Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric and Hyperbolic Function Methods.

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.14(01) No.,pp Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using Trigonometric and Hyperbolic Function Methods Marwan Alquran, Roba Al-omary Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 110, Jordan (Received 1 September 011, accepted 16 May 01) Abstract: In this paper, the Hirota-Satsuma coupled KdV equation and its generalized form are discussed. Trigonometric and hyperbolic function methods such as sine-cosine method, rational sine-cosine method, rational sinh-cosh method and rational tanh-coth method, are used for analytical treatment of these systems. By means of these methods, we have the advantage of reducing the nonlinear problem to a system of algebraic equations that can be solved by any computerized packages. The proposed methods are straightforward, concise and effective, and can be applied to other nonlinear equations and evolutionary systems which arise in mathematical physics. Keywords: Trigonometric-function method; Hyperbolic-function method; generalized Hirota-Satsuma KdV system 1 Introduction Many models in physics are described by nonlinear partial differential equations (PDEs). To understand the nature of such applications, many researchers have stressed their goals to search for explicit solutions. Many powerful methods, such as Hirota bilinear method [1, ], the rational sine-cosine function method [6], the Tanh method [9], the Tanh-Coth method [1], the sine-cosine method [7, 10], the extended Tanh method [4, 1], and many other techniques were used to investigate these types of equations and obtained interesting class of solutions: solitons, kinks and periodic. Solitons are the solutions in the form sech and sech p, the graph of soliton is a wave that goes up only. It is not like periodic solutions sine, cosine, etc, as in trigonometric functions, that goes above and below the horizontal. Kink is also called a soliton; it is in the form tanh not tanh. In kink the limit as x, the answer is a nonzero constant, unlike solitons where the limit goes to 0. In this work, we consider the Hirota-Satsuma coupled KdV equation and the generalized Hirota-Satsuma coupled KdV u t = 1 4 u xxx + uu x 6vv x v t = 1 v xxx uv x, (1) u t = 1 u xxx uu x + (vw) x v t = v xxx + uv x w t = w xxx + uw x. () Hirota and Satsuma [11] presented equation (1) which describes an interaction of two long waves with different dispersion relations. Wu et al. [14] introduced equation () which can be obtained from the four-reduction spectral problem with three potentials. In the literature, Dianchen et al. [5] obtained combined formal solitary wave solutions and combined formal periodic wave solutions to equation (1) by using new improved projective Riccati equations method. Based on the Corresponding author. address: marwan04@just.edu.jo Copyright c World Academic Press, World Academic Union IJNS /650

2 M. Alquran, R. Al-omary: Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using 151 Hirota method and the perturbation technique, the N-soliton solution of equation () is obtained by WU Lian-Ping et al. [1]. The extended tanh method was used in investigating the soliton solutions of (1) and (), see [4] and [1]. Finally, the Homotopy perturbation method was applied [8] to handle (). This paper is organized as follows: In Section, we give a description of the proposed methods. In Section, we apply the methods to equation (1). In Section 4, we apply the methods to equation (). Trigonometric and hyperbolic function methods It is appropriate to assume different ansatzes that involve hyperbolic and trigonometric functions. Some of these ansatzes provide real solutions, whereas others do not. A criterion to specify which ansatze works for real solutions cannot be determined. Only direct substitution can show the existence of real solutions for some ansatzes. In what follows, we present the main steps of such ansatzes. Since we restrict our attention to traveling waves, we use the transformation u(x, t) = u(ζ), where the wave variable ζ = x ct, converts the the nonlinear PDE to an equivalent ODE..1 The sine-cosine method (Ansatze I) The sine-cosine algorithm admits the use of the ansatze [, 10] u(x, t) = λ cos γ (µζ), ζ π µ, () and the ansatze, u(x, t) = λ sin γ (µζ), ζ π µ, (4) where λ, c, µ and γ are parameters that will be determined later. Substituting () or (4) into the reduced ODE gives a polynomial equation of cosine or sine terms. We then collect the coefficients of the resulting triangle functions and set them to zeros to get a system of algebraic equations among the unknowns λ, c, µ, and γ. The problem is now completely reduced to an algebraic one. Having determined λ, c, µ and γ by algebraic calculations or by using any software, the solutions proposed in () and in (4) follow immediately.. The second rational sine-cosine function method (Ansatze II) The second rational triangle functions method can be expressed in the form [] u(x, t) = a 0 + b 0 sin(µζ) 1 + a 1 sin(µζ) (5) and where a 0, a 1, b 0, and µ are parameters that will be determined. u(x, t) = a 0 + b 0 cos(µζ) 1 + a 1 cos(µζ), (6). The first rational hyperbolic function method (Ansatze III) The first rational hyperbolic function method has the form [] u(x, t) = a 1 + λf(µζ), (7) where f(µζ) takes any of the hyperbolic functions sech, csch, tanh, coth. The approach is simply used by applying the equation, setting the coefficients of the resulting hyperbolic functions to zero, and solving the resulting equations to determine the parameters a, λ and µ. IJNS homepage:

3 15 International Journal of Nonlinear Science, Vol.14(01), No., pp The second rational hyperbolic function method (Ansatze IV) The second rational hyperbolic function method has the form [] where f(µζ) takes any of the hyperbolic functions. u(x, t) = f(µζ) 1 + λf(µζ), (8) The coupled Hirota Satsuma KdV equation The Hirota Satsuma system is given by u t = 1 4 u xxx + uu x 6vv x, v t = 1 v xxx uv x. (9) Using the wave variable ζ = x ct transforms (9) into the ODEs cu u + u v = 0 (10) and Substituting (11) into (10) gives u = v v c. (11) 0 = (v ) v (5) + v v v (4) + v (v () ) 8c(v ) v () (v ) v () + 1c (v ) 7(v ) v. (1) First, we solve (1) by using the second rational sine method; Ansatze (III). Substituting (5) into (1) results in 0 = 7a 0 1c 8cµ + 48a 1cµ µ 4 1a 1µ 4 + sin(µζ)(144a 0 b 0 4a 1 c + a 1 cµ + 4a 1 µ 4 ) + sin (µζ)(7b 0 1a 1c 8a 1cµ a 1µ 4 ). (1) The above equation is satisfied only if the following system of algebraic equations hold and c = 1, µ = 1 Solving this system gives 0 = 7a a 1 1a 1 0 = 144a 0 b 0 + 1a 1 0 = 7b 0 1a 1. (14) Thus, the solution of (9) is a 0 = , b 0 = ± 4 4, a 1 = ± 7. (15) Substituting (16) into (11), we get v 1 (x, t) = 1 4 ± 5 4 sin(x t) 1 ±. (16) 15 sin(x t) 7 u 1 (x, t) = 7 ( cos((x t))) 4 ( sin(x t) ). (17) IJNS for contribution: editor@nonlinearscience.org.uk

4 M. Alquran, R. Al-omary: Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using 15 Now, we apply the second rational cosine method. Substituting (6) into (1) gives the same system (14). Therefore, the solution of (9) is Substituting (18) into (11), we get v (x, t) = 1 4 ± 5 4 cos(x t) 1 ±. (18) 15 cos(x t) 7 u (x, t) = 7 ( cos((x t))) 4 ( cos(x t) ). (19) Second, we apply the cosine method Ansatze (I). Substituting () into (1) gives 0 = 8µ 4 + 6γµ 4 + 7γ µ 4 6γ µ 4 + γ 4 µ 4 + cos (µζ)( 16cµ + 4cγµ 8cγ µ + 8µ 4 + 8γ µ + 6γ µ 4 γ 4 µ 4 1γµ 4 ) + cos 4 (µζ)(1c + 8cγ µ + γ 4 µ 4 ) 7λ cos 4+γ (µζ). (0) The above equation is satisfied only if the following system of algebraic equations holds and c = 1 0 = 8 + 6γ + 7γ 6γ + γ 4 0 = 16µ + 4γµ 8γ µ + 8µ 4 + 8γ µ + 6γ µ 4 γ 4 µ 4 1γµ 4 7λ 0 = 1 + 8γ µ + γ 4 µ 4 0 = + γ. (1) Solving the above system gives two solution sets λ =, µ = ±i, γ = 1, λ = 10, µ = ±i 6, γ = 1. () Thus, the solutions of (9) are v (x, t) = ± sech( (x t)), v 4 (x, t) = ± 10 sech( 6(x t)). () Substituting () into (11) gives u (x, t) = sech ( (x t)), u 4 (x, t) = 6 sech ( 6(x t)). (4) By the sine method, substituting (4) into (1) gives the same system (1). Therefore, two more solutions follow v 5 (x, t) = ±i csch( (x t)), v 6 (x, t) = ±i 10 csch( 6(x t)). (5) Substituting (5) into (11) gives u 5 (x, t) = csch ( (x t)), u 6 (x, t) = 6 csch ( 6(x t)). (6) Third, we apply Ansatze (IV) with f(ζ) = sech (ζ). Substituting (8) into (1) gives 0 = 7 + 1c λ 8cλ µ + 48cµ + 1µ 4 + λ µ 4 + cosh(µζ)(4λ c + cλµ 4λµ 4 ) + cosh (µζ)(1c 8cµ + µ 4 ). (7) IJNS homepage:

5 154 International Journal of Nonlinear Science, Vol.14(01), No., pp The above equation is satisfied only if the following system of algebraic equations holds 0 = 7 + 1c λ 8cλ µ + 48cµ + 1µ 4 + λ µ 4 0 = 4λ c + cλµ 4λµ 4 Solving the above system gives four solution sets 0 = 1c 8cµ + µ 4. (8) c = 1, µ = ±i 4, λ = 0, c = 1, µ = ± 4, λ = 0, c = 1 10, µ = ±i 4 5, λ = 0, c = 1 10, µ = ± 4 5, λ = 0. (9) Thus, the solutions of (9) are v 7 (x, t) = sec( 4 ζ), v 8 (x, t) = sech( 4 5 ζ), v 9 (x, t) = sec( 4 5 ζ), v 10 (x, t) = sech( 4 ζ), ζ = x ct. (0) Substituting (0) into (11) gives u 7 (x, t) = sec ( 4 ζ), u 8 (x, t) = sech( 4 5 ζ), u 9 (x, t) = sec( 4 5 ζ), ( u 10 (x, t) = sech ( 4 ) ζ) 1, ζ = x ct. (1) 4 Generalized coupled Hirota Satsuma KdV equation In this section we consider a three-component evolutionary system of KdV equation of order. Such system is called the generalized Hirota Satsuma system given by u t = 1 u xxx uu x + (vw) x, v t = v xxx + uv x, Using the wave variable ζ = x ct transforms () into the ODEs w t = w xxx + uw x. () 0 = cu + 1 u (u ) + (vw), 0 = cv v + uv, 0 = cw w + uw. () IJNS for contribution: editor@nonlinearscience.org.uk

6 M. Alquran, R. Al-omary: Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using 155 Considering the second and third equations of system (), we obtain (v w ) x = (w v ) x. (4) Integrating the obtained ODE in (4) twice and setting the constant of integration to zero, we get Now, we simplify the second equation in () to get v = w. (5) u = v() v c. (6) Substituting (6) into the first equation () and then integrating and setting the constant of integration to zero, yield 0 = (v ) v (5) v v v (4) v (v () ) + 4c(v ) v () + (v ) v () c (v ) + 18(v ) vw. (7) First, we solve (7) by using Ansatze (V). Suppose v(x, t) = a + λ tanh(µζ) w(x, t) = b + λ tanh(µζ). (8) Now substituting (8) into (7), we get 0 = 18ab c 8cµ + 8µ 4 + (18aλ + 18bλ) tanh(µζ) + (18λ + 4cµ 4µ 4 ) tanh (µζ). (9) The above equation is satisfied only if the following system of algebraic equations holds 0 = 18ab c 8cµ + 8µ 4, 0 = 18aλ + 18bλ, 0 = 18λ + 4cµ 4µ 4. (40) The above system gives a = ( c 8cµ + 8µ 4 ), b = ± c 8cµ + 8µ 4, λ = ± ( cµ + µ 4 ). (41) where c and µ are arbitrary constants. Thus, the solutions of () are ( c 8cµ + 8µ 4 ) v 1 (x, t) = ± ( cµ + µ 4 ) tanh(µ(x ct)), w 1 (x, t) = ± c 8cµ + 8µ 4 Substituting (4) into (6), we get ± ( cµ + µ 4 ) tanh(µ(x ct)). (4) u 1 (x, t) = 1 (4µ c 6µ sech (µ(x ct))). (4) IJNS homepage:

7 156 International Journal of Nonlinear Science, Vol.14(01), No., pp Figure 1: The first obtained solution v(x,t) for equation 4.1. a = 1, µ = 1, λ = ; 5 x 5; t = 0.1, 0., 0., 0.4,.1,.,.,.4 Figure : The first obtained solution w(x,t) for equation 4.1. a = 1, µ = 1, λ = ; 5 x 5; t = 0.1, 0., 0., 0.4,.1,.,.,.4 If we use the coth instead of tanh, we get the same system and also one more solution follows v (x, t) Substituting (44) into (6), we get ( c 8cµ + 8µ 4 ) = ± ( cµ + µ 4 ) coth(µ(x ct)), w (x, t) = ± c 8cµ + 8µ 4 Second, we apply Ansatze (V). Suppose that Substituting (46) into (7) yields ± ( cµ + µ 4 ) coth(µ(x ct)). (44) u (x, t) = 1 (4µ c + 6µ csch (µ(x ct))). (45) v(x, t) = a + λsech (µζ), w(x, t) = b + λsech (µζ). (46) 0 = 18λ 4cµ 1µ 4 + (18aλ + 18bλ) cosh(µζ) + (18λ c + 4cµ µ 4 ) cosh (µζ). (47) IJNS for contribution: editor@nonlinearscience.org.uk

8 M. Alquran, R. Al-omary: Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using 157 Figure : The first obtained solution u(x,t) for equation 4.1. a = 1, µ = 1, λ = ; 5 x 5; t = 0.1, 0., 0., 0.4,.1,.,.,.4 The above equation is satisfied only if the following system of algebraic equations holds Solving the above system yields 0 = 18λ 4cµ 1µ 4 0 = 18aλ + 18bλ 0 = 18ab c + 4cµ µ 4. (48) a = ( c + 4cµ µ 4 b = ± c + 4cµ µ 4, λ = ± where c and µ are arbitrary constants. Thus, the solutions of () are ( c + 4cµ µ 4 ) v (x, t) = ± Substituting (50) into (6), we get w (x, t) = ± c + 4cµ µ 4 ± ), 4cµ + µ 4, (49) 4cµ + µ 4 sech(µ(x ct)), 4cµ + µ 4 sech(µ(x ct)). (50) u (x, t) = 1 (µ c 6µ sech (µ(x ct))). (51) Third, we apply the cosine method; Ansatze (I). From (5), suppose v(x, t) = w(x, t) and then substitute () into (6) to get 0 = 8µ 4 + 6γµ 4 + 7γ µ 4 6γ µ 4 + γ 4 µ 4 + cos (µζ)( 8cµ 4 + 1cγµ 4cγ µ + 8µ 4 1γµ 4 + 6γ µ 4 γ 4 µ 4 ) + cos 4 (µζ)(c + 4cγ µ + γ 4 µ 4 ) 18λ cos 4+γ (µζ). (5) The above equation is satisfied only if the following system of algebraic equations holds and µ is any nonzero constant 0 = 8µ 4 + 6γµ 4 + 7γ µ 4 6γ µ 4 + γ 4 µ 4, 0 = 8cµ 4 + 1cγµ 4cγ µ + 8µ 4 1γµ 4 + 6γ µ 4 γ 4 µ 4 18λ, 0 = c + 4cγ µ + γ 4 µ 4, 0 = + γ. (5) IJNS homepage:

9 158 International Journal of Nonlinear Science, Vol.14(01), No., pp Solving the above system gives Therefore, the solutions of () are Substituting (55) into (6) gives λ = ± µ, c = µ, γ = 1, 10µ λ = ±, c = µ, γ = 1. (54) v 4 (x, t) = w 4 (x, t) = ± µ sec(µ(x + µ t)), 10µ v 5 (x, t) = w 5 (x, t) = ± sec(µ(x + µ t)). (55) u 4 (x, t) = µ sec (µ(x + µ t)) u 5 (x, t) = µ 9 ( sec (µ(x + µ t))). (56) Now, by the sine method, we obtain the same system (5) and thus two more solutions follows Substituting (57) into (6) gives v 6 (x, t) = w 6 (x, t) = ± µ csc(µ(x + µ t)), 10µ v 7 (x, t) = w 7 (x, t) = ± csc(µ(x + µ t)). (57) Figure 4: The second obtained solution v(x,t) for equation 4.1. λ =, µ = 1, β = 1, c = 1; 5 x 5; t = 0.1, 0., 0., 0.4,.1,.,.,.4 u 6 (x, t) = µ csc (µ(x + µ t)), u 7 (x, t) = µ 9 ( csc (µ(x + µ t))). (58) Acknowledgements The authors would like to thank the editor and the anonymous referees for their in-depth reading, criticisms, and insightful comments on an earlier version of this paper. IJNS for contribution: editor@nonlinearscience.org.uk

10 M. Alquran, R. Al-omary: Soliton and Periodic Solutions to the Generalized Hirota-Satsuma Coupled System Using 159 Figure 5: The second obtained solution u(x,t) for equation 4.1. λ =, µ = 1, β = 1, c = 1; 5 x 5; t = 0.1, 0., 0., 0.4,.1,.,.,.4 References [1] Abdul-Majid Wazwaz. Multiple-soliton solutions for the KP equation by Hirota s bilinear method and by the tanhcoth method. Applied Mathematics and Computation, 190(007): [] Abdul-Majid Wazwaz. The Hirota s direct method for multiple-soliton solutions for three model equations of shallow water waves. Applied Mathematics and Computation, 01(008): [] Abdul-Majid Wazwaz. New solitions and kink solutions for the Gardner equation. Communications in nonlinear science and numerical simulation, 1(007): [4] Ahmet Beki. Applications of the extended tanh method for coupled nonlinear evolution equations. Communications in Nonlinear Science and Numerical Simulation, 1(008): [5] Dianchen Lu, Baojian Hong, Lixin Tian. New explicit exact solutions for the generalized Hirota-Satsuma KdV system. Computers and Mathematics with Applications, 5(007): [6] Huabing Jia, Wei Xu. Solitons solutions for some nonlinear evolution equations. Applied Mathematics and Computation, 17 (010): [7] Marwan Alquran. Solitons and periodic solutions to nonlinear partial differential equations by the sine-cosine method. Applied Mathematics and Information Sciences (AMIS), 6(1)(01): [8] Marwan Alquran, Mahmoud Mohammad. Approximate Solutions to System of Nonlinear Partial Differential Equations Using Homotopy Perturbation Method. International Journal of Nonlinear Science (IJNS), 1(4)(011): [9] Marwan Alquran, Kamel Al-Khaled. Sinc and solitary wave solutions to the generalized Benjamin-Bona-Mahony- Burgers equations. Physica Scripta, 8(011): (6pp). [10] Marwan Alquran, Kamel Al-Khaled. The tanh and sine-cosine methods for higher order equations of Korteweg-de Vries type. Physica Scripta, 84(011):05010 (4pp). [11] R. Hirota, J. Satsuma. Soliton solutions of a coupled Kortewege-de Vries equation. Phys. Lett. A., 85 (1981): [1] Sami Shukri, Kamel Al-Khaled. The extended tanh method for solving systems of nonlinear wave equations. Applied Mathematics and Computation, 17(010): [1] WU Lian-Ping, GENG Xian-Guo, ZHANG Xiao-Lin. N-soliton Solution of a Generalized Hirota-Satsuma Coupled KdV Equation and Its Reduction. CHIN. PHYS. LETT., 6(009):(4pp). [14] Y. T. Wu, X. G. Geng, X. B. Hu, S. M. Zhu. A generalized Hirota-Satsuma coupled Kortewege-de Vries equation and miura transformations. Phys. Lett. A., 5(1999): IJNS homepage: