Keywords: Exp-function method; solitary wave solutions; modified Camassa-Holm
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1 International Journal of Modern Mathematical Sciences, 2012, 4(3): International Journal of Modern Mathematical Sciences Journal homepage: ISSN: X Florida, USA Article Exp-function Method for Simplified Modified Camassa Holm Equation Amna Irshad, Muhammad Usman and Syed Tauseef Mohyud-Din* Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt Pakistan * Author to whom correspondence should be addressed; syedtauseefs@hotmail.com Article history: Received 17 September 2012, Received in revised form 17 December 2012, Accepted 20 December 2012, Published 21 December Abstract: Explicit solitary wave solutions of modified Camassa-Holm equation are constructed by using Exp-function method. Numerical results are true reflection of the efficiency and reliability of the proposed algorithm. Keywords: Exp-function method; solitary wave solutions; modified Camassa-Holm equations, nonlinear problems. Mathematics Subject Classification: 35Q79 1. Introduction Partial differential equations [1-32] arise very frequently in physical models. The thorough study of literature reveals [1-32] that most of the physical phenomenon is best modeled by nonlinear differential equations. Inspired and motivated by the ongoing research in this area, we apply a relatively new technique which is called the exp-function method [1, 3-6, 16-27] to find solitary wave solutions of nonlinear modified Camassa-Holm equation. which arise frequently in number of scientific models including fluid mechanics, astrophysics, solid state physics, plasma physics, chemical kinematics, chemical physics, optical fiber and geochemistry, see [16, 29-32] and the references therein. It is to be highlighted that exp-function method has been applied on a wide range of nonlinear diversified physical problems including, high-dimensional nonlinear evolution equation, combined KdV and mkdv, Hybrid-Lattice system and discrete mkdv lattice [1, 3-6, 16-27].
2 Camassa and Holm Equations Camassa and Holm [29] derived a completely integrable wave equation (CH equation) for water waves (1.1) by retaining two terms that are usually neglected in the small amplitude, shallow water limit. Tian and Song [30] investigated a modified Camassa Holm equation (MCH equation) (1.2) and obtained new peaked solitary wave solutions. In addition, Boyd [31] investigated that if the solitary wave varies slowly with then the two extra terms on the right-hand side of (1.1) will be small and the soliton is given to lowest order by the solutions of (1.3) In view of (1.3), Wazwaz [32] investigated a modified form of Camassa -Holm equation, which is simplified from MCH equation and given by (1.4) In this paper, we only consider, (1.5) and for simplicity we call (1.5) simplified MCH equation. 3. Exp-function Method [1, 3-6, 16-27] Consider the following general partial differential equation: ) = 0. (2.1) We first unite the independent variables x and t into one wave variable (2.2) to an ordinary differential equation, ) (2.2) leading The Exp-function method is based on the assumption that traveling wave solutions can be expressed in the following form [30]: ). (2.3) c, d, p, and q are positive integers which are unknown to be further determined, and and are unknown constants. To determine the values of c and p, we balance the linear term of highest order in (2.2) with the highest-order nonlinear term. Similarly to determine the values of d and q, we balance the linear term of lowest order in (2.2) with the lowest-order nonlinear term.
3 Numerical Applications In this section we apply Exp-function Method for solving simplified MCH equation. (3.1) Introducing a transformation as equations Integrating Eq. (3.2) with respect to once, we get ), we can convert (3.1) into ordinary differential (3.2) (3.3) The solution of (3.3) can be expressed in the form, (2.3) as To determine the values of can d p, we balance the highest order linear term of (3.3) with the highest order nonlinear term. Therefore, we have and [ ) ] (3.4) [ ) ] (3.5) are coefficients for simplicity; by balancing the highest order of the exp-function, we obtain, (3.6) Which is turns gives (3.7) To determine the values of d and q, we balance the lowest order linear term of (3.2) with the lowest order nonlinear term. Therefore and [ ) ] (3.8) [ ) ] (3.9) are determined coefficients only for simplicity; Now balancing the lowest order of the expfunction, we obtain (3.10)
4 149 which is turns gives (3.11) Case 4.1.1: p = c = 1, and q = d = 1. Equation (2.3) reduces to (3.12) Substituting (3.12) into (3.3), we have ) (3.13) ), Equating the coefficients of ) to be zero, we obtain { } (3.14) We have the following set of Solution First set: Second set: (3.15) Third set: (3.16) (3.17) Now, substituting (3.15) into (3.12), we get the generalized solitary wave solution (3.18)
5 150 next, substituting (3.16) into (3.12), we get the following generalized solitary wave solution (3.19) finally, substituting (3.17) into (3.12), we get the resulting solution (3.20) are real numbers. Case 3.1.2:, and. Equation (2.3) reduces to (3.21) In (3.21), there are some parameters, we set [ simplified as follows ] for simplicity and the trial function is (3.22) substituting (3.22) into (3.3) we have ) (3.23) ), Equating the coefficients of ) to be zero, we obtain { } (3.24) We have the following set of Solution First set:
6 151 (3.25) Second set: (3.26) Third set: (3.27) Now, substituting (3.25) into (3.21), we get the generalized solitary wave solution (3.28) next, substituting (3.26) into (3.21), we get the following generalized solitary wave solution (3.29) finally, substituting (3.27) into (3.21), we get the resulting solution (3.30) Case 3.1.3:, and Equation (2.3) reduces to (3.31)
7 In (3.31), there are some parameters, we set trial function is simplified as follows for simplicity and the 152 (3.32) First set: Proceeding as before, we obtain the solution sets Second set:. (3.33) Third set: (3.34) Now, substituting (3.33) into (3.31), we get the generalized solitary wave solution (3.35) (3.36) next, substituting (3.34) into (3.31), we get the following generalized solitary wave solution (3.37) finally, substituting (3.35) into (3.31), we get the resulting solution
8 153 (3.38) 4. Conclusions In this paper, we applied exp-function method to obtain generalized solitary solutions of the modified Camassa-Holm equation. Numerical results re-confirm the efficiency and reliability of the proposed Exp-function method. References [1] M. A. Abdou, A. A. Soliman and S. T. Basyony, New application of exp-function method for improved Boussinesq equation. Phys. Lett. A, 369 (2007): [2] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1(1948): [3] S. A. El-Wakil, M. A. Madkour and M. A. Abdou, Application of exp-function method for nonlinear evolution equations with variable co-efficient, Phys. Lett. A, 369 (2007): [4] J. H. He, An elementary introduction of recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B 22 (21) (2008): [5] J. H. He and X. H. Wu, exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30(3)(2006): [6] J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equation using expmethod, Chaos, Solitons & Fractals, 34 (2007): [7] W. Hereman and A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simulation, 43(1997): [8] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, [9] R.M. Miura, The Korteweg de-vries equation: a survey of results, SIAM Rev., 18(1976): [10] W. X. Ma, Complexiton solutions to the Korteweg-de Vries equation, Phys. Lett. A, 301 (2002):
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