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January  2013, 18(1): 163-172. doi: 10.3934/dcdsb.2013.18.163

Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations

Jibin Li 1, and  Fengjuan Chen 1,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

Received  March 2012 Revised  July 2012 Published  September 2012

For the system of KP like equation coupled to a Schrödinger equation, a corresponding four-dimensional travelling wave systems and a two-order linear non-autonomous system are studied by using Congrove's results and dynamical system method. For the four-dimensional travelling wave systems, exact explicit homoclinic orbit families, periodic and quasi-periodic wave solution families are obtained. The existence of homoclinic manifolds to four kinds of equilibria including a hyperbolic equilibrium, a center-saddle and an equilibrium with zero pair of eigenvalues is revealed. For the two-order linear non-autonomous system, the dynamical behavior of the bounded solutions is discussed.
Keywords: quasi-periodic wave solution, homoclinic manifold, center manifold, integrable system., Exact solution.
Mathematics Subject Classification: 34A26, 34C15, 34C23, 34C37, 35C07, 35C08, 37J35, 37K4.
Citation: Jibin Li, Fengjuan Chen. Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 163-172. doi: 10.3934/dcdsb.2013.18.163
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X. B. Hu, The higher order KdV equation with a source and nonlinear superposition formula, Chaos, Solitonsand Fract., 7 (1996), 211-215.

[2]

S. Z. Rida and M. Khalfallah, New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, Commun. nonlinear Sci. Numer. Simulat., 15 (2010), 2818-2827.

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M. C. Cosgrove, Higher-order Painleve equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math., 104 (2000), 1-65.

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A. M. Wazwaz, Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput., 182 (2006), 283-300.

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A. M. Wazwaz, The integrable KdV6 equations: multiple solitons and multiple singular soliton solutions, Appl. Math. Comput., 204 (2008), 963-972.

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J. K. Hale, "Ordinary Differential Equation," second Edition, Robert E.Krieger Publishing Company: Huntington, New York, 1980.

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L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer-Verlag, Berlinm, (1959), vii+271 pp.

show all references

References:
[1]

X. B. Hu, The higher order KdV equation with a source and nonlinear superposition formula, Chaos, Solitonsand Fract., 7 (1996), 211-215.

[2]

S. Z. Rida and M. Khalfallah, New periodic wave and soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, Commun. nonlinear Sci. Numer. Simulat., 15 (2010), 2818-2827.

[3]

M. C. Cosgrove, Higher-order Painleve equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math., 104 (2000), 1-65.

[4]

A. M. Wazwaz, Abundant solitons solutions for several forms of the fifth-order KdV equation by using the tanh method, Appl. Math. Comput., 182 (2006), 283-300.

[5]

A. M. Wazwaz, The integrable KdV6 equations: multiple solitons and multiple singular soliton solutions, Appl. Math. Comput., 204 (2008), 963-972.

[6]

J. K. Hale, "Ordinary Differential Equation," second Edition, Robert E.Krieger Publishing Company: Huntington, New York, 1980.

[7]

L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer-Verlag, Berlinm, (1959), vii+271 pp.

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