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

TL;DR

This paper extends the stability analysis of line solitons in the KP-II equation to broader classes of perturbations, demonstrating their nonlinear stability under less restrictive conditions.

Contribution

It proves the nonlinear stability of 1-line solitons for the KP-II equation with new classes of perturbations, expanding previous stability results.

Findings

Stability of 1-line solitons under polynomial decay perturbations.

Stability under perturbations in combined Sobolev and weighted spaces.

Extension of stability results to more general perturbation classes.

Abstract

The KP-II equation was derived by Kadmotsev and Petviashvili to explain stability of line solitary waves of shallow water. Recently, Mizumachi (Mem. Amer. Math. Soc. 238 (2015)) has proved nonlinear stability of $1$-line solitons for exponentially localized perturbations. In this paper, we prove stability of $1$-line solitons for perturbations in $(1+x^2)^{-1/2-0}H^1(\mathbb{R}^2)$ and perturbations in $H^1(\mathbb{R}^2)\cap \partial_xL^2(\mathbb{R}^2)$.