Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms
type: Item
instance of: scholarly article
title: Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms
zbMATH Open document ID: 1477.34057
author: P. Yu
author: Xianbo Sun
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2019-01-11T00:00:00Z
review text: In this paper, a generalized Benjamin-Bona-Mahony (BBM) equation with two different types of weak dissipative effects is studies, \[(u^3)_t + (u^4)_x + u_{xxx}+\epsilon P_i = 0, \,\,\,\, i=1,\,2. \] The term \(P_1\) describes the weak second and fourth derivative diffusions without Marangoni effect, \[ P_1 = u_{xx} + u_{xxxx}.\] The term \(P_2\), on the other hand, describes a generalized Marangoni effect, \[ P_2 = ((a_0 + a_1u + a_2u^2)u_x)_x, \] where \(a_0\), \( a_1\) and \(a_2\) are bounded parameters. The equation is analyzed through a reduction to the case of \(\epsilon=0\). The paper is devoted to the study of solitary and periodic waves in these two cases, including the issues of existence of the waves, uniqueness, co-existence of solitary and periodic waves, and monotonicity. The proofs are based on the theory of weak Hilbert's 16th problem and bifurcation theory.
reviewed by: Anna Ghazaryan
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 7000401
zbMATH Keywords: generalized Benjamin-Bona-Mahony equation
zbMATH Keywords: isolated periodic traveling wave
zbMATH Keywords: solitary wave
zbMATH Keywords: abelian integral
MaRDI profile type: Publication
full work available at URL: dcdsb.2018341
OpenAlex ID: W2899949020
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Wikidata QID: Q128951752
DOI: 10.3934/DCDSB.2018341
description: scientific article; zbMATH DE number 7000401