S. I. Dejak, B. L. G. Jonsson
Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves
(75K, LATeX 2e)

ABSTRACT.  We study the Korteweg-de Vries-type equation dt
u=-dx(dx^2 u+f(u)-B(t,x)u), where B is a small and
bounded, slowly varying function and f is a nonlinearity. Many
variable coefficient KdV-type equations can be rescaled into this
equation. 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.