899+ 119861119899

120596119861119899 = 100381610038161003816100381611986111989910038161003816100381610038162 119861119899 + 120598 (119861119899+1 minus 2119861119899 + 119861119899minus1) + 119894120574119861119899 + 119860119899(3)

where 119860119899 and 119861119899 are complex-valued and the propagationconstant 120596 isin R3 Solutions of Weakly Coupled Equations

In the uncoupled limit that is when 120598 = 0 chain (1)becomes the equations for the dimer The static equation (3)has been analysed in detail in [29 31] where it was shownthat there is a relation between 120596 and 120574 above which thereis no time-independent solution to (3) (see also the analysisbelow)When 120598 is nonzero but small enough the existence ofsolutions emanating from the uncoupled limit can be shownusing the Implicit Function Theorem The existence analysisof [25] can be adopted here despite the slightly differentnonlinearity as the Jacobian of our system when uncoupledshares a rather similar invertible structure (see also [29 30]that have the same nonlinearity in the governing equationsbut different small coupling terms) However below we will

Advances in Mathematical Physics 3

not state the theorem and instead derive the asymptotic seriesof the solutions

Using perturbation expansion solutions of the coupler(3) for small coupling constant 120598 can be expressed analyticallyas

119860119899 = 119860(0)119899 + 120598119860(1)119899 + sdot sdot sdot 119861119899 = 119861(0)119899 + 120598119861(1)119899 + sdot sdot sdot

(4)

By substituting the above expansions into (3) and collectingthe terms in successive powers of 120598 one obtains at O(1) andO(120598) respectively the equations119860(0)119899 = 119861(0)119899 (120596 minus 119861(0)119899 119861lowast(0)119899 minus 119894120574) 119861(0)119899 = 119860(0)119899 (120596 minus 119860(0)119899 119860lowast(0)119899 + 119894120574) (5)

119860(1)119899 = 119861(1)119899 (120596 minus 2119861(0)119899 119861lowast(0)119899 minus 119894120574) minus 119861(0)119899 2119861lowast(1)119899 minus Δ 2119861(0)119899 119861(1)119899 = 119860(1)119899 (120596 minus 2119860(0)119899 119860lowast(0)119899 + 119894120574) minus 119860(0)119899 2119860lowast(1)119899

minus Δ 2119860(0)119899 (6)

It is well-known that there are two natural fundamentalsolutions representing bright discrete solitons that may existfor any 120598 from the anticontinuum to the continuum limitthat is an intersite (two-excited-site) and onsite (one-excited-site) bright discrete mode Here we will limit our study tothese two fundamental modes

31 Intersite Soliton In the uncoupled limit the mode struc-tures 119860(0)119899 and 119861(0)119899 for the intersite soliton are of the form

119860(0)119899 = 0119890119894120601119886 119899 = 0 10 otherwise

119861(0)119899 = 0119890119894120601119887 119899 = 0 10 otherwise

(7)

with [31]

0 = 1198870 = radic120596 ∓ radic1 minus 1205742sin (120601119887 minus 120601119886) = 120574

(8)

which is an exact solution of (5) Note that (8) will have noreal solution when |120574| gt 1 This is the broken region ofPT-symmetry The parameter 120601119886 can be taken as 0 dueto the gauge phase invariance of the governing equation (1)and henceforth 120601119887 = arcsin 120574 and 120587 minus arcsin 120574 The formerphase corresponds to the so-called symmetric configurationbetween the arms while the latter is called antisymmetricone Herein we also refer to the symmetric and antisym-metric soliton as soliton I and II respectively Equation (8)informs us that 120596 gt radic1 minus 1205742 gt 0 and 120596 gt minusradic1 minus 1205742 are thenecessary conditions for solitons I and II respectively

For the first-order correction due to the weak couplingwriting 119860(1)119899 = 1198991119890119894120601119886 and 119861(1)119899 = 1198871198991119890119894120601119887 and substitutingthem into (6) will yield

1198991 = 1198871198991 =

1(20) 119899 = 0 111198860 119899 = minus1 20 otherwise

(9)

Equations (4) (7) (8) and (9) are the asymptotic expan-sion of the intersite solitons One can continue the samecalculation to obtain higher order corrections Here we limitourselves to the first-order correction only which is sufficientto determine the leading order behaviour of the eigenvalueslater

32 Onsite Soliton For the onsite soliton that is a one-excited-site discrete mode one can perform the same com-putations to obtain the mode structure of the form

119860(0)119899 = 0119890119894120601119886 119899 = 00 otherwise

119861(0)119899 = 1198870119890119894120601119887 119899 = 00 otherwise

(10)

with (8) After writing 119860(1)119899 = 1198991119890119894120601119886 and 119861(1)119899 = 1198991119890119894120601119887 thefirst order correction from (6) is given by

1198991 = 1198871198991 = 11198860 119899 = 0 plusmn10 otherwise (11)

4 Stability Analysis

After we find discrete solitons their linear stability is thendetermined by solving a corresponding linear eigenvalueproblem To do so we introduce the linearisation ansatz119906119899 = (119860119899 + (119870119899 + 119894119871119899)119890120582119905)119890119894120596119905 and V119899 = (119861119899 + (119875119899 +119894119876119899)119890120582119905)119890119894120596119905 || ≪ 1 and substitute this into (1) to obtainthe linearised equations at O()

120582119870119899 = minus (1198602119899 minus 120596) 119871119899 minus 120598 (119871119899+1 minus 2119871119899 + 119871119899minus1) + 120574119870119899minus 119876119899


120582119871119899 = (31198602119899 minus 120596)119870119899 + 120598 (119870119899+1 minus 2119870119899 + 119870119899minus1) + 120574119871119899+ 119875119899

120582119875119899 = minus (R (119861119899)2 + 3I (119861119899)2 minus 120596)119876119899minus 120598 (119876119899+1 minus 2119876119899 + 119876119899minus1)minus (2R (119861119899)I (119861119899) + 120574) 119875119899 minus 119871119899

120582119876119899 = (3R (119861119899)2 +I (119861119899)2 minus 120596)119875119899+ 120598 (119875119899+1 minus 2119875119899 + 119875119899minus1)+ (2R (119861119899)I (119861119899) minus 120574)119876119899 + 119870119899

(12)

which have to be solved for the eigenvalue 120582 and thecorresponding eigenvector [119870119899 119871119899 119875119899 119876119899]119879 As thestabilitymatrix of the eigenvalue problem (12) is real valued120582and minus120582 are also eigenvalues with corresponding eigenvectors[119870119899 119871119899 119875119899 119876119899]119879 and [119870119899 minus119871119899 119875119899 minus119876119899]119879 with120574 rarr minus120574 respectively Therefore we can conclude that thesolution 119906119899 is (linearly) stable only when R(120582) = 0 for alleigenvalues 12058241 Continuous Spectrum The spectrum of (12) will consistof continuous spectrum and discrete spectrum (eigenvalue)To investigate the former we consider the limit 119899 rarr plusmninfinintroduce the plane-wave ansatz 119870119899 = 119890119894119896119899 119871119899 = 119890119894119896119899 119875119899 =119890119894119896119899 and 119876119899 = 119890119894119896119899 119896 isin R and substitute the ansatz into(12) to obtain

120582[[[[[[[

]]]]]]]= [[[[[[

120574 120585 0 minus1minus120585 120574 1 00 minus1 minus120574 1205851 0 minus120585 minus120574

]]]]]]

[[[[[[[

]]]]]]] (13)

where 120585 = 120596 minus 2120598(cos 119896 minus 1) The equation can be solvedanalytically to yield the dispersion relation

1205822 = 4120598120596 (cos 119896 minus 1) minus 41205982 (cos 119896 minus 1)2 minus 1205962 minus 1 + 1205742plusmn (4120598 (cos 119896 minus 1) minus 2120596)radic1 minus 1205742 (14)

The continuous spectrum is therefore given by 120582 isinplusmn[1205821minus 1205822minus] and 120582 isin plusmn[1205821+ 1205822+] with the spectrum bound-aries

1205821plusmn = 119894radic1 minus 1205742 + 1205962 plusmn 2120596radic1 minus 12057421205822plusmn= 119894radic1 minus 1205742 + 8120598120596 + 161205982 + 1205962 + 2radic1 minus 1205742 (plusmn120596 minus 4120598)

(15)

obtained from (14) by setting 119896 = 0 and 119896 = 120587 in the equation42 Discrete Spectrum Following theweak-coupling analysisas in Section 3 we will as well use similar asymptoticexpansions to solve the eigenvalue problem (12) analyticallythat is we write

◻ = ◻(0) + radic120598◻(1) + 120598◻(2) + sdot sdot sdot (16)

with ◻ = 120582119870119899 119871119899 119875119899 119876119899 We then substitute the expansionsinto the eigenvalue problem (12)

At order O(1) one will obtain the stability equation forthe dimer which has been discussed for a general value of 120574in [31] The expression of the eigenvalues is simple but theexpression of the corresponding eigenvectors is not whichmakes the result of [31] rather impractical to use Thereforehere we limit ourselves to the case of small |120574| and expand (16)further as

◻(119895) = ◻(1198950) + 120574◻(1198951) + 1205742◻(1198952) + sdot sdot sdot (17)

119895 = 0 1 2 Hence we have two small parameters thatis 120598 and 120574 that are independent of each other For the sakeof presentation the detailed calculations are shown in theAppendix Here we will only cite the final results

421 Intersite Soliton I The intersite soliton I (ie thesymmetric intersite soliton) has three pairs of eigenvalues forsmall 120598 and 120574 One pair bifurcate from the zero eigenvalueThey are asymptotically given by

120582 = radic120598(2radic120596 minus 1 + 1205742(2radic120596 minus 1) + sdot sdot sdot) + O (120598)

120582 =

(2radic120596 minus 2 minus 1205742 120596 minus 42radic120596 minus 2 + sdot sdot sdot ) + 120598 (radic120596 minus 2 minus 1205742120596

4radic120596 minus 2 + sdot sdot sdot ) + O (120598)32 (2radic120596 minus 2 minus 1205742 120596 minus 42radic120596 minus 2 + sdot sdot sdot ) + 120598(

1radic120596 minus 2 minus 1205742120596

4 (120596 minus 2)32 + sdot sdot sdot ) + O (120598)32

(18)


R(

)

005

01

015

02

025

03

02 04 06 08 10

(a)

1

2

3

4

5

6

7

I()

02 04 06 08 10

1416182

01 02 030

(b)

R(

)

005

01

015

02

025

03

035

04

045

05

02 04 06 08 10

(c)

0 01501005

02 04 06 08 10120598

1

2

3

4

5

6

7

I(120582)

12

14

16

(d)

Figure 1 Eigenvalues of intersite soliton I with 120596 = 12 120574 = 0 (a b) and 05 (c d) Dots are from the numerics and solid lines are theasymptotic approximations in Section 421 The collection of dots forming black regions in the right column corresponds to the continuousspectrum

R(

)

1

2

3

4

5

6

02 04 06 08 10

(a)

R(

)

02 04 06 08 10

05

1

15

2

25

3

35

4

45

5

(b)

Figure 2 The same as in Figure 1 with 120574 = 0 (a) and 120574 = 05 (b) but for 120596 = 52 In this case all the eigenvalues are real


minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(a)

R(

)

051

152

253

354

455

02 04 06 08 10

(b)

485

002 004 0060

2

4

6

8

10

I(120582)

02 04 06 08 10120598

(c)

minus15

minus10

minus5

0

5

10

15

I()

minus4 minus2 0 2 4 6minus6R()

(d)

R(

)

1

2

3

4

5

6

02 04 06 08 10

(e)

2

4

6

8

10

12

14

I(120582)

02 04 06 08 10120598

(f)

Figure 3 The spectra of intersite soliton II with 120596 = 52 (a b c) and 82 (d e f) and 120574 = 05 The most left panels are the spectra in thecomplex plane for 120598 = 1 (b c e f) The eigenvalues as a function the coupling constant Solid blue curves are the asymptotic approximations

422 Intersite Soliton II The intersite soliton II that is theintersite soliton that is antisymmetric between the arms hasthree pairs of eigenvalues given by

120582 = radic120598(2radic120596 + 1 minus 1205742(2radic120596 + 1) + sdot sdot sdot) + O (120598)

120582 =

119894(2radic120596 + 2 minus 1205742 120596 + 42radic120596 + 2 + sdot sdot sdot ) minus 119894120598 (radic120596 + 2 + 120574231205964 + 351205963 + 1361205962 + 208120596 + 1088radic120596 + 2 (1205963 + 61205962 + 12120596 + 8) + sdot sdot sdot ) + O (120598)32

119894 (2radic120596 + 2 minus 1205742 120596 + 42radic120596 + 2 + sdot sdot sdot ) + 119894120598 (1radic120596 + 2 + 1205742

1205964 + 211205963 + 1041205962 + 184120596 + 1088radic120596 + 2 (1205963 + 61205962 + 12120596 + 8) + sdot sdot sdot ) + O (120598)32

(19)

423 Onsite Soliton I The onsite soliton has only oneeigenvalue for small 120598 given asymptotically by

120582 = (2radic120596 minus 2 minus 1205742 120596 minus 42radic120596 minus 2 + sdot sdot sdot)+ 120598( 2radic120596 minus 2 minus 1205742

1205962 (120596 minus 2)32 + sdot sdot sdot) + sdot sdot sdot

(20)

43 Onsite Soliton II As for the second type of the onsitesoliton we have

120582 = 119894 (2radic120596 + 2 minus 1205742 120596 + 4radic120596 + 2 + sdot sdot sdot)+ 2119894120598 ( 1radic120596 + 2 minus 1205742

120596(120596 + 2)32 + sdot sdot sdot) + sdot sdot sdot

(21)


minus4

minus3

minus2

minus1

0

1

2

3

4I()

minus005 0 005 01minus01R()

(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)

Figure 4 (a c) The spectrum of onsite soliton I in the complex plane for 120598 = 02 (b d) The eigenvalue as a function of the coupling and itsapproximation from Section 423 (andashd) are for 120596 = 12 and 52 respectively Here 120574 = 01

5 Numerical Results

We have solved the steady-state equation (3) numericallyusing a Newton-Raphson method and analysed the stabilityof the numerical solution by solving the eigenvalue prob-lem (12) Here we will compare the analytical calculationsobtained above with the numerical results

First we consider the discrete intersite soliton I We showin Figure 1 the spectrum of the soliton as a function of thecoupling constant 120598 for 120596 = 12 and 120574 = 0 05 On thereal axis one can observe that there is only one unstableeigenvalue that bifurcates from the origin As the couplingincreases the bifurcating eigenvalue enters the origin againwhen 120598 rarr infin Hence in that limit we obtain a stable soliton I(ie a stable symmetric soliton)Thedynamics of the nonzeroeigenvalues as a function of the coupling constant is shownin the right panels of the figure where one can see that theeigenvalues are on the imaginary axis and simply enter thecontinuous spectrum as 120598 increases

In Figure 2 we plot the eigenvalues for 120596 large enoughHere in the uncoupled limit all the three pairs of eigenvaluesare on the real axis As the coupling increases two pairs goback toward the origin while one pair remains on the realaxis (not shown here) In the continuum limit 120598 rarr infinwe therefore obtain an unstable soliton I (ie an unstablesymmetric soliton)

In both figures we also plot the approximate eigenvaluesin solid (blue) curves where good agreement is obtained forsmall 120598

From numerical computations we conjecture that if inthe limit 120598 rarr 0 all the nonzero eigenvalues 120582 satisfy 1205822 gt 12058221minus(see (15)) then we will obtain unstable soliton I in the contin-uum limit 120598 rarr infin However when in the anticontinuum limit120598 rarr 0 all the nonzero eigenvalues 120582 satisfy 12058221+ lt 1205822 lt 12058222minuswe may either obtain a stable or an unstable soliton I in thecontinuum limit


minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)

Figure 5 The same as in Figure 4 but for onsite soliton II with 120596 = 52 (a b) and 82 (c d) (a c) are with 120598 = 1Next we consider intersite solitons II (ie antisymmetric

intersite solitons) Shown in Figure 3 is the spectrum of thediscrete solitons for two values of 120596 In both cases there isan eigenvalue bifurcating from the origin For the smallervalue of 120596 (Figures 3(a)ndash3(c)) we have the condition thatall the nonzero eigenvalues 120582 satisfy 1205822 lt 12058222minus in the anti-continuum limit 120598 rarr 0The collision between the eigenvaluesand the continuous spectrum as the coupling increasescreates complex eigenvalues In the second case using larger120596 (Figures 3(d)ndash3(f)) the nonzero eigenvalues 120582 satisfy 1205822 gt12058221minus when 120598 = 0 Even though not seen in the figure thecollision between one of the nonzero eigenvalues and the con-tinuous spectrum also creates a pair of complex eigenvaluesAdditionally in the continuum limit both values of 120596 as wellas the other values of the parameter that we computed for thistype of discrete solitons yield unstable solutions

We also study onsite solitons Shown in Figures 4 and 5 isthe stability of discrete solitons types I and II respectively

Figure 4(a) shows that for (120596minusradic1 minus 1205742) small enough wewill obtain stable discrete solitons For coupling constant 120598

small we indeed show it through our analysis depicted as theblue solid line Numerically we obtain that this soliton is alsostable in the continuum limit 120598 rarr infin However when 120596 islarge enough compared to radic1 minus 1205742 even though initially inthe uncoupled limit the nonzero eigenvalue 120582 satisfies 1205822 lt12058222minus onemay obtain an exponential instability (ie instabilitydue to a real eigenvalue) Figure 4(c) shows the case when thediscrete soliton is already unstable even in the uncoupledlimit due to the nonzero eigenvalue that is already real-valued

Figure 5 shows that the antisymmetric solitons aregenerally unstable due to a quartet of complex eigenvaluesas shown in Figures 5(a) and 5(c) When the coupling isincreased further therewill be an eigenvalue bifurcating fromplusmn1205821minus that will move towards the origin and later becomes apair of real eigenvaluesThese solitons are also unstable in thecontinuum limit

Unlike intersite discrete solitons that are always unstableonsite discrete solitons may be stable In Figure 6 we presentthe (in)stability region of the two types of discrete solitons inthe (120598 120596)-plane for three values of the gain-loss parameter 120574


135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)

Figure 6 The stability region of the onsite soliton type I (a) and II (b) in the (120598 120596)-plane for several values of 120574 The solutions are unstableabove the curves in panel (a) and between the curves in panel (b)

Discrete solitons are unstable above the curves in panel (a)and between the curves in panel (b) Indeed as wementionedbefore for soliton I there is a critical 120596 that depends on 120574below which the soliton is stable in the continuum limitwhile soliton II is always unstable in that limit Anotherdifference between the two figures is that the stability curvesin Figure 6(a) generally correspond to an eigenvalue crossingthe origin that becomes real-valued while the curves in theother panel are due to the appearance of a quartet of complexeigenvalues In general we obtain that the gain-loss term canbe parasitic as it reduces the stability region of the discretesolitons

Finally we present in Figure 7 the time dynamics of someof the unstable solutions shown in the previous figures Whatwe obtain is that typically there are two kinds of dynamicsthat is in the form of travelling discrete solitons or solutionblow-ups The first type was the typical dynamics of theintersite soliton IThe second dynamics is typical for the othertypes of unstable discrete solitons

6 Conclusion

We have presented a systematic method to determine thestability of discrete solitons in a PT-symmetric couplerby computing the eigenvalues of the corresponding lineareigenvalue problem using asymptotic expansions We havecompared the analytical results that we obtained with numer-ical computations where good agreement is obtained Fromthe numerics we have also established the mechanism ofinstability as well as the stability region of the discretesolitons The application of the method in higher dimen-sional PT-symmetric couplers (see eg [32]) is a natural

extension of the problem that is addressed for future workAdditionally we also address the computation of eigenvaluesof discrete solitons in the neighbourhood of broken PT-symmetry as future investigations

Appendix

Analytical Calculation

As mentioned in Section 42 to solve the eigenvalue problem(12) analytically we expand the eigenvalue and eigenvectorasymptotically as

◻ = ◻(0) + radic120598◻(1) + 120598◻(2) + sdot sdot sdot (A1)

with ◻ = 120582119870119899 119871119899 119875119899 119876119899Performing the expansion in 120598 at O(1205980) we obtain the

following set of equations

120582(0)V(0)119899 =[[[[[[[[[[

120574 120596 minus 119860(0)119899 2 0 minus13 (119860(0)119899 )2 minus 120596 120574 1 0

0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)

Figure 7 The typical dynamics of the instability of the discrete solitons in the previous figures Here 120574 = 05 and 120598 = 1 Depicted in (andashh)are |119906119899|2 and |V119899|2 respectively (andashh) The dynamics of intersite soliton I with 120596 = 12 intersite soliton II with 120596 = 52 and onsite solitons Iand II both with 120596 = 52


where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]

1198961 = minus (2R (119861(0)119899 )I (119861(0)119899 ) + 120574) 1198962 = 120596 minusR (119861(0)119899 )2 minus 3I (119861(0)119899 )2 1198963 = 3R (119861(0)119899 )2 +I (119861(0)119899 )2 minus 120596

(A3)

At O(12059812) and O(1205981) we obtain120582(0)V(1)119899 = 1198720V(2)119899 minus 120582(1)V(0)119899 (A4)

120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where

1198964 = minus2 (R (119861(0)119899 )I (119861(1)119899 ) +R (119861(1)119899 )I (119861(0)119899 )) 1198965 = minus2 (R (119861(0)119899 )R (119861(1)119899 ) + 3I (119861(0)119899 )I (119861(1)119899 )) 1198966 = 2 (3R (119861(0)119899 )R (119861(1)119899 ) +I (119861(0)119899 )I (119861(1)119899 ))

(A6)

The steps of finding the coefficients 120582(119895) of the asymptoticexpansions 119895 = 0 1 2 are as follows

(i) Solve the eigenvalue problem (A2) which is a 4 times 4system of equations for 120582(0) and V(0)119899

(ii) Determine 120582(1) by taking the vector inner productof both sides of (A4) with the null-space of theHermitian transpose of the block matrix that consistsof (1198720minus120582(0)1198684) along the diagonal where 1198684 is the 4times4identity matrix

(iii) Solve (A4) for V(1)119899

(iv) Determine 120582(2) by taking the vector inner productof both sides of (A5) with the null-space of theHermitian transpose of the block matrix that consistsof (1198720 minus 120582(0)1198684)

Theprocedure repeats if onewould like to calculate the higherorder terms

The leading order eigenvalue 120582(0) of (A2) has been solvedin [31] However the expression of the corresponding eigen-vector V(0)119899 was very lengthy that makes it almost impracticalto be used to determine the higher order corrections of 120582(119895)Therefore in every equation at order O(120598ℓ) obtained from(12) we also expand the variables in 120574 that is

119872119895 = 1198721198950 + 1205741198721198951 + 12057421198721198952 = sdot sdot sdot ◻(119895) = ◻(1198950) + 120574◻(1198951) + 1205742◻(1198952) + sdot sdot sdot (A7)

where again ◻ = 120582 V119899 and obtain equations at orderO(120574ℓ) The steps to determine 120582(119895119896) and V(119895119896)119899 are the same asmentioned above

Competing Interests

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

Acknowledgments

Alhaji A Bachtiar and Hadi Susanto are grateful to theUniversity of Nottingham for the 2013 Visiting FellowshipScheme and British Council for the 2015 Indonesia SecondCity Partnership Travel Grant

References

[1] N Moiseyev Non-Hermitian Quantum Mechanics CambridgeUniversity Press Cambridge UK 2011

[2] C M Bender and S Boettcher ldquoReal spectra in non-hermitianhamiltonians having PT symmetryrdquo Physical Review Lettersvol 80 no 24 p 5243 1998

[3] C M Bender S Boettcher and P N Meisinger ldquoPT-symmetricquantum mechanicsrdquo Journal of Mathematical Physics vol 40no 5 pp 2201ndash2229 1999

[4] C M Bender ldquoMaking sense of non-Hermitian HamiltoniansrdquoReports on Progress in Physics vol 70 no 6 p 947 2007

[5] A Guo G J Salamo D Duchesne et al ldquoObservation ofPT-symmetry breaking in complex optical potentialsrdquo PhysicalReview Letters vol 103 no 9 Article ID 093902 2009

[6] C E Ruter K GMakris R El-Ganainy D N ChristodoulidesM Segev and D Kip ldquoObservation of parity-time symmetry inopticsrdquo Nature Physics vol 6 no 3 pp 192ndash195 2010

[7] S V Suchkov A A Sukhorukov J Huang S V Dmitriev CLee and Yu S Kivshar ldquoNonlinear switching and solitons inPT-symmetric photonic systemsrdquo Laser amp Photonics Reviewsvol 10 no 2 pp 177ndash213 2016

[8] V V Konotop J Yang and D A Zezyulin ldquoNonlinear wavesinPT-symmetric systemsrdquo Reviews of Modern Physics vol 88no 3 Article ID 035002 2016

[9] J Pickton and H Susanto ldquoIntegrability of PT-symmetricdimersrdquo Physical Review A vol 88 no 6 Article ID 0638402013

[10] P G Kevrekidis D E Pelinovsky and D Y Tyugin ldquoNonlineardynamics in PT-symmetric latticesrdquo Journal of Physics AMathematical andTheoretical vol 46 no 36 Article ID 3652012013


[11] I V Barashenkov G S Jackson and S Flach ldquoBlow-up regimesin thePT -symmetric coupler and the actively coupled dimerrdquoPhysical Review A vol 88 no 5 Article ID 053817 2013

[12] S V Suchkov B A Malomed S V Dmitriev and Y S KivsharldquoSolitons in a chain of parity-time-invariant dimersrdquo PhysicalReview E vol 84 no 4 Article ID 046609 2011

[13] P L Chu B A Malomed and G D Peng ldquoSoliton switchingand propagation in nonlinear fiber couplers analytical resultsrdquoJournal of the Optical Society of America B vol 10 no 8 p 13791993

[14] E M Wright G I Stegeman and S Wabnitz ldquoSolitary-wavedecay and symmetry-breaking instabilities in two-mode fibersrdquoPhysical Review A vol 40 no 8 pp 4455ndash4466 1989

[15] N Akhmediev and J M Soto-Crespo ldquoPropagation dynamicsof ultrashort pulses in nonlinear fiber couplersrdquo Physical ReviewE vol 49 no 5 pp 4519ndash4529 1994

[16] B A Malomed I M Skinner P L Chu and G D PengldquoSymmetric and asymmetric solitons in twin-core nonlinearoptical fibersrdquo Physical Review E vol 53 no 4 pp 4084ndash40911996

[17] R Driben and B A Malomed ldquoStability of solitons in parity-time-symmetric couplersrdquo Optics Letters vol 36 no 22 pp4323ndash4325 2011

[18] F K Abdullaev V V Konotop M Ogren and M P SoslashrensenldquoZeno effect and switching of solitons in nonlinear couplersrdquoOptics Letters vol 36 no 23 pp 4566ndash4568 2011

[19] R Driben and B A Malomed ldquoStabilization of solitons in PTmodels with supersymmetry by periodic managementrdquo EPL(Europhysics Letters) vol 96 no 5 Article ID 51001 2011

[20] N V Alexeeva I V Barashenkov A A Sukhorukov andY S Kivshar ldquoOptical solitons in PT-symmetric nonlinearcouplers with gain and lossrdquo Physical Review A vol 85 no 6Article ID 063837 2012

[21] I V Barashenkov S V Suchkov A A Sukhorukov S VDmitriev and Y S Kivshar ldquoBreathers in PT-symmetricoptical couplersrdquo Physical Review A vol 86 no 5 Article ID053809 2012

[22] P Li L Li and B A Malomed ldquoMultisoliton Newtonrsquos cradlesand supersolitons in regular and parity-time-symmetric non-linear couplersrdquo Physical Review E vol 89 Article ID 0629262014

[23] J DrsquoAmbroise P G Kevrekidis and B A Malomed ldquoStaggeredparity-time-symmetric ladders with cubic nonlinearityrdquo Physi-cal Review E vol 91 no 3 Article ID 033207 11 pages 2015

[24] G Herring P G Kevrekidis B A Malomed R Carretero-Gonzalez and D J Frantzeskakis ldquoSymmetry breaking in lin-early coupled dynamical latticesrdquo Physical Review E StatisticalNonlinear and Soft Matter Physics vol 76 no 6 Article ID066606 2007

[25] A Chernyavsky and D Pelinovsky ldquoBreathers in hamiltonianPT -symmetric chains of coupled pendula under a resonantperiodic forcerdquo Symmetry vol 8 no 7 p 59 2016

[26] C M Bender B K Berntson D Parker and E SamuelldquoObservation of PT phase transition in a simple mechanicalsystemrdquo American Journal of Physics vol 81 no 3 pp 173ndash1792013

[27] A Chernyavsky and D E Pelinovsky ldquoLong-time stabil-ity of breathers in Hamiltonian PT-symmetric latticesrdquohttpsarxivorgabs160602333

[28] R S MacKay and S Aubry ldquoProof of existence of breathersfor time-reversible or Hamiltonian networks of weakly coupledoscillatorsrdquo Nonlinearity vol 7 no 6 pp 1623ndash1643 1994

[29] V V Konotop D E Pelinovsky and D A Zezyulin ldquoDiscretesolitons in PT-symmetric latticesrdquo EPL vol 100 no 5 ArticleID 56006 2012

[30] D E Pelinovsky D A Zezyulin and V V Konotop ldquoNonlinearmodes in a generalized PT-symmetric discrete nonlinearSchrodinger equationrdquo Journal of Physics A Mathematical andTheoretical vol 47 no 8 Article ID 085204 2014

[31] K Li and P G Kevrekidis ldquoPT-symmetric oligomers ana-lytical solutions linear stability and nonlinear dynamics-symmetric oligomers analytical solutions linear stability andnonlinear dynamicsrdquo Physical Review E vol 83 no 6 ArticleID 066608 2011

[32] Z Chen J Liu S Fu Y Li and B A Malomed ldquoDiscretesolitons and vortices on two-dimensional lattices of PTmdashsymmetric couplersrdquo Optics Express vol 22 no 24 pp 29679ndash29692 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


become stable after a tangent (saddle-center) bifurcationWhen one adds a gain and loss term in each arm one obtainsPT-symmetric couplers which have been considered in[17ndash22] In the presence of the linear-gain and loss termsasymmetric solitons cease to exist while antisymmetric soli-tons are always unstable [20] even though those with smallamplitudes can live long due to weak underlying instability[17] Symmetric solitons can be stable in a similar fashion tothose in the system without gain-loss [20]

The stability of bright discrete solitons inPT-symmetriccouplers was discussed in [12] using variational methodswhere it was shown that symmetric onsite solutions can bestable and there is a critical solution amplitude above whichthe PT symmetry is broken The case when the polarity ofthe PT-symmetric dimers is staggered along the chain isconsidered in [23]The same equations without gain and losswere considered in [24] where the symmetric soliton losesits stability through the symmetry-breaking bifurcation at afinite value of the energy similarly to that in the continuouscounterpart [13ndash16] Recently a similarPT chain of dimerswith a slightly different nonlinearity was derived [25] todescribe coupled chains of parametrically driven pendula asa mechanical analogue of PT-symmetric systems [26] Thestability of bright discrete solitons was established throughthe applications of the Hamiltonian energy and an indextheorem The nonlinear long-time stability of the discretesolitons was also established using the Lyapunov method inthe asymptotic limit of a weak coupling between the pendula[27]

In this work we determine the eigenvalues of dis-crete solitons in PT-symmetric couplers analytically usingasymptotic expansions The computation is based on the so-called method of weak coupling or anticontinuum limit Theapplication of the method in the study of discrete solitonswas formulated rigorously in [28] for conservative systemsIt was then applied to PT-symmetric networks in [29 30]However no explicit expression of the asymptotic series ofthe eigenvalues for the stability of discrete solitons has beenpresented before Here in addition to the asymptotic limitof weak coupling between the dimers we also propose toconsider expansions in the coefficient of the gain-loss termsIn this case explicit computations of the asymptotic series ofthe eigenvalues become possible

The manuscript is outlined as follows In Section 2 wepresent the mathematical model In Section 3 we use per-turbation theory for small coupling to analyse the existenceof fundamental localised solutions Such analysis is based onthe concept of the so-called anticontinuum limit approachThe stability of the solitons is then considered analyticallyin Section 4 by solving a corresponding eigenvalue problemIn this section in addition to small coupling the expansionis also performed under the assumption of small coefficientof the gain-loss term due to the nonsimple expression of theeigenvectors of the linearised operatorThe findings obtainedfrom the analytical calculations are then compared with thenumerical counterparts in Section 5 We also produce stabil-ity regions for all the fundamental solitons numerically Inthis section we present the typical dynamics of solitons in theunstable parameter ranges by direct numerical integrations

of the governing equation We present the conclusion inSection 6

2 Mathematical Model

The governing equations describing PT-symmetric chainsof dimers are of the form [12]

119899 = 119894120590 100381610038161003816100381611990611989910038161003816100381610038162 119906119899 + 119894120598Δ 2119906119899 + 120574119906119899 + 119894V119899V119899 = 119894120590 1003816100381610038161003816V11989910038161003816100381610038162 V119899 + 119894120598Δ 2V119899 minus 120574V119899 + 119894119906119899

(1)

The derivative with respect to the evolution variable (ie thepropagation distance if we consider their application in fiberoptics) is denoted by the overdot 119906119899 = 119906119899(119905) and V119899 = V119899(119905)are complex-valued wave function at site 119899 isin Z 120598 gt 0is the constant coefficient of the horizontal linear coupling(coupling constant between two adjacent sites) Δ 2119906119899 =(119906119899+1 minus 2119906119899 + 119906119899minus1) and Δ 2V119899 = (V119899+1 minus 2V119899 + V119899minus1) are thediscrete Laplacian term in one spatial dimension the gain andloss acting on complex variables 119906119899 and V119899 are representedby the positive coefficient 120574 that is 120574 gt 0 The nonlinearitycoefficient is denoted by 120590 which can be scaled to +1withoutloss of generality due to the case of focusing nonlinearitythat we consider Bright discrete soliton solutions satisfy thelocalisation conditions 119906119899 V119899 rarr 0 as 119899 rarr plusmninfin

The focusing system has static localised solutions that canbe obtained from substituting

119906119899 = 119860119899119890119894120596119905V119899 = 119861119899119890119894120596119905 (2)

into (1) to yield the equations

120596119860119899 = 100381610038161003816100381611986011989910038161003816100381610038162 119860119899 + 120598 (119860119899+1 minus 2119860119899 + 119860119899minus1) minus 119894120574119860119899+ 119861119899

120596119861119899 = 100381610038161003816100381611986111989910038161003816100381610038162 119861119899 + 120598 (119861119899+1 minus 2119861119899 + 119861119899minus1) + 119894120574119861119899 + 119860119899(3)

where 119860119899 and 119861119899 are complex-valued and the propagationconstant 120596 isin R3 Solutions of Weakly Coupled Equations

In the uncoupled limit that is when 120598 = 0 chain (1)becomes the equations for the dimer The static equation (3)has been analysed in detail in [29 31] where it was shownthat there is a relation between 120596 and 120574 above which thereis no time-independent solution to (3) (see also the analysisbelow)When 120598 is nonzero but small enough the existence ofsolutions emanating from the uncoupled limit can be shownusing the Implicit Function Theorem The existence analysisof [25] can be adopted here despite the slightly differentnonlinearity as the Jacobian of our system when uncoupledshares a rather similar invertible structure (see also [29 30]that have the same nonlinearity in the governing equationsbut different small coupling terms) However below we will





(4)



minus Δ 2119860(0)119899 (6)



119860(0)119899 = 0119890119894120601119886 119899 = 0 10 otherwise

119861(0)119899 = 0119890119894120601119887 119899 = 0 10 otherwise

(7)

with [31]


(8)



1198991 = 1198871198991 =

1(20) 119899 = 0 111198860 119899 = minus1 20 otherwise

(9)



119860(0)119899 = 0119890119894120601119886 119899 = 00 otherwise

119861(0)119899 = 1198870119890119894120601119887 119899 = 00 otherwise

(10)










(12)


120582[[[[[[[

]]]]]]]= [[[[[[


]]]]]]

[[[[[[[

]]]]]]] (13)





(15)





◻(119895) = ◻(1198950) + 120574◻(1198951) + 1205742◻(1198952) + sdot sdot sdot (17)




120582 =





(18)


R(

)

005

01

015

02

025

03

02 04 06 08 10

(a)

1

2

3

4

5

6

7

I()

02 04 06 08 10

1416182

01 02 030

(b)

R(

)

005

01

015

02

025

03

035

04

045

05

02 04 06 08 10

(c)

0 01501005

02 04 06 08 10120598

1

2

3

4

5

6

7

I(120582)

12

14

16

(d)


R(

)

1

2

3

4

5

6

02 04 06 08 10

(a)

R(

)

02 04 06 08 10

05

1

15

2

25

3

35

4

45

5

(b)



minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(a)

R(

)

051

152

253

354

455

02 04 06 08 10

(b)

485

002 004 0060

2

4

6

8

10

I(120582)

02 04 06 08 10120598

(c)

minus15

minus10

minus5

0

5

10

15

I()


(d)

R(

)

1

2

3

4

5

6

02 04 06 08 10

(e)

2

4

6

8

10

12

14

I(120582)

02 04 06 08 10120598

(f)




120582 =




(19)




(20)




(21)


minus4

minus3

minus2

minus1

0

1

2

3

4I()


(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)


5 Numerical Results







minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(4)



minus Δ 2119860(0)119899 (6)



119860(0)119899 = 0119890119894120601119886 119899 = 0 10 otherwise

119861(0)119899 = 0119890119894120601119887 119899 = 0 10 otherwise

(7)

with [31]


(8)



1198991 = 1198871198991 =

1(20) 119899 = 0 111198860 119899 = minus1 20 otherwise

(9)



119860(0)119899 = 0119890119894120601119886 119899 = 00 otherwise

119861(0)119899 = 1198870119890119894120601119887 119899 = 00 otherwise

(10)










(12)


120582[[[[[[[

]]]]]]]= [[[[[[


]]]]]]

[[[[[[[

]]]]]]] (13)





(15)





◻(119895) = ◻(1198950) + 120574◻(1198951) + 1205742◻(1198952) + sdot sdot sdot (17)




120582 =





(18)


R(

)

005

01

015

02

025

03

02 04 06 08 10

(a)

1

2

3

4

5

6

7

I()

02 04 06 08 10

1416182

01 02 030

(b)

R(

)

005

01

015

02

025

03

035

04

045

05

02 04 06 08 10

(c)

0 01501005

02 04 06 08 10120598

1

2

3

4

5

6

7

I(120582)

12

14

16

(d)


R(

)

1

2

3

4

5

6

02 04 06 08 10

(a)

R(

)

02 04 06 08 10

05

1

15

2

25

3

35

4

45

5

(b)



minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(a)

R(

)

051

152

253

354

455

02 04 06 08 10

(b)

485

002 004 0060

2

4

6

8

10

I(120582)

02 04 06 08 10120598

(c)

minus15

minus10

minus5

0

5

10

15

I()


(d)

R(

)

1

2

3

4

5

6

02 04 06 08 10

(e)

2

4

6

8

10

12

14

I(120582)

02 04 06 08 10120598

(f)




120582 =




(19)




(20)




(21)


minus4

minus3

minus2

minus1

0

1

2

3

4I()


(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)


5 Numerical Results







minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(12)


120582[[[[[[[

]]]]]]]= [[[[[[


]]]]]]

[[[[[[[

]]]]]]] (13)





(15)





◻(119895) = ◻(1198950) + 120574◻(1198951) + 1205742◻(1198952) + sdot sdot sdot (17)




120582 =





(18)


R(

)

005

01

015

02

025

03

02 04 06 08 10

(a)

1

2

3

4

5

6

7

I()

02 04 06 08 10

1416182

01 02 030

(b)

R(

)

005

01

015

02

025

03

035

04

045

05

02 04 06 08 10

(c)

0 01501005

02 04 06 08 10120598

1

2

3

4

5

6

7

I(120582)

12

14

16

(d)


R(

)

1

2

3

4

5

6

02 04 06 08 10

(a)

R(

)

02 04 06 08 10

05

1

15

2

25

3

35

4

45

5

(b)



minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(a)

R(

)

051

152

253

354

455

02 04 06 08 10

(b)

485

002 004 0060

2

4

6

8

10

I(120582)

02 04 06 08 10120598

(c)

minus15

minus10

minus5

0

5

10

15

I()


(d)

R(

)

1

2

3

4

5

6

02 04 06 08 10

(e)

2

4

6

8

10

12

14

I(120582)

02 04 06 08 10120598

(f)




120582 =




(19)




(20)




(21)


minus4

minus3

minus2

minus1

0

1

2

3

4I()


(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)


5 Numerical Results







minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R(

)

005

01

015

02

025

03

02 04 06 08 10

(a)

1

2

3

4

5

6

7

I()

02 04 06 08 10

1416182

01 02 030

(b)

R(

)

005

01

015

02

025

03

035

04

045

05

02 04 06 08 10

(c)

0 01501005

02 04 06 08 10120598

1

2

3

4

5

6

7

I(120582)

12

14

16

(d)


R(

)

1

2

3

4

5

6

02 04 06 08 10

(a)

R(

)

02 04 06 08 10

05

1

15

2

25

3

35

4

45

5

(b)



minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(a)

R(

)

051

152

253

354

455

02 04 06 08 10

(b)

485

002 004 0060

2

4

6

8

10

I(120582)

02 04 06 08 10120598

(c)

minus15

minus10

minus5

0

5

10

15

I()


(d)

R(

)

1

2

3

4

5

6

02 04 06 08 10

(e)

2

4

6

8

10

12

14

I(120582)

02 04 06 08 10120598

(f)




120582 =




(19)




(20)




(21)


minus4

minus3

minus2

minus1

0

1

2

3

4I()


(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)


5 Numerical Results







minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(a)

R(

)

051

152

253

354

455

02 04 06 08 10

(b)

485

002 004 0060

2

4

6

8

10

I(120582)

02 04 06 08 10120598

(c)

minus15

minus10

minus5

0

5

10

15

I()


(d)

R(

)

1

2

3

4

5

6

02 04 06 08 10

(e)

2

4

6

8

10

12

14

I(120582)

02 04 06 08 10120598

(f)




120582 =




(19)




(20)




(21)


minus4

minus3

minus2

minus1

0

1

2

3

4I()


(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)


5 Numerical Results







minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus4

minus3

minus2

minus1

0

1

2

3

4I()


(a)

1

2

3

4

5

6

7

I(120582)

02 04 06 08 10120598

121416

005 01 015 020

(b)

minus15

minus10

minus5

0

5

10

15

I()

0 5minus5R()

(c)

R(

)

35

4

45

5

02 04 06 08 10

(d)


5 Numerical Results







minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus15

minus10

minus5

0

5

10

15I()


(a)

4

6

8

10

12

I(120582)

02 04 06 080120598

53545556

005 01 015 02 0250

(b)

minus15

minus10

minus5

0

5

10

15

I()


(c)

02 04 06 080120598

6

8

10

12

14

16

18

I(120582)

6466687

02 04 060

(d)









135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










135

14

145

15

155

16

165

05 1 150

= 0

= 01

= 05

(a)

= 0

= 01

= 05

0

1

2

3

4

5

6

7

8

9

02 04 06 08 10

(b)




6 Conclusion



Appendix






120582(0)V(0)119899 =[[[[[[[[[[


0 minus1 1198961 11989621 0 1198963 minus1198961

]]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198720

V(0)119899 (A2)


500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(a)

500

1000

1500

2000

t

0 10 20minus10n

01

02

03

04

05

06

07

(b)

1

2

3

4

5

6

t

0 5minus5n

20

40

60

80

100

120

140

160

(c)

1

2

3

4

5

6

t

0 5minus5n

2

4

6

8

10

(d)

1

2

3

4

5

t

0 5minus5n

20406080100120140160180

(e)

1

2

3

4

5

t

0 5minus5n

1

2

3

4

5

6

7

(f)

2

4

6

8

10

12

t

0 5minus5n

20406080100120140160180

(g)

2

4

6

8

10

12

t

0 5minus5n

123456789

(h)



where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

V(119895)119899 =[[[[[[[

119870(119895)119899119871(119895)119899119875(119895)119899119876(119895)119899

]]]]]]]


(A3)


120582(0)1198720 = 1198720V(2)119899 minus 120582(1)V(1)119899 minus 120582(2)V(0)119899

+[[[[[[[

0 2 (1 minus 119860(0)119899 119860(1)119899 ) 0 06119860(0)119899 119860(1)119899 minus 2 0 0 0

0 0 1198964 2 + 11989650 0 1198966 minus 2 minus1198964

]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198721

V(0)119899

+ [[[[[[

0 minus1 0 01 0 0 00 0 0 minus10 0 1 0

]]]]]](V(1)119899+1 + V(1)119899minus1)

(A5)

where


(A6)










Competing Interests


Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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