A Note on the existence of traveling-wave solutions to a boussinesq system

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Abstract

We obtain a one-parameter family (uμ(x,t),ημ(x,t))μ≥μ0=(φμ(x-ωμt),ψμ(x-ωμt))μ≥μ0 of traveling-wave solutions to the Boussinesq system {utx+uux+cηxxx=0(x,t)∈R2ηt+ux+(ηu)x+auxxx=0, in the case a,c<0 with non-null speeds ωμ arbitrarily close to 0 (ωμ-→0-μ→+∞). We show that the L2-size of such traveling-waves satisfies the uniform (in μ) estimate φμ22+ψμ22≤C√|a|+|c|,μ2+ψμ2≤C|a|+|c|, where C is a positive constant. Furthermore, φμ and-ψμ are smooth, non-negative, radially decreasing functions which decay exponentially at infinity.

Original languageEnglish
Pages (from-to)127-136
Number of pages10
JournalDifferential and Integral Equations
Volume29
Issue number1-2
Publication statusPublished - 1 Jan 2016

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