New Exact Solutions for the (3+1)-Dimensional Breaking Soliton Equation

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.

• Mohammad Taghi Darvishi
• Maliheh Najafi
• Mohammad Najafi

• (3+1)-dimensional breaking soliton equation
• Hirota'sbilinear form.

[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] J.H. He, New interpretation of homotopy perturbation method, Int. J.
Mod. Phys. B 20(18) (2006) 2561-2568.
[5] J.H. He, Application of homotopy perturbation method to nonlinear wave
equations, Chaos, Solitons and Fractals 26(3) (2005) 695-700.
[6] J.H. He, Homotopy perturbation method for bifurcation of nonlinear
problems, Int. J. Nonlinear Sci. Numer. Simul. 6(2) (2005) 207-208.
[7] 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.
[8] 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.
[9] J.H. He, Bookkeeping parameter in perturbation methods, Int. J. Nonlin.
Sci. Numer. Simul. 2 (2001) 257-264.
[10] 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.
[11] 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.
[12] M.T. Darvishi, Preconditioning and domain decomposition schemes to
solve PDEs, Int-l J. of Pure and Applied Math. 1(4) (2004) 419-439.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] S.J. Liao, An explicit, totally analytic approximate solution for Blasius
viscous flow problems, Int. J. Non-Linear Mech. 34 (1999) 759-778.
[18] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman & Hall/CRC Press, Boca Raton, 2003.
[19] S.J. Liao, On the homotopy analysis method for nonlinear problems,
Appl. Math. Comput. 147 (2004) 499-513.
[20] 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.
[21] 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.
[22] M.T. Darvishi, F. Khani, A series solution of the foam drainage equation,
Comput. Math. Appl. 58 (2009) 360-368.
[23] 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.
[24] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations,
Chaos, Solitons and Fractals, 30(3) (2006) 700-708.
[25] 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.
[26] 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.
[27] 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.
[28] X.H. Wu, J.H. He, Exp-function method and its application to nonlinear
equations, Chaos, Solitons and Fractals 38(3) (2008) 903-910.
[29] S.-H. Ma, J. Peng, C. Zhang, New exact solutions of the (2+1)-
dimensional breaking soliton system via an extended mapping method,
Chaos Solitons Fractals, 46 (2009) 210-214.
[30] A.M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional
breaking soliton equations, Phys. Scr., 81 (2010) 1-5.
[31] 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.
[32] 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.