Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient

Weiguo  Zhang; Yujiao  Sun; Zhengming  Li; Shengbing  Pei; Xiang  Li

doi:10.3934/dcdsb.2016078

Article Contents

2016, Volume 21, Issue 8: 2883-2903. Doi: 10.3934/dcdsb.2016078

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Bounded traveling wave solutions for MKdV-Burgers equation with the negative dispersive coefficient

1.
School of Science, University of Shanghai for Science and Technology, Shanghai 200093
2.
University of Shanghai for Science and Technology, Shanghai 200093

Received: July 31, 2013

Revised: December 31, 2015

Published: September 2016

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Abstract

This paper studies the bounded traveling wave solutions of MKdV-Burgers equation with the negative dispersive coefficient by the theory of planar dynamical systems, undetermined coefficients method. The global phase portraits under the different parameter conditions, as well as the existent number and conditions of the bounded traveling wave solutions are obtained for the dynamical system corresponding to the traveling wave solutions of MKdV-Burgers equation. The relation is investigated between the profiles of the bounded traveling wave solutions and dissipative coefficient. And a critical value characterizing the scale of dissipative effect, is given, which is different from the one proposed by R.F. Bikbaev in his article. Focusing on the open issue proposed by R.F. Bikbaev, based on the bell and kink profile solitary wave solutions of MKdV-Burgers equation we presented, approximate damped oscillatory solutions of MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.

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Mathematics Subject Classification: Primary: 35Q51, 37C29; Secondary: 37C20.

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