Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Xianbo Sun; Pei Yu

doi:10.3934/dcdsb.2018341

Article Contents

2019, Volume 24, Issue 2: 965-987. Doi: 10.3934/dcdsb.2018341

This issue Previous Article Hermite spectral method for Long-Short wave equations Next Article

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Xianbo Sun and
Pei Yu^,

Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada

Author Bio: E-mail address: xsun244@uwo.ca (X. Sun); E-mail address: pyu@uwo.ca (P. Yu)
* Corresponding author: Pei Yu
* Corresponding author: Pei Yu

Received: April 30, 2017

Revised: August 31, 2017

Early access: November 2018

Published: January 2019

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Abstract

In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the co-existence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

Keywords:

Generalized Benjamin-Bona-Mahony equation,
isolated periodic traveling wave,
solitary wave,
Abelian integral.

Mathematics Subject Classification: Primary: 34C25, 34C60; Secondary: 37C27.

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