Stability of Solitons for the KdV equation in H^s, 0 <= s< 1

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 $H^s$, $0\le s<1$, 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 result of \cite{CKSTT3}, and obtains the same maximal growth rate in $t$. Our argument is based on the {``}$I$-method{\rq\rq} used in \cite{CKSTT3} and other papers of Colliander, Keel, Staffilani, Takaoka and Tao, which pushes these $H^s$ functions to the $H^1$ norm.