Sub-Equations and Exact Traveling Wave Solutions to a Class of High-Order Nonlinear Wave Equations

ZHANG Li-jun; CHEN Li-qun

doi:10.3879/j.issn.1000-0887.2015.05.010

Volume 36 Issue 5

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ZHANG Li-jun, CHEN Li-qun. Sub-Equations and Exact Traveling Wave Solutions to a Class of High-Order Nonlinear Wave Equations[J]. Applied Mathematics and Mechanics, 2015, 36(5): 548-554. doi: 10.3879/j.issn.1000-0887.2015.05.010

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Sub-Equations and Exact Traveling Wave Solutions to a Class of High-Order Nonlinear Wave Equations

doi: 10.3879/j.issn.1000-0887.2015.05.010

ZHANG Li-jun^1,2,
CHEN Li-qun³

¹School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, P.R.China;²International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, NorthWest University, Mafikeng Campus, Mmabatho 2735, South Africa;³Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P.R.China

Funds: The National Natural Science Foundation of China（11101371）

Received Date: 2015-02-17
Rev Recd Date: 2015-04-06
Publish Date: 2015-05-15

Abstract

Abstract

The exact traveling wave solutions to a class of 5th-order nonlinear wave equations were studied with the sub-equation method and the dynamic system analysis approach. The lower-order sub-equations of this class of high-order nonlinear equations were first derived, then the traveling wave solutions were investigated via the various exact solutions to the sub-equations under different parameter conditions. As an example, 2 families of exact valley-form solitary wave solutions and 2 families of smooth periodic traveling wave solutions to the Sawada-Kotera equation were presented. This method can be applied to study the traveling wave solutions to high-order nonlinear wave equations of which the corresponding traveling wave system can be reduced to the nonlinear ODEs involving only even-order derivatives, sum of squares of 1st-order derivatives and polynomial of dependent variables.
- high-order nonlinear wave equation,
- solitary wave solution,
- periodic traveling wave solution,
- sub-equation

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References(9)

References

[1]	Chow S N, Hale J K. Method of Bifurcation Theory[M]. New York: Springer, 1981.
[2]	LI Ji-bin. Singular Traveling Wave Equations: Bifurcations and Exact Solutions[M]. Beijing: Science Press, 2013.
[3]	LI Ji-bin, QIAO Zhi-jun. Explicit solutions of the Kaup-Kupershmidt equation through the dynamical system approach[J]. Journal of Applied Analysis and Computation,2011,1(2): 243-250.
[4]	冯大河, 李继彬. Jaulent-Miodek方程的行波解分支[J]. 应用数学和力学, 2007,28(8): 894-900.(FENG Da-he, LI Ji-bin. Bifurcations of travelling wave solutions for Jaulent-Miodek equations[J]. Applied Mathematics and Mechanics,2007,28(8): 894-900.(in Chinese))
[5]	Sawada K, Kotera T. A method for finding N-soliton solutions for the K.d.V. equation and K.d.V.-like equation[J]. Progress of Theoretical Physics,1974,51(5): 1355-1367.
[6]	Caudrey P J, Dodd R K, Gibbon J D. A new hierarchy of Korteweg-de Vries equations[J]. The Proceedings of the Royal Society of London,1976,351(1666): 407-422.
[7]	Kupershmidt B A. A super Korteweg-de Vires equation: an integrable system[J]. Physics Letters A,1984,102(5/6): 213-215.
[8]	ZHANG Li-jun, Khalqiue C M. Exact solitary wave and periodic wave solutions of the Kaup-Kupershmidt equation[J]. Journal of Applied Analysis and Computation,2015,5(3): 485-495.
[9]	Cosgrove M C. Higher-order Painleve equations in the polynomial class I: Bureau symbol P₂[J]. Studies in Applied Mathematics,2000,104(1): 1-65.