- 07-111 Spyridon Kamvissis and Gerald Teschl
- Stability of the periodic Toda lattice under short range perturbations
May 2, 07
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Abstract. 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.