Low regularity stability of solitons for the KDV equation

Abstract

We study the long-time stability of soliton solutions to the Korteweg-deVries equation. We consider solutions $u$ to the KdV with initial data in $H^s$, $0 \leq s < 1$, that are initially close in $H^s$ norm to a soliton. We prove that the possible orbital instability of these ground states is at most polynomial in time. This is an analogue to the $H^s$ orbital instability results of [7] for the nonlinear Schrödinger equation, and obtains the same maximal growth rate in $t$. Our argument is based on the "<em>I</em>-method" used in [7] and other papers of Colliander, Keel, Staffilani, Takaoka and Tao.

Locations

• Communications on Pure &amp Applied Analysis - View - PDF