  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.
