A Note on Exact Travelling Wave Solutions for Nonlinear PHI-Four Equation

Abstract:

The purpose of this work is to reveal the dynamical behavior of the nonlinear PHI-four equation, which is an interesting and very useful model in particle physics and mathematical physics. As a result, travelling wave solutions of nonlinear PHI-four equation are formally derived by employing hyperbolic tangent method in this paper. This paper shows that hyperbolic tangent method can be a powerful tool in obtaining evolution solutions of nonlinear system.

• Periodical: Applied Mechanics and Materials (Volumes 204-208)
• Pages: 4569-4572
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.204-208.4569
• Online since: October 2012
• Authors: Chun Huan Xiang
• Keywords: Dynamical Behavior, Hyperbolic Tangent Method, Nonlinear Evolution Equation, Nonlinear PHI-Four Equation, Travelling Wave Solution

References

[1] X.B. Hu, A (2+1)-dimensional sinh-Gordon equation and its Pfaffian generalization, Phys.Lett. A, 360(2007), 439-447.

DOI: 10.1016/j.physleta.2006.07.031

[2] J.L. Zhang and Y.M. Wang, Exact solutions to two nonlinear equations, Acta Physica Sinca, 52(2003), 1574-1578.

[3] W.L. Chan and X. Zhang, Symmetries, conservation laws and Hamiltonian structures of the non-isospectral and variable coefficient KdV and MKdV equations, J. Phys. A: Math.Gen., 28(1995), 407-412.

DOI: 10.1088/0305-4470/28/2/016

[4] Z.N. Zhu, Soliton-like solutions of generalized KdV equation with external force term, Acta Physica Sinca, 41(1992), 1561-1566.

DOI: 10.7498/aps.41.1561

[5] T. Brugarino and P. Pantano, The integration of Burgers and Korteweg-de Vries equations with nonuniformities, Phys. Lett.A, 80(1980), 223-224.

DOI: 10.1016/0375-9601(80)90005-5

[6] C. Tian and Redekopp L G, Symmetries and a hierarchy of the general KdV equation, J.Phys.A: Math.Gen., 20(1987), 359-366.

DOI: 10.1088/0305-4470/20/2/021

[7] J.F. Zhang and F.Y. Chen, Truncated Expansion Method and New Exact Soliton-like Solution of the General Variable Coefficient KdV equation, Acta Physica Sinca, 50(2001), 1648-1650.

DOI: 10.7498/aps.50.1648

[8] D.S. Li and H.Q. Zhang, Improved tanh-function method and the new exact solutions for the general variable coefficient KdV equation and MKdV equation, Acta Physica Sinca, 52(2003), 1569-1573.

DOI: 10.7498/aps.52.1569

[9] C. Xiang, Travelling wave solutions of reduced super-KdV equation: A perspective from Lame equation, Applied Mathematics and Computation, 217(2010) 42-47.

DOI: 10.1016/j.amc.2010.05.041

[10] Z.T. Fu, S.D. Liu and S.K. Liu, New exact solutions to Kdv equations with variable coefficients or forcing, Applied Mathematics and Mechanics, 25(2004), 73-79.

DOI: 10.1007/bf02437295

[11] F.J. Chen and J.F. Zhang, Soliton-like solution for the (2+1)-dimensional variable coefficients Kadomtsev-Petviashvili equation, Acta Armamentar, 24(2003), 389-391.

[12] M. Malfliet, E. Wieers, A nonlinear theory of charged-particle stopping in non-idealplasmas, J. Plasma. Phys., 56(1996), 441-443.

[13] W. Malfliet, The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations, J. Comput. Appl. Math., 164(2004), 529-541.

DOI: 10.1016/s0377-0427(03)00645-9

[14] A.M. Wazwaz, Analytical study on nonlinear variants of the RLW and the PHI-four equation, Commun. Nonlinear Sci. Numer. Simulat., 12(2007), 314-327.

DOI: 10.1016/j.cnsns.2005.03.001

[15] J.M. Dye and A. Parker, An inverse scatting scheme for the regularized long-wave equation, J. Math. Phys., 41(2000), 2889-2904.

DOI: 10.1063/1.533278