# Stability of the periodic Toda lattice under short range perturbations - Nonlinear Sciences > Exactly Solvable and Integrable Systems

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Abstract: We consider the stability of the periodic Toda lattice (and slightly moregenerally of the algebro-geometric finite-gap lattice) under a short rangeperturbation. We prove that the perturbed lattice asymptotically approaches amodulated lattice.More precisely, let $g$ be the genus of the hyperelliptic curve associatedwith the unperturbed solution. We show that, apart from the phenomenon of thesolitons 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 inthe same isospectral torus. In between there are $g+1$ regions where theperturbed solution is asymptotically close to a modulated lattice whichundergoes a continuous phase transition (in the Jacobian variety) and whichinterpolates between these isospectral solutions. In the special case of thefree lattice ($g=0$) the isospectral torus consists of just one point and werecover the known result.Both the solutions in the isospectral torus and the phase transition areexplicitly characterized in terms of Abelian integrals on the underlyinghyperelliptic curve.Our method relies on the equivalence of the inverse spectral problem to amatrix Riemann-Hilbert problem defined on the hyperelliptic curve andgeneralizes the so-called nonlinear stationary phase-steepest descent methodfor Riemann-Hilbert problem deformations to Riemann surfaces.

Author: ** Spyridon Kamvissis, Gerald Teschl**

Source: https://arxiv.org/