  We consider the stability of (quasi-)periodic solutions of soliton equations
under short range perturbations and give a complete description of the long
time asymptotics in this situation. We show that, apart from the phenomenon of
the solitons travelling on the quasi-periodic background, the perturbed
solution asymptotically approaches a modulated solution. We use the Toda
lattice as a model but the same methods and ideas are applicable to all soliton
equations in one space dimension.
  More precisely, let $g$ be the genus of the hyperelliptic Riemann surface
associated with the unperturbed solution. We show that the $n/t$-pane contains
$g+2$ areas where the perturbed solution is close to a quasi-periodic solution
in the same isospectral torus. In between there are $g+1$ regions where the
perturbed solution is asymptotically close to a modulated lattice which
undergoes a continuous phase transition (in the Jacobian variety) and which
interpolates between these isospectral solutions. In the special case of the
free solution ($g=0$) the isospectral torus consists of just one point and we
recover the classical result.
  Both the solutions in the isospectral torus and the phase transition are
explicitly characterized in terms of Abelian integrals on the underlying
hyperelliptic Riemann surface.
