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Contemporary Mathematics Volume 00, 0000

Summary: Contemporary Mathematics
Volume 00, 0000
Concentration Compactness and the Stability of
Solitary-Wave Solutions to Nonlocal Equations
John P. Albert
Abstract. In their proof of the stability of standing-wave solutions of nonlin-
ear Schršodinger equations, Cazenave and Lions used the principle of concentra-
tion compactness to characterize the standing waves as solutions of a certain
variational problem. In this article we first review the techniques introduced
by Cazenave and Lions, and then discuss their application to solitary-wave
solutions of nonlocal nonlinear wave equations. As an example of such an ap-
plication, we include a new result on the stability of solitary-wave solutions of
the Kubota-Ko-Dobbs equation for internal waves in a stratified fluid.
1. Introduction
The first mathematical treatment of the problem of stability of solitary waves
was published in 1871 by Joseph Boussinesq [Bou], who at the time was 29 years old
and just beginning a long and distinguished career in mathematical physics. The
solitary waves he was concerned with are water waves with readily recognizable
hump-like profiles, which are often produced by disturbances in a shallow channel
and which can undergo strong interactions and travel long distances without evident


Source: Albert, John - Department of Mathematics, University of Oklahoma


Collections: Mathematics