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

TL;DR

This paper analyzes the long-time behavior of perturbed finite-gap solutions to the Korteweg-de Vries equation, revealing a detailed asymptotic structure with soliton and oscillatory regions characterized by spectral data.

Contribution

It applies nonlinear steepest descent to derive explicit long-time asymptotics for perturbed finite-gap KdV solutions, including a nonlinear dispersion relation and phase modulation.

Findings

The x/t plane splits into g+1 soliton regions and g+1 oscillatory regions.

Soliton regions are dominated by traveling solitons on finite-gap backgrounds.

Oscillatory regions are described by modulated finite-gap solutions plus dispersive tails.

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.