Stability of the periodic Toda lattice under short range perturbations

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.Comment: 38 pages, 1 figure. This version combines both the original version and arXiv:0805.384

Long-Time Asymptotics of Perturbed Finite-Gap Korteweg-de Vries Solutions

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of solutions of the Korteweg--de Vries equation which are decaying perturbations of a quasi-periodic finite-gap background solution. We compute a nonlinear dispersion relation and show that the $x/t$ plane splits into $g+1$ soliton regions which are interlaced by $g+1$ oscillatory regions, where $g+1$ is the number of spectral gaps. In the soliton regions the solution is asymptotically given by a number of solitons travelling on top of finite-gap solutions which are in the same isospectral class as the background solution. In the oscillatory region the solution can be described by a modulated finite-gap solution plus a decaying dispersive tail. The modulation is given by phase transition on the isospectral torus and is, together with the dispersive tail, explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve.Comment: 45 pages. arXiv admin note: substantial text overlap with arXiv:0705.034