{"title":"New Exact Solutions for the (3+1)-Dimensional Breaking Soliton Equation","authors":"Mohammad Taghi Darvishi, Maliheh Najafi, Mohammad Najafi","volume":39,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":403,"pagesEnd":407,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10134","abstract":"

In this work, we obtain some analytic solutions for the (3+1)-dimensional breaking soliton after obtaining its Hirota-s bilinear form. Our calculations show that, three-wave method is very easy and straightforward to solve nonlinear partial differential equations.<\/p>\r\n","references":" J.H. He, Variational iteration method-a kind of non-linear analytical\r\ntechnique: some examples, Int. J. Non-linear Mech. 34(4) (1999) 699-\r\n708.\r\n M.T. Darvishi, F. Khani, A.A. Soliman, The numerical simulation for stiff\r\nsystems of ordinary differential equations, Comput. Math. Appl. 54(7-8)\r\n(2007) 1055-1063.\r\n M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order\r\nKorteweg-de Vries equations, Chaos, Solitons and Fractals 39 (2009)\r\n2484-2490.\r\n J.H. He, New interpretation of homotopy perturbation method, Int. J.\r\nMod. Phys. B 20(18) (2006) 2561-2568.\r\n J.H. He, Application of homotopy perturbation method to nonlinear wave\r\nequations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.\r\n J.H. He, Homotopy perturbation method for bifurcation of nonlinear\r\nproblems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.\r\n M.T. Darvishi, F. Khani, Application of He-s homotopy perturbation\r\nmethod to stiff systems of ordinary differential equations, Zeitschrift fur\r\nNaturforschung A, 63a (1-2) (2008) 19-23.\r\n M.T. Darvishi, F. Khani, S. Hamedi-Nezhad, S.-W. Ryu, New modification\r\nof the HPM for numerical solutions of the sine-Gordon and coupled sine-\r\nGordon equations, Int. J. Comput. Math. 87(4) (2010) 908-919.\r\n J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin.\r\nSci. Numer. Simul. 2 (2001) 257-264.\r\n M.T. Darvishi, A. Karami, B.-C. Shin, Application of He-s parameterexpansion\r\nmethod for oscillators with smooth odd nonlinearities, Phys.\r\nLett. A 372(33) (2008) 5381-5384.\r\n B.-C. Shin, M.T. Darvishi, A. Karami, Application of He-s parameterexpansion\r\nmethod to a nonlinear self-excited oscillator system, Int. J.\r\nNonlin. Sci. Num. Simul. 10(1) (2009) 137-143.\r\n M.T. Darvishi, Preconditioning and domain decomposition schemes to\r\nsolve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.\r\n M.T. Darvishi, S. Kheybari and F. Khani, A numerical solution of the\r\nKorteweg-de Vries equation by pseudospectral method using Darvishi-s\r\npreconditionings, Appl. Math. Comput. 182(1) (2006) 98-105.\r\n M.T. Darvishi, M. Javidi, A numerical solution of Burgers- equation\r\nby pseudospectral method and Darvishi-s preconditioning, Appl. Math.\r\nComput. 173(1) (2006) 421-429.\r\n M.T. Darvishi, F. Khani and S. Kheybari, Spectral collocation solution\r\nof a generalized Hirota-Satsuma KdV equation, Int. J. Comput. Math.\r\n84(4) (2007) 541-551.\r\n M.T. Darvishi, F. Khani, S. Kheybari, Spectral collocation method\r\nand Darvishi-s preconditionings to solve the generalized Burgers-Huxley\r\nequation, Commun., Nonlinear Sci. Numer. Simul. 13(10) (2008) 2091-\r\n2103.\r\n S.J. Liao, An explicit, totally analytic approximate solution for Blasius\r\nviscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.\r\n S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis\r\nMethod, Chapman & Hall\/CRC Press, Boca Raton, 2003.\r\n S.J. Liao, On the homotopy analysis method for nonlinear problems,\r\nAppl. Math. Comput. 147 (2004) 499-513.\r\n S.J. Liao, A new branch of solutions of boundary-layer flows over an\r\nimpermeable stretched plate, Int. J. Heat Mass Transfer 48 (2005) 2529-\r\n2539.\r\n S.J. Liao, A general approach to get series solution of non-similarity\r\nboundary-layer flows, Commun. Nonlinear Sci. Numer. Simul. 14(5)\r\n(2009) 2144-2159.\r\n M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,\r\nComput. Math. Appl. 58 (2009) 360-368.\r\n J.H. He, M.A. Abdou, New periodic solutions for nonlinear evolution\r\nequations using Exp-function method, Chaos, Solitons and Fractals 34\r\n(2007) 1421-1429.\r\n J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,\r\nChaos, Solitons and Fractals, 30(3) (2006) 700-708.\r\n J.H. He, X.H. Wu, Construction of solitary solution and compacton-like\r\nsolution by variational iteration method, Chaos, Solitons and Fractals, 29\r\n(2006) 108-113.\r\n F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, S.-W. Ryu, New solitary\r\nwave and periodic solutions of the foam drainage equation using the Expfunction\r\nmethod, Nonlin. Anal.: Real World Appl. 10 (2009) 1904-1911.\r\n B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions\r\nof the Nizhnik-Novikov-Vesselov equation using the Exp-function method,\r\nComput. Math. Appl. 58(11\/12) (2009) 2147-2151.\r\n X.H. Wu, J.H. He, Exp-function method and its application to nonlinear\r\nequations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.\r\n S.-H. Ma, J. Peng, C. Zhang, New exact solutions of the (2+1)-\r\ndimensional breaking soliton system via an extended mapping method,\r\nChaos Solitons Fractals, 46 (2009) 210-214.\r\n A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional\r\nbreaking soliton equations, Phys. Scr., 81 (2010) 1-5.\r\n Z.D. Dai, S.Q. Lin, D.L. Li, G. Mu, The three-wave method for nonlinear\r\nevalution equations, Nonl. Sci. Lett. A, 1(1) (2010) 77-82.\r\n C.J. Wang, Z.D. Dai, L. Liang, Exact three-wave solution for higher\r\ndimensional KDV-type equation, Appl. Math. Comput., 216 (2010) 501-\r\n505.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 39, 2010"}