 
Summary: STABILITY OF SOLITARYWAVE
SOLUTIONS TO LONGWAVE EQUATIONS
WITH GENERAL DISPERSION
John Albert Felipe Linares
1. Introduction
In this report we consider nonlinear dispersive systems of the form
ut + D(f(u)  Mu)x = 0, (1)
where u(x, t) = (u1(x, t), . . . , un(x, t)) is a map from R × R to Rn
, D is a
constant diagonal matrix with positive entries, f : Rn
Rn
is nonlinear, and
the dispersion operator M acts as a Fourier multiplier operator in the x variable.
More precisely, for each fixed t, Mu is a vectorvalued function defined by
Mu(k, t) = m(k)u(k, t), k R,
where circumflexes denote Fourier transforms with respect to x,
g(k) =
R
eikx
g(x) dx.
