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

Abstract

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt−Luxx=B(±|u|p−1u)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.TÜBİTA

Some remarks on the stability and instability properties of solitary waves for the double dispersion equation

“…In this section, we will briefly state the results that were obtained rigorously in [1] for the existence and stability of travelling wave solutions of the general class (2). The basic idea of the existence proof for travelling waves is to show that travelling wave solutions are minimizers of a constrained variational problem.…”

“…The basic idea of the existence proof for travelling waves is to show that travelling wave solutions are minimizers of a constrained variational problem. The concentration-compactness lemma of Lions [14,15] is the main tool in obtaining the results, and for a complete understanding of the results we refer the reader to [1]. We now state the three main results of [1].…”

“…The first observation in [1] is that travelling wave solutions of the general class (2) with power nonlinearities exist for two different regimes. That is, travelling wave solution u(x,t) = ϕ c (x − ct) of Eq.…”

“…The present paper provides an overview of results obtained in [1,2] for travelling wave solutions of the double dispersion equation…”

“…The double dispersion equation (1) was introduced as a mathematical model of nonlinear dispersive waves in various contexts (see for instance [3][4][5][6] and the references therein). It belongs to the following general class of nonlocal nonlinear wave equations:…”