S.I. Dejak and I.M. Sigal
Long-Time Dynamics of KdV Solitary Waves over a Variable Bottom
(157K, AMS-TeX)

ABSTRACT.  We study the variable bottom generalized Korteweg-de 
Vries (bKdV) equation dt u=-dx(dx^2 u+f(u)-b(t,x)u), 
where f is a nonlinearity and b is a small, bounded and 
slowly varying function related to the varying depth of a channel 
of water. Many variable coefficient KdV-type equations, including 
the variable coefficient, variable bottom KdV equation, can be 
rescaled into the bKdV. We study the long time behaviour of 
solutions with initial conditions close to a stable, b=0 
solitary wave. We prove that for long time intervals, such 
solutions have the form of the solitary wave, whose centre and 
scale evolve according to a certain dynamical law involving the 
function b(t,x), plus an H^1-small fluctuation.