Research Article A Numerical Solution for Hirota-Satsuma …downloads.hindawi.com/journals/aaa/2014/819367.pdf · A Numerical Solution for Hirota-Satsuma Coupled KdV Equation M.S.IsmailandH.A.Ashi
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Research ArticleA Numerical Solution for Hirota-SatsumaCoupled KdV Equation
M S Ismail and H A Ashi
Department of Mathematics College of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia
Correspondence should be addressed to M S Ismail msismailkauedusa
Received 6 February 2014 Accepted 16 July 2014 Published 17 August 2014
Academic Editor Fuding Xie
Copyright copy 2014 M S Ismail and H A Ashi 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
A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupledKorteweg-de Vries equation using cubic 119861-splines as test functions and a linear 119861-spline as trial functions The implicit midpointrule is used to advance the solution in time Newtonrsquos method is used to solve the block nonlinear pentadiagonal system we haveobtainedThe resulting schemes are of second order accuracy in both directions space and timeThe vonNeumann stability analysisof the schemes shows that the two schemes are unconditionally stable The single soliton solution and the conserved quantities areused to assess the accuracy and to show the robustness of the schemes The interaction of two solitons three solitons and birth ofsolitons is also discussed
1 Introduction
In 1981 Hirota and Satsuma [1] introduced the coupledKorteweg-de Vries equation (CKdV) as follows
120597119906
120597119905= 119886(
1205973119906
1205971199093+ 6119906
120597119906
120597119909) + 2119887V
120597V120597119909
120597V120597119905
= minus1205973V1205971199093
minus 3119906120597V120597119909
(1)
where 119886 and 119887 are arbitrary constants The CKdV equationdescribes interactions of two longwaveswith different disper-sion relations Namely it is connected withmost types of longwaves with weak dispersion internal acoustic and planetarywaves in geophysical hydrodynamics [2 3] By using Hirotamethod [1 4] the single solitary wave solution of this systemis
119906 (119909 119905) = 21205822sech2 (120585) V (119909 119905) =
1
2radic119908sech (120585)
120585 = 120582 (119909 minus 1205822119905) +
1
2 log (119908) 119908 =
minus119887
8 (4119886 + 1) 1205824
(2)
where 120582 is an arbitrary constant The two and three solitonssolutions for 119886 = 12 can be found in [1] The CKdV systemhas three conserved quantities
1198681= int
infin
minusinfin
119906119889119909
1198682= int
infin
minusinfin
(1199062+2
3119887V2)119889119909
1198683= int
infin
minusinfin
[(1 + 119886) (1199063minus1
21199062
119909) + 119887 (119906V2 minus V2
119909)] 119889119909
(3)
The proof can be found in [1 5 6]The CKdV equation has been discussed analytically by
many authors Kaya and Inan [7] used Adomian decompo-sition method to solve this system Fan used tanh methodto find some traveling wave solution [8] Assas [9] solvedthis system using variational iteration method Abbasbandy[10] used homotopy analysis method to solve the generalizedCKdV system
The numerical solutions of coupled nonlinear systemsare very interesting and important in applied science forexample the coupled nonlinear Schrodinger equation whichadmits soliton solution and it has many applications in
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 819367 9 pageshttpdxdoiorg1011552014819367
2 Abstract and Applied Analysis
communication and optical fibers this system has beensolved numerically by Ismail using finite difference and finiteelementmethods [11ndash13]TheCKdVhas been also considerednumerically by some researchers Halim et al [2 3] haveintroduced a numerical scheme for general CKdV systemsThe scheme is valid for solving an arbitrary number ofequations with arbitrary constant coefficients the methodis applied to the Hirota system and compared with itsknown explicit solution investigating the influence of initialconditions and grid sizes on accuracyWazwaz [14] produceda finite difference scheme for the periodic initial boundaryproblem of the CKdV system Ismail [6] solved this systemusing collocation method and quintic splines Kutluay andUcar [15] solved this system using a quadratic B-splineGalerkin approach
In this work we are going to solve the CKdV equationusing Petrov-Galerkin method [4] We choose the cubic119861-spline as test functions and the linear 119861-spline as trial(basis) functions Implicit midpoint rule is used in the timedirection Newtonrsquos method is used to solve the nonlinearblock pentadigonal system obtained from the schemes wehave derived The von Neumann stability analysis shows thatthe scheme is unconditionally stable Regarding the accuracythe scheme is of second order in space and time
The paper is organized as follows In Section 2 Petrov-Galerkin method is used to derive a numerical method fortheCKdVequation a coupled nonlinear pentadigonal systemis obtained Analysis of the method is given in Section 3 InSection 4 product approximation technique is used to derivea second method for solving the CKdV equation Numericalresults of various tests are contained in Section 5 We recapand sum up our conclusions in Section 6
2 Numerical Method
To derive numerical method for the CKdV system weconsider the initial boundary value problem
120597119906
120597119905minus 119886(
1205973119906
1205971199093+ 6119906
120597119906
120597119909) minus 2119887
120597V120597119909
= 0
(119909 119905) isin [119909119871 119909119877] times (0 119879]
120597V120597119905
+1205973V1205971199093
+ 3119906120597V120597119909
= 0 (119909 119905) isin [119909119871 119909119877] times (0 119879]
(4)
subject to the initial conditions119906 (119909 0) = 119892 (119909) V (119909 0) = 0 (5)
and the boundary conditions119906 (119909119871 119905) = 119906 (119909
119877 119905) = 0
119906119909(119909119871 119905) = 119906
119909(119909119877 119905) = 0
(6)
A standard weak formulation is obtained by multiplying (4)by a twice differentiable test function120595(119909) and integrating byparts to obtain
(119906119905 120595) minus 6119886 (119906119906
119909 120595) minus 119886 (119906
119909 120595119909119909) minus 2119887 (VV
119909 120595) = 0
(V119905 120595) + 3 (119906V
119909 120595) + (V
119909 120595119909119909) = 0
(7)
where ( ) denotes the usual 1198712inner product
(119891 119892) = int
119909119877
119909119871
119891 (119909) 119892 (119909) 119889119909 (8)
The space interval [119909119871 119909119877] is discretized by uniform (119873+
The system (17)-(18) is a nonlinear block penta diagonalsystem in the unknown vectors U119899+1 and V119899+1 and is solvedusing Newtonrsquos methodWe denote this method by Scheme 1
3 Analysis of the Method
In this section we will discuss the stability and the accuracyof Scheme 1
31 Stability of the Scheme To study the stability of Scheme 1von Neumann stability analysis will be used and this can beonly applied for linear finite difference schemes accordinglywe consider the linear version of the proposed method
(119880119899+1
119898minus2+ 26119880
119899+1
119898minus1+ 66119880
119899+1
119898+ 26119880
119899+1
119898+1+ 119880119899+1
119898+2)
minus (119880119899
119898minus2+ 26119880
119899
119898minus1+ 66119880
119899
119898+ 26119880
119899
119898+1+ 119880119899
119898+2)
+ 1199011(119880lowast
119898+2minus 2119880lowast
119898+1+ 2119880lowast
119898minus1minus 119880lowast
119898minus2)
+ 1199012119865 (119880119880
lowast) + 1199013119865 (119881119881
lowast) = 0
(119881119899+1
119898minus2+ 26119881
119899+1
119898minus1+ 66119881
119899+1
119898+ 26119881
119899+1
119898+1+ 119881119899+1
119898+2)
minus (119881119899
119898minus2+ 26119881
119899
119898minus1+ 66119881
119899
119898+ 26119881
119899
119898+1+ 119881119899
119898+2)
+ 1199014(119881lowast
119898+2minus 2119881lowast
119898+1+ 2119881lowast
119898minus1minus 119881lowast
119898minus2)
+ 1199015119865 (119880119881
lowast) = 0
(20)
where
119865 (119880119881) = 119880 [119881119898+2
+ 10119881119898+1
minus 10119881119898minus1
minus 119881119898minus2
] (21)
1199011015840
2= 51199012 11990110158403= 51199013 and 1199011015840
5= 51199015 119880 and 119881 are assumed
to be constant on the whole range We assume the solution ofthe linearized scheme (20) to be of the form
119880119899
119898= 119880119899119890119894120573119898ℎ
119881119899
119898= 119881119899119890119894120573119898ℎ
(22)
where120573 is a real constant Direct substitution of (22) into (20)will lead us to the following system
The system (23) can be written in a matrix vector form as
Ψ119899+1
= 119861Ψ119899 (25)
4 Abstract and Applied Analysis
where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider
119861 = [1205741+ 1198941198881
1198882
0 1205741+ 1198941198883
]
minus1
[1205741minus 1198941198881
minus1198882
0 1205741minus 1198941198883
] (26)
where
1198881= 11990111205742+ 1199011015840
21198801205743 119888
2= 1199011015840
31198811205743
1198883= 11990141205742+ 1199011015840
51198801205743
(27)
von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are
1205821=1205741minus 1198941198881
1205741+ 1198941198881
1205822=1205741minus 1198941198883
1205741+ 1198941198883
(28)
which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense
32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899
119898and 119881119899
119898in
(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use
of the following Taylor series expansions at the point (119909119898 119905119899)
and hence the scheme is of second order in both directionsspace and time
4 Product Approximation Technique
A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform
120597119906
120597119905= 119886(
1205973119906
1205971199093+ 3
120597 (1199062)
120597119909) + 119887
120597 (V2)120597119909
(34)
120597V120597119905
= minus1205973V1205971199093
minus 3119906120597V120597119909
(35)
The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner
1199062(119909 119905) =
119873
sum
119898=1
1198802
119898(119905) 120601119898(119909)
V2 (119909 119905) =119873
sum
119898=1
1198812
119898(119905) 120601119898(119909)
(36)
By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme
Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2
5 Numerical Results
To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868
1 1198682 and 119868
3using
the trapezoidal rule The 119871infin(119906) and 119871
infin(V) error norms are
defined as
119871infin(119906) =
1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119880119899
119898minus 119906119899
119898
1003816100381610038161003816
119871infin (V) =
1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119881119899
119898minus V119899119898
1003816100381610038161003816
(40)
are used to examine the accuracy of the proposed schemes
51 Single Soliton In the first test we choose the initialconditions
119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =
1
2radic119908sech (120585)
120585 = 120582119909 +1
2 log (119908) 119908 =
minus119887
8 (4119886 + 1) 1205824
(41)
which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10
minus7 119909119871= minus25 119909
119877= 25 119886 = minus0125
119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871
infin(119880) 119871
infin(119881) are displayed in Tables 1 and 2
for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899
119898and 119881119899
119898for 119905 = 0 1 2 20
By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method
52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as
which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582
1= 10 120582
2= 06 119910
1= 10
and 1199102= 30 In Table 4 we present the conserved quantities
during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]
To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582
1= 10 120582
2= 06
Abstract and Applied Analysis 7
020
4060
80100
0
20
40
60
80
0123
minus1
x
t
u
Figure 3 Numerical solution 119880119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
020
4060
80100
010
2030
4050
6070
80
012
x
t
v
minus1
Figure 4 Numerical solution 119881119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
x
t
Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)
020
4060
80100
010
2030
4050
0
1
2
3
4
5
xt
u
minus1
Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =
0495
020
4060
80100
010
2030
4050
0
05
1
15
2
25
xt
v
minus05
Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
communication and optical fibers this system has beensolved numerically by Ismail using finite difference and finiteelementmethods [11ndash13]TheCKdVhas been also considerednumerically by some researchers Halim et al [2 3] haveintroduced a numerical scheme for general CKdV systemsThe scheme is valid for solving an arbitrary number ofequations with arbitrary constant coefficients the methodis applied to the Hirota system and compared with itsknown explicit solution investigating the influence of initialconditions and grid sizes on accuracyWazwaz [14] produceda finite difference scheme for the periodic initial boundaryproblem of the CKdV system Ismail [6] solved this systemusing collocation method and quintic splines Kutluay andUcar [15] solved this system using a quadratic B-splineGalerkin approach
In this work we are going to solve the CKdV equationusing Petrov-Galerkin method [4] We choose the cubic119861-spline as test functions and the linear 119861-spline as trial(basis) functions Implicit midpoint rule is used in the timedirection Newtonrsquos method is used to solve the nonlinearblock pentadigonal system obtained from the schemes wehave derived The von Neumann stability analysis shows thatthe scheme is unconditionally stable Regarding the accuracythe scheme is of second order in space and time
The paper is organized as follows In Section 2 Petrov-Galerkin method is used to derive a numerical method fortheCKdVequation a coupled nonlinear pentadigonal systemis obtained Analysis of the method is given in Section 3 InSection 4 product approximation technique is used to derivea second method for solving the CKdV equation Numericalresults of various tests are contained in Section 5 We recapand sum up our conclusions in Section 6
2 Numerical Method
To derive numerical method for the CKdV system weconsider the initial boundary value problem
120597119906
120597119905minus 119886(
1205973119906
1205971199093+ 6119906
120597119906
120597119909) minus 2119887
120597V120597119909
= 0
(119909 119905) isin [119909119871 119909119877] times (0 119879]
120597V120597119905
+1205973V1205971199093
+ 3119906120597V120597119909
= 0 (119909 119905) isin [119909119871 119909119877] times (0 119879]
(4)
subject to the initial conditions119906 (119909 0) = 119892 (119909) V (119909 0) = 0 (5)
and the boundary conditions119906 (119909119871 119905) = 119906 (119909
119877 119905) = 0
119906119909(119909119871 119905) = 119906
119909(119909119877 119905) = 0
(6)
A standard weak formulation is obtained by multiplying (4)by a twice differentiable test function120595(119909) and integrating byparts to obtain
(119906119905 120595) minus 6119886 (119906119906
119909 120595) minus 119886 (119906
119909 120595119909119909) minus 2119887 (VV
119909 120595) = 0
(V119905 120595) + 3 (119906V
119909 120595) + (V
119909 120595119909119909) = 0
(7)
where ( ) denotes the usual 1198712inner product
(119891 119892) = int
119909119877
119909119871
119891 (119909) 119892 (119909) 119889119909 (8)
The space interval [119909119871 119909119877] is discretized by uniform (119873+
The system (17)-(18) is a nonlinear block penta diagonalsystem in the unknown vectors U119899+1 and V119899+1 and is solvedusing Newtonrsquos methodWe denote this method by Scheme 1
3 Analysis of the Method
In this section we will discuss the stability and the accuracyof Scheme 1
31 Stability of the Scheme To study the stability of Scheme 1von Neumann stability analysis will be used and this can beonly applied for linear finite difference schemes accordinglywe consider the linear version of the proposed method
(119880119899+1
119898minus2+ 26119880
119899+1
119898minus1+ 66119880
119899+1
119898+ 26119880
119899+1
119898+1+ 119880119899+1
119898+2)
minus (119880119899
119898minus2+ 26119880
119899
119898minus1+ 66119880
119899
119898+ 26119880
119899
119898+1+ 119880119899
119898+2)
+ 1199011(119880lowast
119898+2minus 2119880lowast
119898+1+ 2119880lowast
119898minus1minus 119880lowast
119898minus2)
+ 1199012119865 (119880119880
lowast) + 1199013119865 (119881119881
lowast) = 0
(119881119899+1
119898minus2+ 26119881
119899+1
119898minus1+ 66119881
119899+1
119898+ 26119881
119899+1
119898+1+ 119881119899+1
119898+2)
minus (119881119899
119898minus2+ 26119881
119899
119898minus1+ 66119881
119899
119898+ 26119881
119899
119898+1+ 119881119899
119898+2)
+ 1199014(119881lowast
119898+2minus 2119881lowast
119898+1+ 2119881lowast
119898minus1minus 119881lowast
119898minus2)
+ 1199015119865 (119880119881
lowast) = 0
(20)
where
119865 (119880119881) = 119880 [119881119898+2
+ 10119881119898+1
minus 10119881119898minus1
minus 119881119898minus2
] (21)
1199011015840
2= 51199012 11990110158403= 51199013 and 1199011015840
5= 51199015 119880 and 119881 are assumed
to be constant on the whole range We assume the solution ofthe linearized scheme (20) to be of the form
119880119899
119898= 119880119899119890119894120573119898ℎ
119881119899
119898= 119881119899119890119894120573119898ℎ
(22)
where120573 is a real constant Direct substitution of (22) into (20)will lead us to the following system
The system (23) can be written in a matrix vector form as
Ψ119899+1
= 119861Ψ119899 (25)
4 Abstract and Applied Analysis
where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider
119861 = [1205741+ 1198941198881
1198882
0 1205741+ 1198941198883
]
minus1
[1205741minus 1198941198881
minus1198882
0 1205741minus 1198941198883
] (26)
where
1198881= 11990111205742+ 1199011015840
21198801205743 119888
2= 1199011015840
31198811205743
1198883= 11990141205742+ 1199011015840
51198801205743
(27)
von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are
1205821=1205741minus 1198941198881
1205741+ 1198941198881
1205822=1205741minus 1198941198883
1205741+ 1198941198883
(28)
which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense
32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899
119898and 119881119899
119898in
(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use
of the following Taylor series expansions at the point (119909119898 119905119899)
and hence the scheme is of second order in both directionsspace and time
4 Product Approximation Technique
A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform
120597119906
120597119905= 119886(
1205973119906
1205971199093+ 3
120597 (1199062)
120597119909) + 119887
120597 (V2)120597119909
(34)
120597V120597119905
= minus1205973V1205971199093
minus 3119906120597V120597119909
(35)
The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner
1199062(119909 119905) =
119873
sum
119898=1
1198802
119898(119905) 120601119898(119909)
V2 (119909 119905) =119873
sum
119898=1
1198812
119898(119905) 120601119898(119909)
(36)
By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme
Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2
5 Numerical Results
To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868
1 1198682 and 119868
3using
the trapezoidal rule The 119871infin(119906) and 119871
infin(V) error norms are
defined as
119871infin(119906) =
1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119880119899
119898minus 119906119899
119898
1003816100381610038161003816
119871infin (V) =
1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119881119899
119898minus V119899119898
1003816100381610038161003816
(40)
are used to examine the accuracy of the proposed schemes
51 Single Soliton In the first test we choose the initialconditions
119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =
1
2radic119908sech (120585)
120585 = 120582119909 +1
2 log (119908) 119908 =
minus119887
8 (4119886 + 1) 1205824
(41)
which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10
minus7 119909119871= minus25 119909
119877= 25 119886 = minus0125
119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871
infin(119880) 119871
infin(119881) are displayed in Tables 1 and 2
for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899
119898and 119881119899
119898for 119905 = 0 1 2 20
By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method
52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as
which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582
1= 10 120582
2= 06 119910
1= 10
and 1199102= 30 In Table 4 we present the conserved quantities
during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]
To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582
1= 10 120582
2= 06
Abstract and Applied Analysis 7
020
4060
80100
0
20
40
60
80
0123
minus1
x
t
u
Figure 3 Numerical solution 119880119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
020
4060
80100
010
2030
4050
6070
80
012
x
t
v
minus1
Figure 4 Numerical solution 119881119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
x
t
Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)
020
4060
80100
010
2030
4050
0
1
2
3
4
5
xt
u
minus1
Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =
0495
020
4060
80100
010
2030
4050
0
05
1
15
2
25
xt
v
minus05
Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
The system (17)-(18) is a nonlinear block penta diagonalsystem in the unknown vectors U119899+1 and V119899+1 and is solvedusing Newtonrsquos methodWe denote this method by Scheme 1
3 Analysis of the Method
In this section we will discuss the stability and the accuracyof Scheme 1
31 Stability of the Scheme To study the stability of Scheme 1von Neumann stability analysis will be used and this can beonly applied for linear finite difference schemes accordinglywe consider the linear version of the proposed method
(119880119899+1
119898minus2+ 26119880
119899+1
119898minus1+ 66119880
119899+1
119898+ 26119880
119899+1
119898+1+ 119880119899+1
119898+2)
minus (119880119899
119898minus2+ 26119880
119899
119898minus1+ 66119880
119899
119898+ 26119880
119899
119898+1+ 119880119899
119898+2)
+ 1199011(119880lowast
119898+2minus 2119880lowast
119898+1+ 2119880lowast
119898minus1minus 119880lowast
119898minus2)
+ 1199012119865 (119880119880
lowast) + 1199013119865 (119881119881
lowast) = 0
(119881119899+1
119898minus2+ 26119881
119899+1
119898minus1+ 66119881
119899+1
119898+ 26119881
119899+1
119898+1+ 119881119899+1
119898+2)
minus (119881119899
119898minus2+ 26119881
119899
119898minus1+ 66119881
119899
119898+ 26119881
119899
119898+1+ 119881119899
119898+2)
+ 1199014(119881lowast
119898+2minus 2119881lowast
119898+1+ 2119881lowast
119898minus1minus 119881lowast
119898minus2)
+ 1199015119865 (119880119881
lowast) = 0
(20)
where
119865 (119880119881) = 119880 [119881119898+2
+ 10119881119898+1
minus 10119881119898minus1
minus 119881119898minus2
] (21)
1199011015840
2= 51199012 11990110158403= 51199013 and 1199011015840
5= 51199015 119880 and 119881 are assumed
to be constant on the whole range We assume the solution ofthe linearized scheme (20) to be of the form
119880119899
119898= 119880119899119890119894120573119898ℎ
119881119899
119898= 119881119899119890119894120573119898ℎ
(22)
where120573 is a real constant Direct substitution of (22) into (20)will lead us to the following system
The system (23) can be written in a matrix vector form as
Ψ119899+1
= 119861Ψ119899 (25)
4 Abstract and Applied Analysis
where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider
119861 = [1205741+ 1198941198881
1198882
0 1205741+ 1198941198883
]
minus1
[1205741minus 1198941198881
minus1198882
0 1205741minus 1198941198883
] (26)
where
1198881= 11990111205742+ 1199011015840
21198801205743 119888
2= 1199011015840
31198811205743
1198883= 11990141205742+ 1199011015840
51198801205743
(27)
von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are
1205821=1205741minus 1198941198881
1205741+ 1198941198881
1205822=1205741minus 1198941198883
1205741+ 1198941198883
(28)
which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense
32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899
119898and 119881119899
119898in
(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use
of the following Taylor series expansions at the point (119909119898 119905119899)
and hence the scheme is of second order in both directionsspace and time
4 Product Approximation Technique
A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform
120597119906
120597119905= 119886(
1205973119906
1205971199093+ 3
120597 (1199062)
120597119909) + 119887
120597 (V2)120597119909
(34)
120597V120597119905
= minus1205973V1205971199093
minus 3119906120597V120597119909
(35)
The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner
1199062(119909 119905) =
119873
sum
119898=1
1198802
119898(119905) 120601119898(119909)
V2 (119909 119905) =119873
sum
119898=1
1198812
119898(119905) 120601119898(119909)
(36)
By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme
Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2
5 Numerical Results
To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868
1 1198682 and 119868
3using
the trapezoidal rule The 119871infin(119906) and 119871
infin(V) error norms are
defined as
119871infin(119906) =
1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119880119899
119898minus 119906119899
119898
1003816100381610038161003816
119871infin (V) =
1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119881119899
119898minus V119899119898
1003816100381610038161003816
(40)
are used to examine the accuracy of the proposed schemes
51 Single Soliton In the first test we choose the initialconditions
119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =
1
2radic119908sech (120585)
120585 = 120582119909 +1
2 log (119908) 119908 =
minus119887
8 (4119886 + 1) 1205824
(41)
which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10
minus7 119909119871= minus25 119909
119877= 25 119886 = minus0125
119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871
infin(119880) 119871
infin(119881) are displayed in Tables 1 and 2
for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899
119898and 119881119899
119898for 119905 = 0 1 2 20
By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method
52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as
which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582
1= 10 120582
2= 06 119910
1= 10
and 1199102= 30 In Table 4 we present the conserved quantities
during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]
To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582
1= 10 120582
2= 06
Abstract and Applied Analysis 7
020
4060
80100
0
20
40
60
80
0123
minus1
x
t
u
Figure 3 Numerical solution 119880119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
020
4060
80100
010
2030
4050
6070
80
012
x
t
v
minus1
Figure 4 Numerical solution 119881119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
x
t
Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)
020
4060
80100
010
2030
4050
0
1
2
3
4
5
xt
u
minus1
Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =
0495
020
4060
80100
010
2030
4050
0
05
1
15
2
25
xt
v
minus05
Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
where Ψ = [119880 119881]119905 and 119861 is the (2 times 2)matrix Consider
119861 = [1205741+ 1198941198881
1198882
0 1205741+ 1198941198883
]
minus1
[1205741minus 1198941198881
minus1198882
0 1205741minus 1198941198883
] (26)
where
1198881= 11990111205742+ 1199011015840
21198801205743 119888
2= 1199011015840
31198811205743
1198883= 11990141205742+ 1199011015840
51198801205743
(27)
von Neumann stability condition for the system (25) statesthat for the scheme to be stable the maximum modulus ofthe eigenvalues of the amplification matrix 119861 should be lessthan or equal to one The eigenvalues of the matrix 119861 are
1205821=1205741minus 1198941198881
1205741+ 1198941198881
1205822=1205741minus 1198941198883
1205741+ 1198941198883
(28)
which hasmodulus equal to one and hence the scheme underconsideration is unconditionally stable in the linearizedsense
32 Accuracy of the Scheme In order to study the accuracy ofthe scheme we replace the numerical solution 119880119899
119898and 119881119899
119898in
(17) and (18) by the exact solutions 119906119899119898and V119899119898 By making use
of the following Taylor series expansions at the point (119909119898 119905119899)
and hence the scheme is of second order in both directionsspace and time
4 Product Approximation Technique
A modified Petrov-Galerkin method for solving the CKdVsystem (1) can be achieved by using the product approxi-mation technique where we used special approximation tothe nonlinear terms in the differential system In order toapply this technique we rewrite the CKdV in the followingform
120597119906
120597119905= 119886(
1205973119906
1205971199093+ 3
120597 (1199062)
120597119909) + 119887
120597 (V2)120597119909
(34)
120597V120597119905
= minus1205973V1205971199093
minus 3119906120597V120597119909
(35)
The product approximation technique is used to approx-imate the nonlinear terms in (34) in the following man-ner
1199062(119909 119905) =
119873
sum
119898=1
1198802
119898(119905) 120601119898(119909)
V2 (119909 119905) =119873
sum
119898=1
1198812
119898(119905) 120601119898(119909)
(36)
By using the same procedure in deriving Scheme 1 and theapproximation (37) we can obtain after some manipulationsthe following scheme
Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2
5 Numerical Results
To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868
1 1198682 and 119868
3using
the trapezoidal rule The 119871infin(119906) and 119871
infin(V) error norms are
defined as
119871infin(119906) =
1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119880119899
119898minus 119906119899
119898
1003816100381610038161003816
119871infin (V) =
1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119881119899
119898minus V119899119898
1003816100381610038161003816
(40)
are used to examine the accuracy of the proposed schemes
51 Single Soliton In the first test we choose the initialconditions
119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =
1
2radic119908sech (120585)
120585 = 120582119909 +1
2 log (119908) 119908 =
minus119887
8 (4119886 + 1) 1205824
(41)
which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10
minus7 119909119871= minus25 119909
119877= 25 119886 = minus0125
119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871
infin(119880) 119871
infin(119881) are displayed in Tables 1 and 2
for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899
119898and 119881119899
119898for 119905 = 0 1 2 20
By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method
52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as
which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582
1= 10 120582
2= 06 119910
1= 10
and 1199102= 30 In Table 4 we present the conserved quantities
during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]
To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582
1= 10 120582
2= 06
Abstract and Applied Analysis 7
020
4060
80100
0
20
40
60
80
0123
minus1
x
t
u
Figure 3 Numerical solution 119880119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
020
4060
80100
010
2030
4050
6070
80
012
x
t
v
minus1
Figure 4 Numerical solution 119881119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
x
t
Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)
020
4060
80100
010
2030
4050
0
1
2
3
4
5
xt
u
minus1
Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =
0495
020
4060
80100
010
2030
4050
0
05
1
15
2
25
xt
v
minus05
Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
Again the method is unconditionally stable accordingto von Neumann stability analysis and it is of secondorder in both directions The scheme produced a nonlinearblock pentadiagonal system and its solution obtained usingNewtonrsquosmethodWehave noticed that the accuracy has beenimproved in the first equation (38) as we will see in the nextsection We will denote the scheme obtained by using theproduct approximation technique by Scheme 2
5 Numerical Results
To gain insight into the performance of the proposedschemes we perform different numerical tests like sin-gle soliton two and three solitons interaction and birthof solitons The conservation properties of the proposedschemes are examined by calculating 119868
1 1198682 and 119868
3using
the trapezoidal rule The 119871infin(119906) and 119871
infin(V) error norms are
defined as
119871infin(119906) =
1003817100381710038171003817U119899minus u1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119880119899
119898minus 119906119899
119898
1003816100381610038161003816
119871infin (V) =
1003817100381710038171003817V119899minus k1198991003817100381710038171003817infin = max
119898
1003816100381610038161003816119881119899
119898minus V119899119898
1003816100381610038161003816
(40)
are used to examine the accuracy of the proposed schemes
51 Single Soliton In the first test we choose the initialconditions
119906 (119909 0) = 21205822sech2 (120585) V (119909 0) =
1
2radic119908sech (120585)
120585 = 120582119909 +1
2 log (119908) 119908 =
minus119887
8 (4119886 + 1) 1205824
(41)
which represents a single soliton solution at 119905 = 0 To studythe behavior of numerical solution using Scheme 1 andScheme 2 we choose the set of parameters as ℎ = 005119896 = 001 tol = 10
minus7 119909119871= minus25 119909
119877= 25 119886 = minus0125
119887 = minus3 and 120582 = 05 The conserved quantities and theerror norms 119871
infin(119880) 119871
infin(119881) are displayed in Tables 1 and 2
for Scheme 1 and Scheme 2 respectively It is clear from thesetables that our schemes are highly accurate In addition theschemes preserve the conserved quantities exactly during theevolution of the numerical solution from 119905 = 0 to 119905 = 10 Theexecution time required to produce Table 1 is 2328 secondand 2171 second to produce Table 2 We have noticed thatScheme 2 has an upper hand over Scheme 1 with respect toaccuracy and CPU time In Figures 1 and 2 we display thenumerical solution of 119880119899
119898and 119881119899
119898for 119905 = 0 1 2 20
By choosing the set of values 119896 = 001 119886 = 05 119887 = minus30120582 = 05 and 119905 = 1 we perform a comparison of Scheme 1 andScheme 2 with Ismail [6] and we display this in Table 3 wecan easily see that the three methods produce highly accurateresults with some credits for collocation method
52 Two Solitons Interaction To study the interaction of twosolitons we choose the initial conditions as
which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582
1= 10 120582
2= 06 119910
1= 10
and 1199102= 30 In Table 4 we present the conserved quantities
during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]
To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582
1= 10 120582
2= 06
Abstract and Applied Analysis 7
020
4060
80100
0
20
40
60
80
0123
minus1
x
t
u
Figure 3 Numerical solution 119880119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
020
4060
80100
010
2030
4050
6070
80
012
x
t
v
minus1
Figure 4 Numerical solution 119881119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
x
t
Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)
020
4060
80100
010
2030
4050
0
1
2
3
4
5
xt
u
minus1
Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =
0495
020
4060
80100
010
2030
4050
0
05
1
15
2
25
xt
v
minus05
Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
which represents the sum of two single solitons we assignthe value of the parameters 119888 = 0 119889 = 100 ℎ = 005119896 = 001 119886 = 05 119887 = minus30 120582
1= 10 120582
2= 06 119910
1= 10
and 1199102= 30 In Table 4 we present the conserved quantities
during the interaction scenario and show that our numericalmethods achieved the goal of conserving these quantitiesThe interaction scenario is presented in Figures 3 and 4The contours of the interaction process are given in Figure 5We have noticed that the taller (faster) wave collides withthe shorter (slower) wave and leaves the interaction regionwithout any disturbance in their identitiesThis phenomenonindicates the interaction scenario is elastic [1]
To examine the interaction scenario for 119886 = 12 we usethe set of parameters 119886 = 0495 119887 = minus3 120582
1= 10 120582
2= 06
Abstract and Applied Analysis 7
020
4060
80100
0
20
40
60
80
0123
minus1
x
t
u
Figure 3 Numerical solution 119880119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
020
4060
80100
010
2030
4050
6070
80
012
x
t
v
minus1
Figure 4 Numerical solution 119881119899 with parameters 120582
1= 10 120582
2=
06 1199101= 10 and 119910
2= 30
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
x
t
Figure 5 Contours of the numerical solution119880119899 (interaction of twosolitons)
020
4060
80100
010
2030
4050
0
1
2
3
4
5
xt
u
minus1
Figure 6 Inelastic interaction numerical solution 119880119899 with 119886 =
0495
020
4060
80100
010
2030
4050
0
05
1
15
2
25
xt
v
minus05
Figure 7 Inelastic interaction numerical solution 119881119899 with 119886 =
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
results are demonstrated in Figures 6 and 7 the amplitudesof the two solitons have changed after the interaction whichindicates that the interaction scenario is inelastic We havetested other values of 119886 in all cases we have found thatthe interaction is inelastic and this in agreement with [1]which they claim the CKdV equation is integrable only for119886 = 12
53 Three Solitons Interaction To study the interaction ofthree solitons we choose the initial condition as a sumof three well separated single solitons in the followingform
119906 (119909 0) =
3
sum
119895=1
119906119895(119909 0) V (119909 0) =
3
sum
119895=1
V119895(119909 0) (44)
8 Abstract and Applied Analysis
020
4060
80100
010
2030
4050
6070
80
0123
x
t
u
minus1
Figure 8 Interaction of three solitons numerical solution 119880119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
020
4060
80100
020
4060
80
0020406081
121416
xt
v
minus02
Figure 9 Interaction of three solitons numerical solution 119881119899 withparameters 120582
1= 1 120582
2= 06 and 120582
3= 03
where
119906119895 (119909 0) = 2120582
2
119895sech2 (120585
119895) V
119895 (119909 0) =1
2radic119908119895
sech (120585119895)
120585119895= 120582119895(119909 minus 119910
119895) +
1
2 log (119908119895)
119908119895=
minus119887
8 (4119886 + 1) 1205824
119895
119895 = 1 2 3
(45)
In this test we choose the set of parameters 119909119871= 0 119909
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
The simulation of the three solitons interaction scenariois given in Figures 8 and 9 respectively for the time duration119905 = 0 4 80 It is very clear to see how the solitonwith the largest amplitude 119906
1interacts with the other two
solitons and leaves the interaction regionunchanged in shapeThe three solitons appear after the interaction in the reverseorder compared to the initial state In Table 5 we display theconserved quantities during the interaction scenario
020
4060
80
05
1015
2025
024
minus40
minus20
minus2
x
t
u
Figure 10 Birth of solitons with parameters 119886 = 05 and 119887 = minus30
Table 5 Three solitons interaction the conserved quantities
54 Birth of Solitons In this test we choose the initial condi-tion
119906 (119909 0) = exp (minus0011199092)
V (119909 0) = exp (minus0011199092) (46)
with the following set of parameters 119886 = 05 119887 = minus30 0 le
119909 le 100 ℎ = 005 and 119896 = 001 We have noticed as timeevolves a birth of four solitons with different amplitudes andthis can be easily seen in Figure 10 The conserved quantitiesare given in Table 6 which is almost conserved
6 Conclusion
In this work we have derived two numerical schemes forsolving the Hirota-Satsuma CKdV system The resultingschemes are nonlinear implicit and unconditionally stableThe schemes show almost similar results Single soliton solu-tion and conserved quantities are used to assess the accuracyand the efficiency the derived schemes We have noticedthat the schemes accomplished the aim of preserving theconserved quantities while maintaining small errors norm
Abstract and Applied Analysis 9
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ
during the simulation The features of an elastic interactionhave been shown in the simulation of two and three solitonsinteraction using the proposed schemes for 119886 = 12 andinelasticity occurs for 119886 = 12
To sum up the derived methods are qualified and can beadopted for solving any CKdV like systems successfully dueto their effective performance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R Hirota and J Satsuma ldquoSoliton solutions of a coupledKorteweg-de Vries equationrdquo Physics Letters A vol 85 no 8-9pp 407ndash408 1981
[2] A A Halim S P Kshevetskii and S B Leble ldquoNumericalintegration of a coupled Korteweg-de Vries systemrdquo Computersamp Mathematics with Applications vol 45 no 4-5 pp 581ndash5912003
[3] A A Halim and S B Leble ldquoAnalytical and numerical solutionof a coupled KdV-MKdV Systemrdquo Chaos Solitons and Fractalsvol 19 no 1 pp 99ndash108 2004
[4] J M Sanz-Serna and I Christie ldquoPetrov-Galerkin methods fornonlinear dispersive wavesrdquo Journal of Computational Physicsvol 39 no 1 pp 94ndash102 1981
[5] L Debenath Nonlinear Partial Differential EquationsBirkhauser Boston Mass USA 1997
[6] M S Ismail ldquoNumerical solution of a coupled Korteweg-deVries equations by collocation methodrdquo Numerical Methods forPartial Differential Equations vol 25 no 2 pp 275ndash291 2009
[7] D Kaya and I E Inan ldquoExact and numerical traveling wavesolutions for nonlinear coupled equations using symbolic com-putationrdquoAppliedMathematics and Computation vol 151 no 3pp 775ndash787 2004
[8] E G Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002
[9] L M B Assas ldquoVariational iteration method for solvingcoupled-KdV equationsrdquo Chaos Solitons and Fractals vol 38no 4 pp 1225ndash1228 2008
[10] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007
[11] M S Ismail ldquoNumerical solution of coupled nonlinearSchrodinger equation by Galerkin methodrdquo Mathematics andComputers in Simulation vol 78 no 4 pp 532ndash547 2008
[12] M S Ismail and T R Taha ldquoA linearly implicit conservativescheme for the coupled nonlinear Schrodinger equationrdquoMath-ematics and Computers in Simulation vol 74 no 4-5 pp 302ndash311 2007
[13] M S Ismail and S Z Alamri ldquoHighly accurate finite differencemethod for coupled nonlinear Schrodinger equationrdquo Interna-tional Journal of Computer Mathematics vol 81 no 3 pp 333ndash351 2004
[14] A Wazwaz ldquoThe KdV equationrdquo in Handbook of DifferentialEquations Evolutionary Equations VOL IV Handb Differ