Some Exact Solutions of the (2+1)-Dimensional Breaking Soliton Equation using the Three-wave Method

Authors: Mohammad Taghi Darvishi, Mohammad Najafi

Abstract:

This paper considers the (2+1)-dimensional breaking soliton equation in its bilinear form. Some exact solutions to this equation are explicitly derived by the idea of three-wave solution method with the assistance of Maple. We can see that the new idea is very simple and straightforward.

Keywords: Soliton solution, computerized symbolic computation, painleve analysis, (2+1)-dimensional breaking soliton equation, Hirota's bilinear form.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079360

References:

[1] J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int. J. Non-linear Mech. 34(4) (1999) 699- 708.
[2] M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff systems of ordinary differential equations, Comput. Math. Appl. 54(7-8) (2007) 1055-1063.
[3] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order Korteweg-de Vries equations, Chaos, Solitons and Fractals 39 (2009) 2484-2490.
[4] S. Abbasbandy, M.T. Darvishi, A numerical solution of Burgers- equation by modified Adomian method, Appl. Math. Comput. 163 (2005) 1265- 1272.
[5] S. Abbasbandy, M.T. Darvishi, A numerical solution of Burgers- equation by time discretization of Adomian-s decomposition method, Appl. Math. Comput. 170 (2005) 95-102.
[6] J.H. He, New interpretation of homotopy perturbation method, Int. J. Mod. Phys. B 20(18) (2006) 2561-2568.
[7] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[8] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[9] M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation method to stiff systems of ordinary differential equations, Zeitschrift fur Naturforschung A, 63a (1-2) (2008) 19-23.
[10] M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.
[11] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin. Sci. Numer. Simul. 2 (2001) 257-264.
[12] M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion method for oscillators with smooth odd nonlinearities, Phys. Lett. A 372(33) (2008) 5381-5384.
[13] B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion method to a nonlinear self-excited oscillator system, Int. J. Nonlin. Sci. Num. Simul. 10(1) (2009) 137-143.
[14] M.T. Darvishi, Preconditioning and domain decomposition schemes to solve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.
[15] M.T. Darvishi, S. Kheybari and F. Khani, A numerical solution of the Korteweg-de Vries equation by pseudospectral method using Darvishi-s preconditionings, Appl. Math. Comput. 182(1) (2006) 98-105.
[16] M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation by pseudospectral method and Darvishi-s preconditioning, Appl. Math. Comput. 173(1) (2006) 421-429.
[17] M.T. Darvishi, F. Khani and S. Kheybari, Spectral collocation solution of a generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math. 84(4) (2007) 541-551.
[18] M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method and Darvishi-s preconditionings to solve the generalized Burgers-Huxley equation, Commun., Nonlinear Sci. Numer. Simul. 13(10) (2008) 2091- 2103.
[19] S.J. Liao, An explicit, totally analytic approximate solution for Blasius viscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.
[20] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[21] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499-513.
[22] S.J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. Heat Mass Transfer 48 (2005) 2529- 2539.
[23] S.J. Liao, A general approach to get series solution of non-similarity boundary-layer flows, Commun. Nonlinear Sci. Numer. Simul. 14(5) (2009) 2144-2159.
[24] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation, Comput. Math. Appl. 58 (2009) 360-368.
[25] A. Aziz, F. Khani, M.T. Darvishi, Homotopy analysis method for variable thermal conductivity heat flux gage with edge contact resistance, Zeitschrift fuer Naturforschung A, 65a(10) (2010) 771-776.
[26] J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons and Fractals 34 (2007) 1421-1429.
[27] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30(3) (2006) 700-708.
[28] J.H. He, X.H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals, 29 (2006) 108-113.
[29] F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary wave and periodic solutions of the foam drainage equation using the Expfunction method, Nonlin. Anal.: Real World Appl. 10 (2009) 1904-1911.
[30] B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions of the Nizhnik-Novikov-Vesselov equation using the Exp-function method, Comput. Math. Appl. 58(11/12) (2009) 2147-2151.
[31] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear equations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.
[32] Z.-D. Dai, S.-Q. Lin. D.-L. Li, G. Mu, The three-wave method for nonlinear evalution equations, Nonl. Sci. Lett. A 1(1) (2010) 77-82.
[33] C.-J. Wang, Z.-D. Dai, L. Liang, Exact three-wave solution for higher dimensional KDV-type equation, Appl. Math. Comput. 216 (2010) 501- 505.
[34] S. Ting, G.X. Guo, M.Y. Ling, Wronskian form of N-Soliton solution for the (2+1)-dimensional breaking soliton equation, Chin. Phys. Lett. 24(2) (2007) 305-307.
[35] W.-Q. Yong, Bilinear B¨acklund transformation and explicit solutions for a nonlinear evolution equation, Chin. Phys. B 19(4) (2010).
[36] J. Hietarinta, Hirota-s bilinear method and soliton solutions, Physics AUC, 15(1) (2005) 31-37.