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

Abstract

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.

45 pages. arXiv admin note: substantial text overlap with arXiv:0705.0346

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• Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra
• Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results
• Scattering theory for Schrödinger operators on steplike, almost periodic infinite-gap backgrounds
• Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation
• Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background