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Instability of periodic traveling waves for the symmetric regularized long wave equation

Part of: Partial differential equations Representations of solutions Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  11 January 2016

Jaime Angulo Pava  and
Carlos Alberto Banquet Brango
Jaime Angulo Pava
Affiliation:
Department of Mathematics, Instituto de Matemática e Estatística, Universidade de São Paulo, CEP 05508-090, São Paulo, SP, Brazil, angulo@ime.usp.br
Carlos Alberto Banquet Brango
Affiliation:
Departamento de Matemáticas y Estadística, Universidad de CórdobaCódigo, Postal 230002, Colombia, cbanquet@correo.unicordoba.edu.co
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Abstract

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We prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

MSC classification

Secondary: 35Q51: Soliton-like equations 35B35: Stability 35B10: Periodic solutions 35C07: Traveling wave solutions
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 235 - 268
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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