 
Summary: Contemporary Mathematics
Volume 00, 0000
Concentration Compactness and the Stability of
SolitaryWave Solutions to Nonlocal Equations
John P. Albert
Abstract. In their proof of the stability of standingwave 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 solitarywave
solutions of nonlocal nonlinear wave equations. As an example of such an ap
plication, we include a new result on the stability of solitarywave solutions of
the KubotaKoDobbs 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
humplike profiles, which are often produced by disturbances in a shallow channel
and which can undergo strong interactions and travel long distances without evident
