Low regularity stability of solitons for the KDV equation

S. Raynor; G. Staffilani

doi:10.3934/cpaa.2003.2.277

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2003, Volume 2, Issue 3: 277-296. Doi: 10.3934/cpaa.2003.2.277

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Low regularity stability of solitons for the KDV equation

S. Raynor^1, and
G. Staffilani^1,

1.
Massachusetts Institute of Technology

Received: January 2003

Revised: April 2003

Published: September 2003

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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 "I-method" used in [7] and other papers of Colliander, Keel, Staffilani, Takaoka and Tao.

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Mathematics Subject Classification: 35Q53, 42B35, 37K10, 37B25.

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