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.