Low regularity stability of solitons for the KdV equation
The authors investigate the long-time stability of soliton solutions to the Korteweg-de Vries (KdV) equation. They deal with solutions \(u\) to the KdV equation with initial data in \(H^s\) (with \(0 <s< 1\)), that are initially close in \(H^s\) norm to a soliton. It is proven that the possible orbital instability of these ground states is almost polynomial in time. This is an analogue to the \(H^s\) orbital instability results achieved in [\textit{J. Colliander, M. Keel, G. Staffilani, H. Takaoka} and \textit{T. Tao}, Discrete Contin. Dyn. Syst. 9, No.~1, 31--54 (2003; Zbl 1028.35141)] for the nonlinear Schrödinger equation. The main result of the paper is contained in Theorem 1.1. In Section 2 some notations are introduced and certain estimates are quoted that will be used in the remaining sections. Section 3 is devoted to make a first attempt at proving the main Theorem, obtaining a weaker version of the main result of the paper. Section 4 contains a second pass of the main Theorem, involving a refining of the techniques of Section 3. Finally, the proof of the main Theorem is completed in Section 5.
scientific article; zbMATH DE number 2053140
scientific article