  We consider the stability of the periodic Toda lattice (and slightly more
generally of the algebro-geometric finite-gap lattice) under a short range
perturbation. We prove that the perturbed lattice asymptotically approaches a
modulated lattice.
  More precisely, let $g$ be the genus of the hyperelliptic curve associated
with the unperturbed solution. We show that, apart from the phenomenon of the
solitons travelling on the quasi-periodic background, the $n/t$-pane contains
$g+2$ areas where the perturbed solution is close to a finite-gap 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 lattice ($g=0$) the isospectral torus consists of just one point and we
recover the known result.
  Both the solutions in the isospectral torus and the phase transition are
explicitly characterized in terms of Abelian integrals on the underlying
hyperelliptic curve.
  Our method relies on the equivalence of the inverse spectral problem to a
matrix Riemann--Hilbert problem defined on the hyperelliptic curve and
generalizes the so-called nonlinear stationary phase/steepest descent method
for Riemann--Hilbert problem deformations to Riemann surfaces.
