Stability and instability of the KdV solitary wave under the KP-I flow

TL;DR

This paper analyzes the stability of KdV solitary waves under the KP-I flow, proving stability for subcritical speeds and instability for supercritical speeds, with detailed constructions of converging solutions.

Contribution

It establishes the orbital stability of KdV solitons for subcritical speeds and constructs solutions demonstrating instability at supercritical speeds under the KP-I flow.

Findings

KdV solitons are orbitally stable for speeds c<c* under KP-I flow.

Supercritical speeds c>c* lead to instability, with solutions converging to the soliton.

Results extend to the generalized KP-I equation.

Abstract

We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.