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Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

Published online by Cambridge University Press:  03 June 2015

Xiang Li ,
Weiguo Zhang  and
Yan Zhao
Xiang Li*
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Weiguo Zhang*
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Yan Zhao*
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China
*
Email: lixiang1121@yeah.net
Corresponding author. Email: zwgzwm@yahoo.com.cn
Email: zhaoyanem@hotmail.com
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Abstract

In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.

Keywords

35Q5137C29Nonlinear wave equationplanar dynamical systemexact solutionsapproximate damped oscillatory solutionserror estimate
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 559 - 574
Copyright
Copyright © Global-Science Press 2012

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