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Solitary wave solutions for a coupled pair of mKdV equations
Applied Mathematics and Computation 217 (2010) 1540–1548
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Solitary wave solutions for a coupled pair of mKdV equations H. Triki a, M.S. Ismail b,* a b
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Department of Mathematics, College of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
Keywords: Modified Korteweg-de Vries equation Soliton solution Finite difference method
a b s t r a c t Propagation of weakly nonlinear long waves is studied within the framework of a system of two coupled modified Korteweg-de Vries equations. We investigate analytically and numerically the various families of soliton states for the considered model. By scaling the functions and variables we find that the resulting coupled pair of equations has only one combined parameter. This parameter depends on the wave speed and the coupling coefficient. Explicit analytical expressions for both of the symmetric and antisymmetric states are determined. Numerical method is derived to solve the proposed system, many numerical tests have been conducted to study the behavior of the solution, and the existence of the asymmetric soliton states is displayed numerically. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Studies of nonlinear wave models have drawn considerable attentions in recent years. This can be considered as one of the fundamental objects of nature [1]. The most famous examples are the nonlinear Schrödinger [2,3], the Korteweg-de Vries [4,5] and the sine-Gordon [6] equations. The application of such models cover various branches of physics like nonlinear optics, fluid dynamics, condensed matter physics and so on. The solutions of these nonlinear evolution equations are known to be in the form of soliton envelope. The existence of soliton states implies perfect balance between nonlinearity and dispersion effects which usually requires specific conditions. It should be noted that the nonlinear optics is the field, where all soliton features are exhibited to a great extent [1]. In 1967, Zabusky [7] showed how the modified Korteweg-de Vries (mKdV) equation may model the oscillations of a lattice of particles connected by nonlinear springs as the Fermi-Pasta-Ulam model does. Further, the evolution of internal waves at the interface of two layers of equal depth is also described by the mKdV equation [8]. The mKdV equation and some other nonlinear evolution equations are gradually putting together to form a list of soliton equations, which are usually to be found in nonlinear optics theory. In a recent study [9] shows that the propagation of optical pulses in nematic optical fibers is described by a complex modified KdV equation when we go beyond the usual weakly nonlinear limit of a Kerr medium. Wave propagation and switching in a nonlinear multicore fiber arrays which is described by coupled nonlinear Schrödinger equations have appeared in a number of papers [10–13]. In particular, Akhmediev and Ankiewicz [11] have made an interesting study of the various families of soliton states dynamics in nonlinear fiber couplers. The authors have found that in addition to the symmetric and antisymmetric soliton states, there exists an other family of asymmetric soliton states which bifurcate from the first ones. The aim of this paper is to find various soliton solutions for a model of two coupled mKdV equations, governing the mutual interaction in addition to nonlinear self-interaction of two solitary waves in a mKdV system. A numerical method
* Corresponding author. E-mail address: [email protected] (M.S. Ismail). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.047
H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
1541
is presented,which is implicit, unconditionally stable, and of second order accuracy in space and second order accuracy in time. This method is used to study the behavior of the solutions. The paper is organized as follows. In Section 2, we present the considering model of two coupled mKdV equations, and then we derive the symmetric and antisymmetric soliton solutions. In Section 3, we present a numerical method for solving the coupled systems. Numerical results with various initial conditions are reported in Section 4. Concluding remarks are given in Section 5.
2. Model and equations We consider the coupled pair of mKdV equations [14]:
ut þ uxxx þ a½ðu2 þ v 2 Þux ¼ 0;
v t þ v xxx þ a½ðu2 þ v 2 Þv x ¼ 0;
ð1Þ
where u and v are real functions of the spatial coordinate x and the time t; a is a real coupling parameter, and subscripts t and x denote differentiation. derivatives. System (1) describes the interaction of two orthogonally polarized transverse waves [14], where u and v represent y-polarized and z-polarized transverse waves, respectively, propagating in the x-direction in an xyz Cartesian coordinate system [14]. Note that the coupling here occurs through cubic nonlinear terms. The numerical solution of the above two coupled nonlinear wave equations has attracted more interest in recent years. Interestingly, Ismail has recently solved numerically this system by using Petrov–Galerkin method [15], and also by using collocation method with quintic splines [16]. The split-step Fourier method has been used by Muslu and Erabay [14] to solve the complex mKdV equation which is the basic model of the considered system. Suppose that we are in a physical situation where the linear coupling occurs in the system (1). Under this condition, we obtain the new modified system
ut þ uxxx þ a½ðu2 þ v 2 Þux þ bv x ¼ 0;
v t þ v xxx þ a½ðu2 þ v 2 Þv x þ bux ¼ 0;
ð2Þ
where b is the linear coupling coefficient. In the absence of the effect of v, Eq. (2) reduces to the single mKdV equation and has the soliton solution
uðx; tÞ ¼
rffiffiffi 2
g sec hgðx g2 tÞ; a
ð3Þ
where g is a free parameter. Note that the amplitude of this solitary wave (soliton) is proportional to its speed. In the linear limit (i.e. as a tends to zero), the dispersion relation corresponding to the coupled mKdV Eq. (2) is obtained by assuming iðkxxtÞ solutions in the form u ! aeiðkxxtÞ and v ! be , with a and b being arbitrary constants different from to zero (see, for instance, [7]). As a result, we find 3
3
2
ðx þ k Þðx þ k Þ b2 k ¼ 0:
ð4Þ
This yields two solution branches
xðkÞ ¼ k3 kb:
ð5Þ
Accordingly, the phase velocity is given by
Vp ¼
x
2
¼ k b;
k
ð6Þ
which shows a simple quadratic dependence of the phase velocity on the wave vector k. The mKdV system (2) has the three conservation laws, namely,
I1 ¼
Z
þ1
1 þ1
I2 ¼
Z
u dx;
ð7Þ
ðu2 þ v 2 Þdx
ð8Þ
1
and
I3 ¼
1 2
Z
þ1
1
½aðu2 þ v 2 Þ2 2ðu2x þ v 2x Þ 2buv dx;
ð9Þ
which remain constant in time. In order to prove the first conserved quantity I1 , integrate both sides of the first equation of the system (2) with respect to x from 1 to 1:
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@ @t
Z
þ1
u dx þ 1
Z
þ1
1
ðuxxx þ aðu3 þ v 2 uÞx þ bv x Þdx ¼ 0:
v and all partial derivatives approach to zero as x ! 1, this will lead us to
Now by assuming that u and
@ @t
Z
ð10Þ
þ1
u dx ¼ 0:
ð11Þ
1
Hence the first conserved quantity is obtained. For the second conserved quantity I2 , we multiply the first equation of the system (2) by 2u and the second one by 2v and add them. Then integrating the resulting equation with respect to x from 1 to 1:
@ @t
Z
þ1
ðu2 þ v 2 Þdx þ
Z
1
þ1
2uuxxx þ 2vv xxx þ
1
3a 2 dx ¼ 0 ðu þ v 2 Þ2 þ 2buv 2 x
ð12Þ
and by making use of the vanishing boundary conditions, the second term in Eq. (12) will vanish to end with
@ @t
Z
þ1
ðu2 þ v 2 Þdx ¼ 0:
ð13Þ
1
which give the second conserved quantity. The proof of the third conserved quantity I3 is given in the Appendix A. The mKdV equation possesses a variety of soliton solutions, including traveling wave solutions and propagating ‘‘breather” solitons [17]. In our study, we will focus on the traveling wave solutions of Eq. (2) of the form:
uðx; tÞ ¼ uðx ctÞ uðXÞ;
ð14Þ
v ðx; tÞ ¼ v ðx ctÞ v ðXÞ; where cð> 0Þ is the wave speed and u(X), v(X) are real functions that decrease to zero at infinity. By inserting (14) into (2), one gets the following set of ordinary differential equations:
cuX þ uXXX þ a½ðu2 þ v 2 ÞuX þ bv X ¼ 0; cv X þ v XXX þ a½ðu2 þ v 2 Þv X þ buX ¼ 0:
ð15Þ
By integrating with respect to the space variable X, one obtains
cu þ uXX þ a½ðu2 þ v 2 Þu þ bv ¼ 0; cv þ v XX þ a½ðu2 þ v 2 Þv þ bu ¼ 0:
ð16Þ
By introducing the transformations:
u¼
rffiffiffi c
a
f;
v¼
rffiffiffi c
a
g;
n¼
pffiffiffi cX
ð17Þ
and substituting them in Eq. (16) yields
fnn þ ðf 2 þ g 2 Þf f þ pg ¼ 0; g nn þ ðf 2 þ g 2 Þg g þ pg ¼ 0;
ð18Þ
where the integration constant is taken as zero for the solitary waves case of solution. Eq. (18) has only one combined parameter, p ¼ b=c: The soliton solutions of this system are the symmetric solutions
f ðnÞ ¼ gðnÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 psech 1 pn
ð19Þ
and the antisymmetric soliton solutions:
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi f ðnÞ ¼ gðnÞ ¼ 1 þ psech 1 þ pn :
ð20Þ
We see that the symmetric states exist when p ¼ b=c < 1, and the antisymmetric ones exist when p ¼ b=c > 1: By using the change of variables (14) and (17), the symmetric soliton solutions of Eq. (2) are
uðx; tÞ ¼ v ðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffi cb
a
npffiffiffiffiffiffiffiffiffiffiffi o sech c bðx ctÞ
ð21Þ
and the antisymmetric soliton solutions for all c > 0 are
uðx; tÞ ¼ v ðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffi o cþb sech c þ bðx ctÞ :
a
ð22Þ
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The total energy for these soliton solutions, where u ¼ v for the symmetric and u ¼ v for the antisymmetric soliton solutions:
I2 ¼
Z
þ1
ðu2 þ v 2 Þdx ¼ 2
1
Z
þ1
u2 dx:
ð23Þ
1
By using the solutions (21) and (22), we obtain
pffiffiffiffiffiffiffiffiffiffiffiffi I2 ¼ 4 1 p; K
ð24Þ
pffiffiffi where K ¼ c=a, where the upper sign for the symmetric case and the lower sign for antisymmetric states. 3. Numerical method In this section, we present a numerical solution for (2). The finite difference method is proposed
h nþ1 i h nþ1 i h nþ1 i nþ12 nþ12 nþ12 nþ12 n nþ12 2 2 2 U nþ1 ¼ 0; m U m þ p1 U mþ2 2U mþ1 þ 2U m1 U m2 þ p2 fmþ1 fm1 þ p3 V mþ1 V h nþ1 i h i h i 1 1 1 1 1 1 nþ2 nþ2 nþ2 nþ2 nþ2 nþ2 n nþ12 2 ¼ 0; V nþ1 m V m þ p1 V mþ2 2V mþ1 þ 2V m1 V m2 þ p2 g mþ1 g m1 þ p3 U mþ1 U
ð25Þ ð26Þ
where
f ðu; v Þ ¼ u3 þ uv 2 ;
gðu; v Þ ¼ v 3 þ v u2
n U nþ1 V nþ1 þ V nm nþ1 m þ Um ; Vm 2 ¼ m 2 2 k ak bk p1 ¼ 3 ; p2 ¼ ; p3 ¼ 2h 2h 2h nþ1
Um 2 ¼
ð27Þ
The proposed scheme is a nonlinear one in U nþ1 and V nþ1 m m . The scheme after imposing the Dirichlet boundary condition will produce a block nonlinear penta-diagonal system, where Newton’s method is used to solve this system. Concerning the accuracy the proposed scheme is of second order in space and time. Using the von Neumann stability, the proposed scheme is unconditionally stable in the linearized sense. 4. Numerical results We are going to test the numerical scheme and the behavior of the derived solution using the following problems. 4.1. Symmetric soliton solution In this test, we choose the initial condition as
uðx; 0Þ ¼ v ðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffi o cb sech c bx :
ð28Þ
a
The following parameters are used: h ¼ 0:1; k ¼ 0:1; t ¼ 0; 5; . . . ; 20; a ¼ 1; c ¼ 1; b ¼ 0:5: We calculate the error and the conserved quantities, and we present them in Table 1. We have noticed that the scheme conserved the conserved quantities exactly. For the interaction of two solitons, we choose as the initial conditions
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 b c2 b uðx; 0Þ ¼ sech c1 bðx s1 Þ þ sech c2 bðx s2 Þ
ð29Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 b c2 b v ðx; 0Þ ¼ sech c1 bðx s1 Þ þ sech c2 bðx s2 Þ :
ð30Þ
a
and
a
a
a
Table 1 Symmetric soliton solution. t
L1
I1
I2
0 5 10 15 20
0.0 0.00166 0.00310 0.00453 0.00599
2.82843 2.82843 2.82843 2.82843 2.82843
0.472502 0.472501 0.472499 0.472499 0.472499
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H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
In this test, we choose the set of parameters h ¼ 0:1; k ¼ 0:1; a ¼ 1:0; b ¼ 0:5; c1 ¼ 1:0; c2 ¼ 0:7; s1 ¼ 10; s2 ¼ 10; t ¼ 0; 5; 10; . . . ; 150; the interaction scenario displayed in Fig. 1 and in Fig. 2 we displayed the contours of the test. The interaction is elastic, the two waves leave the interaction region unchanged in shape.
1
u
0.5 0 −0.5
200
160 150
140 120
100
100 80
t
x
50
60 40
0
20 0
−50
Fig. 1. Interaction of two symmetric solitons.
150
100
50
0 −20
0
20
40
60
80
100
120
Fig. 2. The contours of interaction of two symmetric solitons.
140
160
180
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H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
4.2. Antisymmetric soliton solution In this test, we choose the initial condition as
uðx; 0Þ ¼ v ðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffi o cþb sech c þ bx :
ð31Þ
a
For this test, we choose the set of parameters: h ¼ 0:1; k ¼ 0:1; t ¼ 0; 5; . . . ; 20; a ¼ 1; b ¼ 0:5; c ¼ 1:0. We display in Table 2 the error and the conserved quantities. Table 2 indicates how the numerical method conserved the conserved quantities exactly. For the interaction of two solitons, we choose the following initial conditions:
uðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 þ b c2 þ b sech c1 þ bðx s1 Þ þ sech c2 bðx s2 Þ
a
ð32Þ
a
and
v ðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 þ b c2 þ b sech c1 þ bðx s1 Þ sech c2 þ bðx s2 Þ :
a
ð33Þ
a
In this test, the following parameters: h ¼ 0:1; k ¼ 0:1; a ¼ 1:0; b ¼ 0:5; c1 ¼ 1:0; c2 ¼ 0:1; s1 ¼ 10; s2 ¼ 10; :are selected. The interaction scenario displayed in Fig. 3 and the contours of the same test in Fig. 4. The elastic collision is observed, the two waves left the interaction without any change in shapes.
Table 2 Antisymmetric soliton solution. t
L1
I1
I2
0 5 10 15 20
0.0000 0.0127 0.0241 0.03615 0.04776
4.89898 4.89896 4.89896 4.89896 4.89896
2.46653 2.46667 2.46665 2.46665 2.46665
0.2 0 −0.2
v
−0.4 −0.6 −0.8 −1 −1.2 −1.4 70 60 50 40
t
50
30 20
0
10 0
−50
Fig. 3. Interaction of two antisymmetric solitons.
x
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60
50
40
30
20
10
0 −50
−40
−30
−20
−10
0
10
20
30
40
50
Fig. 4. Contours of two antisymmetric solitons.
1.2 1 0.8
u
0.6 0.4 0.2 0 −0.2 45 40 35 30
t
25
50
20 15
0
10 5 0
x
−50 Fig. 5. Birth of solitons U.
5. Asymmetric states (birth of solitons) In addition to the symmetric and antisymmetric soliton solutions, there exists another type of soliton solutions called asymmetric states with unequal amplitudes (i.e. uðx; tÞ–v ðx; tÞ) which are not known in explicit form. These solitons in general will bifurcate from the symmetric and antisymmetric states.
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0.5
v
0
−0.5
−1 50 40 30
t
20 50 10
0 0
x
−50 Fig. 6. Birth of solitons V.
Our aim here is to compute the evolution of the asymmetric soliton solutions by solving the considered system (2) numerically. The initial conditions for the coupled pair of mKdV equations are taken in the form 2
uðx; 0Þ ¼ Aecx ;
v ðx; 0Þ ¼ Becx
2
ð34Þ
:
The following parameters are selected: h ¼ 0:1; k ¼ 0:1; a ¼ 1:0; b ¼ 0:5; A ¼ 0:5; B ¼ 0:5; c ¼ 0:01: In Figs. 5 and 6, we display the numerical solutions of u and v, respectively, we show the evolution of the initial Gaussian pulses (34), and how these pulses break up into a train of three solitary waves as the time evolved from t ¼ 0; 5; 10; . . . ; 45: 6. Conclusion In this work, we have derived the exact symmetric and antisymmetric soliton solutions for a coupled pair of mKdV equations in the case where both of the linear and nonlinear coupling terms are present. Conditions for the existence of such solutions have also been reported. Numerical tests for single and interaction of two solitons are given. We have shown the existence of asymmetric soliton solutions numerically by using the Gaussian pulses as initial conditions. Acknowledgements The authors are very grateful to the referees for their valuable comments and suggestions, which have improved the paper considerably. Appendix A.
I3 ¼
1 2
In this appendix, we shall prove that I3 is a conserved quantity. Consider
Z
þ1
1
½aðu2 þ v 2 Þ2 2ðu2x þ v 2x Þ 2buv dx:
ðA:1Þ
We begin by differentiating Eq. (A.1) with respect to the time variable t:
Z þ1 @I3 1 ½aðu2 þ v 2 Þ2 2ðu2x þ v 2x Þ 2buv dx ¼ @t 2 1 @t Z þ1 ¼ f2aðu2 þ v 2 Þðuut þ vv t Þ 2ðux uxt þ v x v xt Þ 2bðut v þ uv t Þgdx 1
which is equivalent to
ðA:2Þ
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H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
@I3 ¼ @t
Z
þ1
ut f2auðu2 þ v 2 Þ 2uxx bv gdx þ
1
Now by assuming that u and will lead us to
Z
þ1
v t f2av ðu2 þ v 2 Þ 2v xx bugdx:
ðA:3Þ
1
v and all partial derivatives approach zero as x ! 1 for solitary wave case of solutions, this
@I3 ¼ 0: @t
ðA:4Þ
Hence the third conserved quantity is obtained. References [1] Andrey I. Maimistov, Completely integrable models of nonlinear optics, Pramana J. Phys. 57 (5&6) (2001) 953. [2] A. Ankiewicz, N. Akhmediev, Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts, IEE Proc. Optoelectron. 150 (6) (2003) 519–526. [3] Min Du, Andrew K. Chan, Charles K. Chui, A novel approach to solving the nonlinear Schrödinger equation by the coupled amplitude-phase formulation, IEEE J. Quantum Electron. 3 (1) (1995) 177. [4] H. Triki, A. El Akrmi, M.K. Rabia, Soliton solutions in three linearly coupled Korteweg-de Vries equations, Opt. Commun. 201 (2002) 448. [5] G. Gottwald, R. Grimshaw, B. Malomed, Parametric envelope solitons in coupled Korteweg-de Vries equations, Phys. Lett. A 227 (1997) 47–54. [6] Yuri S. Kivshar, Dmitry E. Pelinovsky, Thierry Cretegny, Michel Peyrard, Internal modes of solitary waves, Phys. Rev. Lett. 80 (23) (1998) 5032–5035. [7] P.G. Grazini, Solitons, London Mathematical Society Lecture Note Series 85, First Published 1985, Cambridge University Press, 1985, pp. 4–23. [8] Jianke Yang, Complete eigenfunctions of linearized integrable equations expanded around a soliton solution, J. Math. Phys. 41 (9) (2000) 6614. [9] R.F. Rodriguez, J.A. Reyes, A. Espinosa-Ceron, J. Fujioka, B.A. Malomed, Standard and embedded solitons in nematic optical fibers, Phys. Rev. E 68 (2003) 036606. [10] J.M. Soto-Crespo, Nail Akhmediev, Stability of the soliton states in a nonlinear fiber coupler, Phys. Rev. E 48 (6) (1993) 4710–4715. [11] Nail Akhmediev, Adrian Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett. 70 (16) (1993) 2395– 2398. [12] A.V. Buryak, N.N. Akhmediev, Stationary pulse propagation in N-core nonlinear fiber arrays, IEEE J. Quantum Electron. 31 (4) (1995) 682–688. [13] Nail Akhmediev, J.M. Soto-Crespo, Propagation dynamics of ultrashort pulses in nonlinear fiber couplers, Phys. Rev. E 49 (5) (1994) 4519–4529. [14] G.M. Muslu, H.A. Erbay, A split-step Fourier method for the complex modified Korteweg-de Vries equation, Comput. Math. Appl. 45 (2003) 503–514. [15] M.S. Ismail, Numerical solution of complex modified Korteweg-de Vries equation by Petrov–Galerkin method, Appl. Math. Comput. 202 (2008) 520– 531. [16] M.S. Ismail, Numerical solution of complex modified Korteweg-de Vries equation by collocation method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 749–759. [17] Simon Clarke, Roger Grimshaw, Peter Miller, Efim Pelinovsky, Tatiana Talipova, On the generation of solitons and breathers in the modified Kortewegde Vries equation, Chaos 10 (2) (2000) 383.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Solitary wave solutions for a coupled pair of mKdV equations H. Triki a, M.S. Ismail b,* a b
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000 Annaba, Algeria Department of Mathematics, College of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
Keywords: Modified Korteweg-de Vries equation Soliton solution Finite difference method
a b s t r a c t Propagation of weakly nonlinear long waves is studied within the framework of a system of two coupled modified Korteweg-de Vries equations. We investigate analytically and numerically the various families of soliton states for the considered model. By scaling the functions and variables we find that the resulting coupled pair of equations has only one combined parameter. This parameter depends on the wave speed and the coupling coefficient. Explicit analytical expressions for both of the symmetric and antisymmetric states are determined. Numerical method is derived to solve the proposed system, many numerical tests have been conducted to study the behavior of the solution, and the existence of the asymmetric soliton states is displayed numerically. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Studies of nonlinear wave models have drawn considerable attentions in recent years. This can be considered as one of the fundamental objects of nature [1]. The most famous examples are the nonlinear Schrödinger [2,3], the Korteweg-de Vries [4,5] and the sine-Gordon [6] equations. The application of such models cover various branches of physics like nonlinear optics, fluid dynamics, condensed matter physics and so on. The solutions of these nonlinear evolution equations are known to be in the form of soliton envelope. The existence of soliton states implies perfect balance between nonlinearity and dispersion effects which usually requires specific conditions. It should be noted that the nonlinear optics is the field, where all soliton features are exhibited to a great extent [1]. In 1967, Zabusky [7] showed how the modified Korteweg-de Vries (mKdV) equation may model the oscillations of a lattice of particles connected by nonlinear springs as the Fermi-Pasta-Ulam model does. Further, the evolution of internal waves at the interface of two layers of equal depth is also described by the mKdV equation [8]. The mKdV equation and some other nonlinear evolution equations are gradually putting together to form a list of soliton equations, which are usually to be found in nonlinear optics theory. In a recent study [9] shows that the propagation of optical pulses in nematic optical fibers is described by a complex modified KdV equation when we go beyond the usual weakly nonlinear limit of a Kerr medium. Wave propagation and switching in a nonlinear multicore fiber arrays which is described by coupled nonlinear Schrödinger equations have appeared in a number of papers [10–13]. In particular, Akhmediev and Ankiewicz [11] have made an interesting study of the various families of soliton states dynamics in nonlinear fiber couplers. The authors have found that in addition to the symmetric and antisymmetric soliton states, there exists an other family of asymmetric soliton states which bifurcate from the first ones. The aim of this paper is to find various soliton solutions for a model of two coupled mKdV equations, governing the mutual interaction in addition to nonlinear self-interaction of two solitary waves in a mKdV system. A numerical method
* Corresponding author. E-mail address: [email protected] (M.S. Ismail). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.06.047
H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
1541
is presented,which is implicit, unconditionally stable, and of second order accuracy in space and second order accuracy in time. This method is used to study the behavior of the solutions. The paper is organized as follows. In Section 2, we present the considering model of two coupled mKdV equations, and then we derive the symmetric and antisymmetric soliton solutions. In Section 3, we present a numerical method for solving the coupled systems. Numerical results with various initial conditions are reported in Section 4. Concluding remarks are given in Section 5.
2. Model and equations We consider the coupled pair of mKdV equations [14]:
ut þ uxxx þ a½ðu2 þ v 2 Þux ¼ 0;
v t þ v xxx þ a½ðu2 þ v 2 Þv x ¼ 0;
ð1Þ
where u and v are real functions of the spatial coordinate x and the time t; a is a real coupling parameter, and subscripts t and x denote differentiation. derivatives. System (1) describes the interaction of two orthogonally polarized transverse waves [14], where u and v represent y-polarized and z-polarized transverse waves, respectively, propagating in the x-direction in an xyz Cartesian coordinate system [14]. Note that the coupling here occurs through cubic nonlinear terms. The numerical solution of the above two coupled nonlinear wave equations has attracted more interest in recent years. Interestingly, Ismail has recently solved numerically this system by using Petrov–Galerkin method [15], and also by using collocation method with quintic splines [16]. The split-step Fourier method has been used by Muslu and Erabay [14] to solve the complex mKdV equation which is the basic model of the considered system. Suppose that we are in a physical situation where the linear coupling occurs in the system (1). Under this condition, we obtain the new modified system
ut þ uxxx þ a½ðu2 þ v 2 Þux þ bv x ¼ 0;
v t þ v xxx þ a½ðu2 þ v 2 Þv x þ bux ¼ 0;
ð2Þ
where b is the linear coupling coefficient. In the absence of the effect of v, Eq. (2) reduces to the single mKdV equation and has the soliton solution
uðx; tÞ ¼
rffiffiffi 2
g sec hgðx g2 tÞ; a
ð3Þ
where g is a free parameter. Note that the amplitude of this solitary wave (soliton) is proportional to its speed. In the linear limit (i.e. as a tends to zero), the dispersion relation corresponding to the coupled mKdV Eq. (2) is obtained by assuming iðkxxtÞ solutions in the form u ! aeiðkxxtÞ and v ! be , with a and b being arbitrary constants different from to zero (see, for instance, [7]). As a result, we find 3
3
2
ðx þ k Þðx þ k Þ b2 k ¼ 0:
ð4Þ
This yields two solution branches
xðkÞ ¼ k3 kb:
ð5Þ
Accordingly, the phase velocity is given by
Vp ¼
x
2
¼ k b;
k
ð6Þ
which shows a simple quadratic dependence of the phase velocity on the wave vector k. The mKdV system (2) has the three conservation laws, namely,
I1 ¼
Z
þ1
1 þ1
I2 ¼
Z
u dx;
ð7Þ
ðu2 þ v 2 Þdx
ð8Þ
1
and
I3 ¼
1 2
Z
þ1
1
½aðu2 þ v 2 Þ2 2ðu2x þ v 2x Þ 2buv dx;
ð9Þ
which remain constant in time. In order to prove the first conserved quantity I1 , integrate both sides of the first equation of the system (2) with respect to x from 1 to 1:
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@ @t
Z
þ1
u dx þ 1
Z
þ1
1
ðuxxx þ aðu3 þ v 2 uÞx þ bv x Þdx ¼ 0:
v and all partial derivatives approach to zero as x ! 1, this will lead us to
Now by assuming that u and
@ @t
Z
ð10Þ
þ1
u dx ¼ 0:
ð11Þ
1
Hence the first conserved quantity is obtained. For the second conserved quantity I2 , we multiply the first equation of the system (2) by 2u and the second one by 2v and add them. Then integrating the resulting equation with respect to x from 1 to 1:
@ @t
Z
þ1
ðu2 þ v 2 Þdx þ
Z
1
þ1
2uuxxx þ 2vv xxx þ
1
3a 2 dx ¼ 0 ðu þ v 2 Þ2 þ 2buv 2 x
ð12Þ
and by making use of the vanishing boundary conditions, the second term in Eq. (12) will vanish to end with
@ @t
Z
þ1
ðu2 þ v 2 Þdx ¼ 0:
ð13Þ
1
which give the second conserved quantity. The proof of the third conserved quantity I3 is given in the Appendix A. The mKdV equation possesses a variety of soliton solutions, including traveling wave solutions and propagating ‘‘breather” solitons [17]. In our study, we will focus on the traveling wave solutions of Eq. (2) of the form:
uðx; tÞ ¼ uðx ctÞ uðXÞ;
ð14Þ
v ðx; tÞ ¼ v ðx ctÞ v ðXÞ; where cð> 0Þ is the wave speed and u(X), v(X) are real functions that decrease to zero at infinity. By inserting (14) into (2), one gets the following set of ordinary differential equations:
cuX þ uXXX þ a½ðu2 þ v 2 ÞuX þ bv X ¼ 0; cv X þ v XXX þ a½ðu2 þ v 2 Þv X þ buX ¼ 0:
ð15Þ
By integrating with respect to the space variable X, one obtains
cu þ uXX þ a½ðu2 þ v 2 Þu þ bv ¼ 0; cv þ v XX þ a½ðu2 þ v 2 Þv þ bu ¼ 0:
ð16Þ
By introducing the transformations:
u¼
rffiffiffi c
a
f;
v¼
rffiffiffi c
a
g;
n¼
pffiffiffi cX
ð17Þ
and substituting them in Eq. (16) yields
fnn þ ðf 2 þ g 2 Þf f þ pg ¼ 0; g nn þ ðf 2 þ g 2 Þg g þ pg ¼ 0;
ð18Þ
where the integration constant is taken as zero for the solitary waves case of solution. Eq. (18) has only one combined parameter, p ¼ b=c: The soliton solutions of this system are the symmetric solutions
f ðnÞ ¼ gðnÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 1 psech 1 pn
ð19Þ
and the antisymmetric soliton solutions:
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi f ðnÞ ¼ gðnÞ ¼ 1 þ psech 1 þ pn :
ð20Þ
We see that the symmetric states exist when p ¼ b=c < 1, and the antisymmetric ones exist when p ¼ b=c > 1: By using the change of variables (14) and (17), the symmetric soliton solutions of Eq. (2) are
uðx; tÞ ¼ v ðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffi cb
a
npffiffiffiffiffiffiffiffiffiffiffi o sech c bðx ctÞ
ð21Þ
and the antisymmetric soliton solutions for all c > 0 are
uðx; tÞ ¼ v ðx; tÞ ¼
rffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffi o cþb sech c þ bðx ctÞ :
a
ð22Þ
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The total energy for these soliton solutions, where u ¼ v for the symmetric and u ¼ v for the antisymmetric soliton solutions:
I2 ¼
Z
þ1
ðu2 þ v 2 Þdx ¼ 2
1
Z
þ1
u2 dx:
ð23Þ
1
By using the solutions (21) and (22), we obtain
pffiffiffiffiffiffiffiffiffiffiffiffi I2 ¼ 4 1 p; K
ð24Þ
pffiffiffi where K ¼ c=a, where the upper sign for the symmetric case and the lower sign for antisymmetric states. 3. Numerical method In this section, we present a numerical solution for (2). The finite difference method is proposed
h nþ1 i h nþ1 i h nþ1 i nþ12 nþ12 nþ12 nþ12 n nþ12 2 2 2 U nþ1 ¼ 0; m U m þ p1 U mþ2 2U mþ1 þ 2U m1 U m2 þ p2 fmþ1 fm1 þ p3 V mþ1 V h nþ1 i h i h i 1 1 1 1 1 1 nþ2 nþ2 nþ2 nþ2 nþ2 nþ2 n nþ12 2 ¼ 0; V nþ1 m V m þ p1 V mþ2 2V mþ1 þ 2V m1 V m2 þ p2 g mþ1 g m1 þ p3 U mþ1 U
ð25Þ ð26Þ
where
f ðu; v Þ ¼ u3 þ uv 2 ;
gðu; v Þ ¼ v 3 þ v u2
n U nþ1 V nþ1 þ V nm nþ1 m þ Um ; Vm 2 ¼ m 2 2 k ak bk p1 ¼ 3 ; p2 ¼ ; p3 ¼ 2h 2h 2h nþ1
Um 2 ¼
ð27Þ
The proposed scheme is a nonlinear one in U nþ1 and V nþ1 m m . The scheme after imposing the Dirichlet boundary condition will produce a block nonlinear penta-diagonal system, where Newton’s method is used to solve this system. Concerning the accuracy the proposed scheme is of second order in space and time. Using the von Neumann stability, the proposed scheme is unconditionally stable in the linearized sense. 4. Numerical results We are going to test the numerical scheme and the behavior of the derived solution using the following problems. 4.1. Symmetric soliton solution In this test, we choose the initial condition as
uðx; 0Þ ¼ v ðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffi o cb sech c bx :
ð28Þ
a
The following parameters are used: h ¼ 0:1; k ¼ 0:1; t ¼ 0; 5; . . . ; 20; a ¼ 1; c ¼ 1; b ¼ 0:5: We calculate the error and the conserved quantities, and we present them in Table 1. We have noticed that the scheme conserved the conserved quantities exactly. For the interaction of two solitons, we choose as the initial conditions
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 b c2 b uðx; 0Þ ¼ sech c1 bðx s1 Þ þ sech c2 bðx s2 Þ
ð29Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 b c2 b v ðx; 0Þ ¼ sech c1 bðx s1 Þ þ sech c2 bðx s2 Þ :
ð30Þ
a
and
a
a
a
Table 1 Symmetric soliton solution. t
L1
I1
I2
0 5 10 15 20
0.0 0.00166 0.00310 0.00453 0.00599
2.82843 2.82843 2.82843 2.82843 2.82843
0.472502 0.472501 0.472499 0.472499 0.472499
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H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
In this test, we choose the set of parameters h ¼ 0:1; k ¼ 0:1; a ¼ 1:0; b ¼ 0:5; c1 ¼ 1:0; c2 ¼ 0:7; s1 ¼ 10; s2 ¼ 10; t ¼ 0; 5; 10; . . . ; 150; the interaction scenario displayed in Fig. 1 and in Fig. 2 we displayed the contours of the test. The interaction is elastic, the two waves leave the interaction region unchanged in shape.
1
u
0.5 0 −0.5
200
160 150
140 120
100
100 80
t
x
50
60 40
0
20 0
−50
Fig. 1. Interaction of two symmetric solitons.
150
100
50
0 −20
0
20
40
60
80
100
120
Fig. 2. The contours of interaction of two symmetric solitons.
140
160
180
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4.2. Antisymmetric soliton solution In this test, we choose the initial condition as
uðx; 0Þ ¼ v ðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffi o cþb sech c þ bx :
ð31Þ
a
For this test, we choose the set of parameters: h ¼ 0:1; k ¼ 0:1; t ¼ 0; 5; . . . ; 20; a ¼ 1; b ¼ 0:5; c ¼ 1:0. We display in Table 2 the error and the conserved quantities. Table 2 indicates how the numerical method conserved the conserved quantities exactly. For the interaction of two solitons, we choose the following initial conditions:
uðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 þ b c2 þ b sech c1 þ bðx s1 Þ þ sech c2 bðx s2 Þ
a
ð32Þ
a
and
v ðx; 0Þ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o rffiffiffiffiffiffiffiffiffiffiffiffiffi npffiffiffiffiffiffiffiffiffiffiffiffiffi o c1 þ b c2 þ b sech c1 þ bðx s1 Þ sech c2 þ bðx s2 Þ :
a
ð33Þ
a
In this test, the following parameters: h ¼ 0:1; k ¼ 0:1; a ¼ 1:0; b ¼ 0:5; c1 ¼ 1:0; c2 ¼ 0:1; s1 ¼ 10; s2 ¼ 10; :are selected. The interaction scenario displayed in Fig. 3 and the contours of the same test in Fig. 4. The elastic collision is observed, the two waves left the interaction without any change in shapes.
Table 2 Antisymmetric soliton solution. t
L1
I1
I2
0 5 10 15 20
0.0000 0.0127 0.0241 0.03615 0.04776
4.89898 4.89896 4.89896 4.89896 4.89896
2.46653 2.46667 2.46665 2.46665 2.46665
0.2 0 −0.2
v
−0.4 −0.6 −0.8 −1 −1.2 −1.4 70 60 50 40
t
50
30 20
0
10 0
−50
Fig. 3. Interaction of two antisymmetric solitons.
x
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60
50
40
30
20
10
0 −50
−40
−30
−20
−10
0
10
20
30
40
50
Fig. 4. Contours of two antisymmetric solitons.
1.2 1 0.8
u
0.6 0.4 0.2 0 −0.2 45 40 35 30
t
25
50
20 15
0
10 5 0
x
−50 Fig. 5. Birth of solitons U.
5. Asymmetric states (birth of solitons) In addition to the symmetric and antisymmetric soliton solutions, there exists another type of soliton solutions called asymmetric states with unequal amplitudes (i.e. uðx; tÞ–v ðx; tÞ) which are not known in explicit form. These solitons in general will bifurcate from the symmetric and antisymmetric states.
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0.5
v
0
−0.5
−1 50 40 30
t
20 50 10
0 0
x
−50 Fig. 6. Birth of solitons V.
Our aim here is to compute the evolution of the asymmetric soliton solutions by solving the considered system (2) numerically. The initial conditions for the coupled pair of mKdV equations are taken in the form 2
uðx; 0Þ ¼ Aecx ;
v ðx; 0Þ ¼ Becx
2
ð34Þ
:
The following parameters are selected: h ¼ 0:1; k ¼ 0:1; a ¼ 1:0; b ¼ 0:5; A ¼ 0:5; B ¼ 0:5; c ¼ 0:01: In Figs. 5 and 6, we display the numerical solutions of u and v, respectively, we show the evolution of the initial Gaussian pulses (34), and how these pulses break up into a train of three solitary waves as the time evolved from t ¼ 0; 5; 10; . . . ; 45: 6. Conclusion In this work, we have derived the exact symmetric and antisymmetric soliton solutions for a coupled pair of mKdV equations in the case where both of the linear and nonlinear coupling terms are present. Conditions for the existence of such solutions have also been reported. Numerical tests for single and interaction of two solitons are given. We have shown the existence of asymmetric soliton solutions numerically by using the Gaussian pulses as initial conditions. Acknowledgements The authors are very grateful to the referees for their valuable comments and suggestions, which have improved the paper considerably. Appendix A.
I3 ¼
1 2
In this appendix, we shall prove that I3 is a conserved quantity. Consider
Z
þ1
1
½aðu2 þ v 2 Þ2 2ðu2x þ v 2x Þ 2buv dx:
ðA:1Þ
We begin by differentiating Eq. (A.1) with respect to the time variable t:
Z þ1 @I3 1 ½aðu2 þ v 2 Þ2 2ðu2x þ v 2x Þ 2buv dx ¼ @t 2 1 @t Z þ1 ¼ f2aðu2 þ v 2 Þðuut þ vv t Þ 2ðux uxt þ v x v xt Þ 2bðut v þ uv t Þgdx 1
which is equivalent to
ðA:2Þ
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H. Triki, M.S. Ismail / Applied Mathematics and Computation 217 (2010) 1540–1548
@I3 ¼ @t
Z
þ1
ut f2auðu2 þ v 2 Þ 2uxx bv gdx þ
1
Now by assuming that u and will lead us to
Z
þ1
v t f2av ðu2 þ v 2 Þ 2v xx bugdx:
ðA:3Þ
1
v and all partial derivatives approach zero as x ! 1 for solitary wave case of solutions, this
@I3 ¼ 0: @t
ðA:4Þ
Hence the third conserved quantity is obtained. References [1] Andrey I. Maimistov, Completely integrable models of nonlinear optics, Pramana J. Phys. 57 (5&6) (2001) 953. [2] A. Ankiewicz, N. Akhmediev, Stability analysis for solitons in planar waveguides, fibres and couplers using Hamiltonian concepts, IEE Proc. Optoelectron. 150 (6) (2003) 519–526. [3] Min Du, Andrew K. Chan, Charles K. Chui, A novel approach to solving the nonlinear Schrödinger equation by the coupled amplitude-phase formulation, IEEE J. Quantum Electron. 3 (1) (1995) 177. [4] H. Triki, A. El Akrmi, M.K. Rabia, Soliton solutions in three linearly coupled Korteweg-de Vries equations, Opt. Commun. 201 (2002) 448. [5] G. Gottwald, R. Grimshaw, B. Malomed, Parametric envelope solitons in coupled Korteweg-de Vries equations, Phys. Lett. A 227 (1997) 47–54. [6] Yuri S. Kivshar, Dmitry E. Pelinovsky, Thierry Cretegny, Michel Peyrard, Internal modes of solitary waves, Phys. Rev. Lett. 80 (23) (1998) 5032–5035. [7] P.G. Grazini, Solitons, London Mathematical Society Lecture Note Series 85, First Published 1985, Cambridge University Press, 1985, pp. 4–23. [8] Jianke Yang, Complete eigenfunctions of linearized integrable equations expanded around a soliton solution, J. Math. Phys. 41 (9) (2000) 6614. [9] R.F. Rodriguez, J.A. Reyes, A. Espinosa-Ceron, J. Fujioka, B.A. Malomed, Standard and embedded solitons in nematic optical fibers, Phys. Rev. E 68 (2003) 036606. [10] J.M. Soto-Crespo, Nail Akhmediev, Stability of the soliton states in a nonlinear fiber coupler, Phys. Rev. E 48 (6) (1993) 4710–4715. [11] Nail Akhmediev, Adrian Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett. 70 (16) (1993) 2395– 2398. [12] A.V. Buryak, N.N. Akhmediev, Stationary pulse propagation in N-core nonlinear fiber arrays, IEEE J. Quantum Electron. 31 (4) (1995) 682–688. [13] Nail Akhmediev, J.M. Soto-Crespo, Propagation dynamics of ultrashort pulses in nonlinear fiber couplers, Phys. Rev. E 49 (5) (1994) 4519–4529. [14] G.M. Muslu, H.A. Erbay, A split-step Fourier method for the complex modified Korteweg-de Vries equation, Comput. Math. Appl. 45 (2003) 503–514. [15] M.S. Ismail, Numerical solution of complex modified Korteweg-de Vries equation by Petrov–Galerkin method, Appl. Math. Comput. 202 (2008) 520– 531. [16] M.S. Ismail, Numerical solution of complex modified Korteweg-de Vries equation by collocation method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 749–759. [17] Simon Clarke, Roger Grimshaw, Peter Miller, Efim Pelinovsky, Tatiana Talipova, On the generation of solitons and breathers in the modified Kortewegde Vries equation, Chaos 10 (2) (2000) 383.