Numerical solution of complex modified Korteweg-de Vries equation by collocation method

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Communications in Nonlinear Science and Numerical Simulation 14 (2009) 749–759 www.elsevier.com/locate/cnsns

Numerical solution of complex modified Korteweg-de Vries equation by collocation method M.S. Ismail Department of Mathematics, College of Science, P.O. Box 80203, King Abdulaziz University, Jeddah 21589, Saudi Arabia Received 18 October 2007; received in revised form 6 December 2007; accepted 6 December 2007 Available online 15 December 2007

Abstract The collocation method using quintic B-spline is derived for solving the complex modified Korteweg-de Vries (CMKdV). The method is based on Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The von Neumann stability is used to prove that the scheme is unconditionally stable. Newton’s method is used to solve the nonlinear block pentadiagonal system obtained. Numerical tests for single, two, and three solitons are used to assess the performance of the proposed scheme. Ó 2007 Elsevier B.V. All rights reserved. PACS: 03.40.Kf; 02.30.Ik Keywords: CMKdV equation; Collocation method; Quintic B-splines; Solitons interaction

1. Introduction In this work, we aim to solve numerically the complex modified Korteweg-de Vries CMKdV equation 2 oW o3 W oðjWj WÞ þ 3 þa ¼ 0; 1 < x < 1; t > 0; ð1Þ ot ox ox where W is a complex valued function of the spatial coordinate x and the time t, a is a real parameter. This equation has been proposed as a model for the nonlinear evolution of plasma waves. The physical model (1) incorporates the propagation of transverse waves in a molecular chain model [10]. The CMKdV equation has a solitary wave solution of the form rffiffiffiffiffi pffiffiffi 2c Wðx; tÞ ¼ sech½ cðx  x0  ctÞ expðihÞ; ð2Þ a which represents a solitary wave positioned at x0 and moving to the right with velocity c and satisfying the boundary conditions u ! 0 as x ! 1. The CMKdV equation has the conserved quantities Z 1 Z 1 Z 1  a jWj4  jWx j2 dx: I1 ¼ Wdx; I 2 ¼ jWj2 dx; I 3 ¼ ð3Þ 1 1 1 2 E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.12.005

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The proof of the conserved quantities can be done by the help of [2]. The CMKdV equation (1) has been solved analytically by sine–cosine and tanh method by Wazwaz [17] and he showed that this equation admits sech-shaped soliton solutions whose amplitudes and velocities are free parameters, and tanh solution (kink type). To avoid complex computation, we transform the CMKdV equation into a nonlinear coupled system by decomposing Wðx; tÞ into its real and imaginary parts, by assuming Wðx; tÞ ¼ uðx; tÞ þ ivðx; tÞ;

i2 ¼ 1;

ð4Þ

where uðx; tÞ and vðx; tÞ are real functions, to obtain the coupled pair of the modified KdV equations ou o3 u o þ þ a ½ðu2 þ v2 Þu ¼ 0; ot ox3 ox ov o3 v o þ 3 þ a ½ðu2 þ v2 Þv ¼ 0: ot ox ox

ð5Þ

These two coupled nonlinear equations describe the interaction of two orthogonally polarized transverse waves [10,17], where u and v represent y-polarized and z-polarized transverse waves, respectively, propagating in the x-direction in an xyz coordinate system. The numerical solution of nonlinear wave equations has been the subject of many studies in recent years. Although many numerical schemes have been proposed for some well- known integrable equations, such as the Korteweg-de Vries (KdV) equation [14–16], there is a little numerical analysis literature for the nonintegrable wave equations, like the coupled nonlinear Schro¨dinger equation which admits soliton solution, and it has many applications in communication and optical fibers, this system has been solved numerically by Ismail [5–8], and the coupled Korteweg-de Vries equation [3,4,9,20]. In [10], Muslu et al., solved this equation using a split step Fourier method. Also Taha [16] solved a different form of the CKMdV equation by the same method. Collocation method has been used to solve different equations in partial differential equations, as an example, equal width and modified equal width equations [1,12,19], Burger’s and modified Burger’s equations [11,13], Korteweg-de Vries-Burger’s equation [18]. In this work, we are going to solve the CMKdV equation numerically using collocation method with quintic splines. Implicit midpoint rule is used in the time direction. The resulting scheme produced a block nonlinear pentadiagonal system, where Newton’s method is used to solve it. von Nuemann stability analysis shows that the scheme is unconditionally stable. The paper is organized as follows. In Section 2, the details of collocation method are given for solving the CMKdV equation, a coupled nonlinear pentadiagonal system is obtained, stability analysis and accuracy are displayed. In Section 3, numerical results for single, two, and three solitons are given. The error and the conserved quantities are produced to asses the efficiency of the proposed method. Concluding remarks contained in Section 4. 2. Collocation method In order to describe the collocation method, we cast the CMKdV equation as   ou o3 u ou ov þ 3 þ a ð3u2 þ v2 Þ þ 2uv ¼ 0; ot ox ox ox   ov o3 v ou 2 2 ov þ þ a 2uv þ ðu þ 3v Þ ¼ 0; ot ox3 ox ox

ð6Þ

which can be written in a vector form as o3 w ow ¼ 0; þ Aðu; vÞ 3 ox ox 2 3u þ v2 t where w ¼ ½u; v ; A ¼ a 2uv w_ þ



2uv : u2 þ 3v2

ð7Þ

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We assume that the solution of the CMKdV equation is negligible outside the interval ½a; b. The interval ½a; b is partitioned into N finite elements of uniformly equal length h by the knots xm , m ¼ 0; 1; 2; . . . ; N such that a ¼ x0 < x1 < x2 <    < xN 1 < xN ¼ b. Let wnm and Wnm to denote the exact and approximate solutions of the CMKdV equation at the point ðxm ; tn Þ, respectively, where xm ¼ mh; t ¼ nk, h and k are the grid step sizes for space and time, respectively. The quintic B-splines Bm ðxÞ; m ¼ 2; 3; . . . ; N þ 2 are defined at knots xm by 8 q0 ¼ ðx  xm3 Þ5 ½xm3 ; xm2 ; > > > > 5 > q ¼ q  6ðx  x Þ ½xm2 ; xm1 ; > m2 1 0 > > > 5 > > q ¼ q1 þ 15ðx  xm1 Þ ½xm1 ; xm ; 1< 2 ð8Þ Bm ðxÞ ¼ 5 q3 ¼ q2  20ðx  xm Þ5 ½xm ; xmþ1 ; h > > > q ¼ q þ 15ðx  x Þ5 ½x ; x ; > mþ1 mþ1 mþ2 > 4 3 > > 5 > > q ¼ q  6ðx  x Þ ½x mþ2 mþ2 ; xmþ3 ; > 5 4 > : 0 otherwise: The set of those B-splines fB2 ; B1 ; . . . ; BN þ1 g defines to form a basis over the interval ½a; b. Numerical solution of Eq. (7) will be derived using the collocation method with a basis of quintic B-splines. A global approximation W to the exact solution w is given by W ðx; tÞ ¼

N þ2 X

dm ðtÞBm ðxÞ;

ð9Þ

m¼2

where Bm ðxÞ are the quintic B-spline and dm ¼ ½d1 ; d2 t are time dependent parameters to be determined from the CMKdV equation and the boundary conditions. The quintic B-spline and its three principle derivatives vanish outside the interval ½xm3 ; xmþ3 . The intervals ½xm ; xmþ1  are identified with the finite elements, which are each covered by quintic B-splines. Over the element ½xm ; xmþ1  the variation of the function Wðx; tÞ is formed from Wðx; tÞ ¼

mþ3 X

dj ðtÞBj ðxÞ:

ð10Þ

j¼m2

In terms of a local coordinate system g which is given by hg ¼ x  xm , where h ¼ xmþ1  xm and 0 6 g 6 1, expression for the element splines are [18] Bm2 ¼ 1  5g þ 10g2  10g3 þ 5g4  g5 ; Bm1 ¼ 26  50g þ 20g2 þ 20g3  20g4 þ 5g5 ; Bm ¼ 66  60g2 þ 30g4  10g5 ; Bmþ1 ¼ 26 þ 50g þ 20g2  20g3  20g4 þ 10g5 ;

ð11Þ

Bmþ2 ¼ 1 þ 5g þ 10g2 þ 10g3 þ 5g4  5g5 ; Bmþ3 ¼ g5 : The nodal values of Wm ; W0 and W000 at the knot xm , are given in terms of the parameters dm by Wm ¼ dm2 þ 26dm1 þ 66dm þ 26dmþ1 þ dmþ2 ; 5 W0m ¼ ðdmþ2 þ 10dmþ1  10dm1  dm2 Þ; h 60 000 Wm ¼ 3 ðdmþ2  2dmþ1 þ 2dm1  dm2 Þ; h

ð12Þ

where the dashes denote differentiation with respect to x. Collocation points are selected to coincide with knots. Substituting the nodal values Wm , W0m , and W000 m into Eq. (7), this will lead us to the coupled first order differential system

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ðd_ m2 þ 26d_ m1 þ 66d_ m þ 26d_ mþ1 þ d_ mþ2 Þ þ þ

60 ðdmþ2  2dmþ1 þ 2dm1  dm2 Þ h3

30 AðWm Þðdmþ2 þ 10dmþ1  10dm1  dm2 Þ ¼ 0; h

ð13Þ

where the dot denotes the derivative with respect to t. The previous differential system can be solved by an implicit schemes like, trapezoidal or implicit midpoint rules, which are usually produced unconditionally stable methods, or by an explicit methods which produced conditionally stable methods. To derive the fully discretized difference equations, we adopt the implicit midpoint rule approximation for the unknown parameters dm , where the following approximations are used dm ¼

dmnþ1 þ dnm ; 2

dnþ1  dnm d_ m ¼ m : k

ð14Þ

This will lead us to the nonlinear system  nþ1   n  nþ1 nþ1 n n n n dmþ2 þ 26dmþ1 þ 66dmnþ1 þ 26dnþ1 m1 þ dm2  dmþ2 þ 26dmþ1 þ 66dm þ 26dm1 þ dm2    n 

nþ1 nþ1 nþ1 n n n þ p1 dnþ1 mþ2  2dmþ1 þ 2dm1  dm2 þ dmþ2  2dmþ1 þ 2dm1  dm2  nþ1   n  Wm þ Wnm  nþ1 nþ1 nþ1 n n n þ p2 A ¼ 0; dmþ2 þ 10dnþ1 mþ1  10dm1  dm2 þ dmþ2 þ 10dmþ1  10dm1  dm2 2 m ¼ 0; 1; 2; . . . ; N :

ð15Þ

Eq. (15) form a nonlinear coupled pentadiagonal system of ð2N þ 2Þ algebraic equations in ð2N þ 10Þ unknowns parameters dm . So in order to obtain a unique solution to this system we have to impose the eight boundary conditions wða; tÞ ¼ wx ða; tÞ ¼ wðb; tÞ ¼ wx ðb; tÞ ¼ 0:

ð16Þ

nþ1 nþ1 nþ1 These conditions enable us to eliminate the parameters dnþ1 2 ; d1 ; dN þ1 and dN þ2 . These parameters can be obtained in the following manner

wða; tÞ ¼ d2 þ 26d1 þ 66d0 þ 26d1 þ d2 ¼ 0; 5 wx ða; tÞ ¼ ðd2 þ 10d1  10d1 þ d2 Þ ¼ 0: h

ð17Þ

Solving these equations for d2 and d1 will give 9 65 165 d0 ; d2 ¼ d2 þ d1 þ 4 2 4 1 9 33 d1 ¼  d2  d1  d0 ; 8 4 8

ð18Þ

and for the boundary conditions at x ¼ b, we have 1 9 33 dN þ1 ¼  dN 2  dN 1  dN ; 8 4 8 9 65 165 dN : dN þ2 ¼ dN 2 þ dN 1 þ 4 2 4

ð19Þ

By eliminating these parameters from (15), this system will be reduced into ð2N þ 2Þ nonlinear equation in ð2N þ 2Þ unknowns. The solution of this system can be obtained by Newton’s method. This method can be easily applied, since the resulting Jacobian is a block pentadiagonal matrix with elements of ð2  2Þ matrices, which can be easily calculated analytically. Before the solution process begin, initial parameters d0m must be determined by using the initial condition and the following derivatives at the boundaries:

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753

wx ða; 0Þ ¼ d02 þ 10d01  10d01  d02 ¼ 0; wxx ða; 0Þ ¼ d02 þ 2d01  6d00 þ 2d01 þ d02 ¼ 0; wðxm ; 0Þ ¼ d0m2 þ 26d0m1 þ 66d0m þ 26d0m¼1 þ d0mþ2 ; wx ðb; 0Þ ¼ d0N þ2 þ 10d0N þ1  10d0N 1  d0N 2 wxx ðb; 0Þ ¼ d0N þ2 þ 2d0N þ1  6d0N þ 2d0N 1 þ

m ¼ 0; 1; 2; . . . ; N ;

ð20Þ

¼ 0; d0N 2 ¼ 0:

This forms a linear block pentadiagonal system for the unknown initial conditions d0 of order ð2N þ 2Þ after eliminating the fictitious values of d. This system can be easily solved using Crout’s method. Once the initial vector of parameters has been calculated, time evaluation of Wnm can be determined from the time evolution of the vectors dn by using the recursion relation nþ1 nþ1 nþ1 nþ1 Wmnþ1 ¼ dmþ2 þ 26dnþ1 mþ1 þ 66dm þ 26dm1 þ dm2 :

2.1. Stability of the scheme To study the stability of the resulting scheme, von Neumann stability analysis is used. This method can only be applied for linear schemes. The linear version of the proposed scheme is given by  nþ1   n  nþ1 nþ1 nþ1 n n n n dmþ2 þ 26dmþ1 þ 66dnþ1 m þ 26dm1 þ dm2  dmþ2 þ 26dmþ1 þ 66dm þ 26dm1 þ dm2  nþ1   n 

nþ1 nþ1 n n n þ p1 dmþ2  2dnþ1 mþ1 þ 2dm1  dm2 þ dmþ2  2dmþ1 þ 2dm1  dm2

 nþ1   n  nþ1 nþ1 n n n þ p2 C dmþ2 þ 10dnþ1 ¼ 0; ð21Þ mþ1  10dm1  dm2 þ dmþ2 þ 10dmþ1  10dm1  dm2 where C = constant. We assume the solution of (21) to be of the form dnm ¼ dn eibmh :

ð22Þ

By substituting (22) into (21) and after some manipulation, we get ½c1 þ ip1 c2 þ ip2 Cc3 dnþ1 ¼ ½c1  ip1 c2  ip2 Cc3 dn ;

ð23Þ

where c1 ¼ 66 þ 2 cosð2bhÞ þ 52 cosðbhÞ; c2 ¼ 2 sinð2bhÞ  4 sinðbhÞ; c3 ¼ 2 sinð2bhÞ þ 20 sinðbhÞ;

ð24Þ

which can be written in a matrix vector form as dnþ1 ¼ Bdn ; where B is the ð2  2Þ matrix  1  c1 þ ic1 0 c1  ic1 B¼ 0 c1 þ ic1 0

ð25Þ  0 ; c1  ic1

ð26Þ

and c1 ¼ p1 c2 þ p2 Cc3 . von Neumann stability condition for the system (25) is the maximum modulus of the eigenvalues of the matrix B is to be less than or equal to one. Direct calculation of these eigenvalues gives k1;2 ¼

c1  ic1 ; c1 þ ic1

ð27Þ

where the modulus of these eigenvalues equal to one. This means the scheme which we have derived is unconditionally stable according to von Neumann stability analysis, which means that no restriction on the grid sizes of h and k, but we should choose them in such a way the accuracy of the scheme is not degraded.

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3. Numerical results To gain insight into the performance of the suggested, we calculate the error using the exact solution and we calculate the conserved quantities by using trapezoidal rule. In all calculations we choose h ¼ 0:1; k ¼ 0:01, and Tol = 0.5E06 for checking the convergence of Newton’s method, where L1 error norm is used which is defined by L1 ¼ max fjjWn jj  jjW n jjg: 1
3.1. Single soliton In this test we choose the initial condition from the exact solution rffiffiffiffiffi pffiffiffi 2c Wðx; 0Þ ¼ sech½ cðx  x0 Þ expðihÞ: a We choose a ¼ 0; b ¼ 60; x0 ¼ 20. We consider the following tests (i) We first consider a y-polarized solitary wave solution with a ¼ 2; h ¼ p=2; c ¼ 1. In Table 1, we calculate the error and the conserved quantities, and it is very clear the method produced an accurate results and keep all the conserved quantities almost constant. In Fig. 1 we display the numerical solution at t ¼ 0; 1; 2; . . . ; 20. (ii) In the second test, we consider a solitary wave solution with a ¼ 1; h ¼ p=4; c ¼ 1. In Table 2, we calculate the error and the conserved quantities, the results indicate the ability of the scheme to display the properties of CMKdV equation. In Fig. 2 we display the numerical solution at t ¼ 0; 1; 2; . . . ; 20.

3.2. Two solitons interaction In this test we choose the initial condition as the sum of two solitary waves of the form rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffi 2c1 2c2 Wðx; 0Þ ¼ sech½ c1 ðx  x1 Þ expðih1 Þ þ sech½ c2 ðx  x2 Þ expðih2 Þ; a a where x 2 ½0; 100, x1 ¼ 25 and x2 ¼ 50 are the initial location of the two solitary waves. The following numerical tests will be discussed (i) We first study the interaction of two orthogonally polarized solitary  waves, namely, the interaction between a y-polarized ðh1 ¼ 0Þ solitary wave and a z-polarized h2 ¼ p2 solitary wave. We choose the set of parameters a ¼ 2; c1 ¼ 2; c2 ¼ 0:5. The interaction scenario at t ¼ 0; 1; 2; . . . ; 30 is given in Fig. 3, we note that there is a tail following the shorter one after the interaction and this is agree with [10].

Table 1 Single soliton with a ¼ 2; c ¼ 1; h ¼ p=2 T

L1

I1

I2

I3

0.0 5.0 10.0 15.0 20.0

0.000000 0.000055 0.000108 0.000163 0.000218

3.141591 3.141592 3.141591 3.141593 3.141589

2.000000 1.999999 1.999999 1.999998 1.999998

0.669765 0.669764 0.669764 0.669763 0.669763

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755

1

0.8

|u|

0.6

0.4

0.2

0 20 60

15 50 10

40 30 5

20 10 0

y

0

x

Fig. 1. Single soliton: a ¼ 2; c ¼ 1; h ¼ p=2.

Table 2 Single soliton with a ¼ 1; c ¼ 1; h ¼ p=4 T

L1

I2

I3

0.0 5.0 10.0 15.0 20.0

0.000000 0.000078 0.000154 0.000231 0.000308

4.000000 3.999999 3.999998 3.999998 3.999997

1.339529 1.339529 1.339528 1.339528 1.339526

1.5

|u|

1

0.5

0 20 60

15 50 10

40 30

t

5

20 10 0

0

x

Fig. 2. Single soliton: a ¼ 1; c ¼ 1; h ¼ p=4.

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1.5

|u|

1

0.5

0 30 25

100

20

80

15

60 10

40 5

20 0

t

0

x

Fig. 3. Interaction of two orthogonally solitary waves c1 ¼ 2; c2 ¼ 0:5; h1 ¼ 0; h2 ¼ p=2.

(ii) In the second test [10], we study the interaction between two y-polarized solitary waves, namely ðh1 ¼ h2 ¼ 0Þ, and the parameters a ¼ 2; c1 ¼ 2; c2 ¼ 0:5. Fig. 4 shows the evolution of the modulus of the numerical solution at t ¼ 0; 1; 2; . . . ; 30, the interaction is elastic, and it is in agreement with the fact that the CMKdV equation reduces to the single MKdV equation. The contours are given in Fig. 5   (iii) Lastly, we consider, the interaction of two solitary waves, h1 ¼ h2 ¼ p4 and the parameters a ¼ 2; c1 ¼ 2; c2 ¼ 0:5. The conserved quantities are given in Table 3, which are almost constant during the interaction simulation. Fig. 6 shows the evolution of the modulus of the numerical solution at t ¼ 0; 1; 2; . . . ; 30. The interaction scenario is elastic.

1.5

|u|

1

0.5

0 30 25

100

20

80

15

60 10

t

40 5

20 0

0

x

Fig. 4. Interaction of two y-polarized solitary waves c1 ¼ 2; c2 ¼ 0:5; h1 ¼ 0; h2 ¼ 0.

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757

30

25

20

15

10

5

0

0

10

20

30

40

50

60

70

80

90

100

Fig. 5. Contour of two y-polarized solitary waves c1 ¼ 2; c2 ¼ 0:5; h1 ¼ 0; h2 ¼ 0.

Table 3 Conserved quantities with h1 ¼ h2 ¼ p4 T

I2

I3

0.0 5.0 10.0 15.0 20.0 25.0 30.0

4.242641 4.242642 4.242642 4.242642 4.242642 4.242642 4.242643

2.139322 2.139325 2.139065 2.123464 2.138992 2.139325 2.139328

3.3. Three solitons interaction In this test we choose the initial condition Wðx; 0Þ ¼

3 X

Wj ðx; 0Þ;

j¼1

where rffiffiffiffiffiffi 2cj pffiffiffiffi Wj ðx; 0Þ ¼ sech½ cj ðx  xj Þ expðihj Þ; a which represent a sum of three single solitons located at x1 ; x2 and x3 . In this test we choose the parameters a ¼ 0; b ¼ 120; x1 ¼ 10:0; x2 ¼ 30:0; x3 ¼ 50; c1 ¼ 1:0; c2 ¼ 0:5; c3 ¼ 0:3; h1 ¼ h2 ¼ h3 ¼ 0. Fig. 7 displays the interaction scenario at t ¼ 0; 2; 4; . . . ; 80, which shows how the three solitons are traveling with different velocities in the same direction where the fastest soliton is located at x1 ¼ 10, and the second soliton is located at x2 ¼ 30, while the slowest one is located at x3 ¼ 50, as the time progress the

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1.5

|u|

1

0.5

0 30 25 100

20

80

15

60 10

40

t

5

20 0

x

0

Fig. 6. Interaction of two solitary waves c1 ¼ 2; c2 ¼ 0:5; h1 ¼ p=4; h2 ¼ p=4.

|u|

1

0.5

0 80 70 60 50

t

120 40

100 80

30 60

20 40 10

20 0

x

0

Fig. 7. Three solitons interaction.

Table 4 Conserved quantities of three solitons T

I1

I2

0.0 20 40 60 80

4.510006 4.510005 4.510006 4.510008 4.510005

1.015897 1.015783 1.015031 1.015660 1.015752

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interaction occurred and the fastest soliton interact and leave the interaction unchanged in shape with reverse locations, the fastest soliton become a head of the other two. The conserved quantities recorded in Table 4. 4. Conclusion In this work, we have solved the CMKdV equation using collocation method with quintic splines. The resulting scheme produced a nonlinear block penta diagonal system. Newton’s method is used to solve this system. Single soliton, the interaction of two and three solitons are used to asses the performance of the method, we have noticed that the scheme produced highly accurate results, and keeps the conserved quantities are almost constants during the simulation. The interaction is elastic for certain values of h. As a conclusion we can say the derived method is highly accurate and can simulate the solution of the CMKdV equation in a very nice way. The suggested method can be also used in a very efficient way for solving coupled KdV like equations. Acknowledgement The author thanks the referees for their valuable comments and suggestions. References [1] Dag I, Saka B, Irk D. Galerkin method for the numerical solution of the RLW equation using quintic B-splines. Comput Appl Math 2006;190:532–47. [2] Debenath L. Nonlinear partial differential equations. Boston: Birkha¨user; 1997. [3] Halim AA, Kshevetskii SP, Leble SB. Numerical integration of a coupled Korteweg-de Vries System. Comput Math Appl 2003;45:581–91. [4] Halim AA, Leble SB. Analytical and numerical solution of coupled KdV–MKdV system. Chaos Soliton Fract 2004;19:99–108. [5] Ismail MS. Numerical solution of coupled nonlinear Schro¨dinger equation by Galerkin method. Math Comp Simul [in press]. [6] Ismail MS, Taha Thiab R. A linearly implicit conservative scheme for the coupled nonlinear Schro¨dinger equation. Math Comp Simul 2007;74:302–11. [7] Ismail MS, Alamri SZ. Highly accurate finite difference method for coupled nonlinear Schro¨dinger equation. Int J Comp Math 2004;81(3):333–51. [8] Ismail MS, Taha Thiab R. Numerical simulation of coupled nonlinear Schro¨dinger equation. Math Comp Simul 2001;56:547–62. [9] Kaya D, Inan I. Exact and numerical traveling wave solutions for nonlinear coupled equation using symbolic computation. Appl Math Comput 2004;151:775–87. [10] Muslu GM, Erabay HA. A split-step fourier method for the complex modified Korteweg-de Vries equation. Comput Math Appl 2003;45:503–14. [11] Ramadan MA, El Danaf TS. Numerical treatment for the modified Burger’s equation. Math Comput Simul 2005;70:90–8. [12] Saka B. Algorithms for numerical solution of the modified equal width wave equation using collocation method. Math Comput Model 2007;45:1096–117. [13] Saka B, Dag I. Quartic B-spline collocation method to the numerical solutions of Burger’s equation. Chaos Soliton Fract 2007;32:1125–37. [14] Taha TR, Ablowitz MJ. Analytical and numerical aspects of certain nonlinear evolution equations. IV. Numerical, Korteweg-de Vries equation. J comput Phys 1984;55:231–53. [15] Taha TR, Ablowitz MJ. Analytical and numerical aspects of certain nonlinear evolution equations. IV. Numerical, modified Korteweg-de Vries equation. J Comput Phys 1988;77:540–8. [16] Taha TR. Numerical simulations of the complex modified Korteweg-de Vries equation. Math Comput Simul 1994;37:461–7. [17] Wazwaz AM. The tanh and the sine–cosine methods for the complex modified KdV and the generalized KdV equations. Comput Math Appl 2005;49:1101–12. [18] Zaki SI. A quintic B-spline finite elements scheme for the KdVB equation. Comput Meth Appl Mech Eng 2000;188:121–34. [19] Zaki SI. Solitary wave interactions for the modified equal width equation. Comput Phys Commun 2000;126:219–31. [20] Zhu S. A difference scheme for coupled KdV equation. Commun Nonlinear Sci Numer Simul 1999;4(1):63–9.