Stability of line solitons for the KP-II equation in $\R^2$

TL;DR

This paper proves the nonlinear stability of line solitons for the KP-II equation in two dimensions, showing how localized perturbations affect the amplitude and phase shift over time.

Contribution

It establishes the stability of line solitons under exponentially localized transverse perturbations and describes their evolution via 1D wave equations with diffraction.

Findings

Amplitude converges to initial value

Phase shifts propagate at finite speed

Local crest behavior described by wave equations

Abstract

We prove nonlinear stability of line soliton solutions of the KP-II equation with respect to transverse perturbations that are exponentially localized as $x\to\infty$. We find that the amplitude of the line soliton converges to that of the line soliton at initial time whereas jumps of the local phase shift of the crest propagate in a finite speed toward $y=\pm\infty$. The local amplitude and the phase shift of the crest of the line solitons are described by a system of 1D wave equations with diffraction terms.