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

2003/07/07 by Sarah Raynor, S. Raynor, Gigliola Staffilani +3
Mathematics · #35Q53 #37B25 #37K10 #42B35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35Q53 #msc:37B25 #msc:37K10 #msc:42B35

paper · pdf · doi:10.48550/arxiv.math/0307084

arxiv created 2003/07/07 · openalex publication_date 2003/07/07 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

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 ≤ s &lt; 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 \citeCKSTT3, and obtains the same maximal growth rate in t. Our argument is based on the ``I-method\rq\rq used in \citeCKSTT3 and other papers of Colliander, Keel, Staffilani, Takaoka and Tao, which pushes these H^s functions to the H¹ norm.

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