Qualitative analysis to the traveling wave solutions ofKakutani-Kawahara equation and its approximate damped oscillatorysolution

Weiguo Zhang; Yan Zhao; Xiang Li

doi:10.3934/cpaa.2013.12.1075

Article Contents

2013, Volume 12, Issue 2: 1075-1090. Doi: 10.3934/cpaa.2013.12.1075

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Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution

1.
School of Science, University of Shanghai for Science and Technology, Shanghai 200093

Received: July 31, 2011

Revised: October 31, 2012

Published: February 2013

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Abstract

In this paper, we apply the theory of planar dynamical systems to make a qualitative analysis to the traveling wave solutions of nonlinear Kakutani-Kawahara equation $u_t+uu_x+bu_{x x x}-a(u_t+uu_x)_x=0$ ($b>0, a\ge0$) and obtain the existent conditions of the bounded traveling wave solutions. In dispersion-dominant case, we find that the unique bounded traveling wave solution of this equation has not only oscillatory property but also damped property. Furthermore, according to the evolution of orbits in the global phase portraits, we present an approximate damped oscillatory solution for this equation by the undetermined coefficients method. Finally, by the idea of homogenization principles, we obtain an integral equation which reflects the relation between this approximate damped oscillatory solution and its exact solution, thereby having the error estimate. The error is an infinitesimal decreasing in exponential form. From the results in this paper, it can be seen that the damped oscillatory solution of Kakutani-Kawahara equation in dispersion-dominant case still remains some properties of solitary wave solution for KdV equation.

Keywords:

Nonlinear dispersive-dissipative equation,
qualitative analysis,
damped oscillatory solution,
approximate solution,
error estimate.

Mathematics Subject Classification: Primary: 35Q53; Secondary: 35C07.

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