Existence and stability of traveling waves for a class of nonlocal nonlinear equations

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Erbay, Hüsnü A. and Erbay, Saadet and Erkip, Albert (2015) Existence and stability of traveling waves for a class of nonlocal nonlinear equations. Journal of Mathematical Analysis and Applications, 425 (1). pp. 307-336. ISSN 0022-247X (Print) 1096-0813 (Online)

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Official URL: http://dx.doi.org/10.1016/j.jmaa.2014.12.039


In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: u_tt−Lu_xx=B(±|u|^(p−1)u)_xx, p>1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operators L and B. Members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations. For instance, all Boussinesq-type equations and the so-called double-dispersion equation are members of the class. We first establish the existence of traveling wave solutions to the nonlocal wave equations considered. We then obtain results on the orbital stability or instability of traveling waves. For the case L=I, corresponding to a class of Klein-Gordon-type equations, we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow-up.

Item Type:Article
Uncontrolled Keywords:Solitary waves; Orbital stability; Boussinesq equation; Double dispersion equation; Concentration-compactness; Klein–Gordon equation
Subjects:Q Science > QA Mathematics > QA299.6-433 Analysis
ID Code:26651
Deposited By:Albert Erkip
Deposited On:02 Feb 2015 15:57
Last Modified:29 Jul 2019 11:14

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