Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation

Khaled El Dika

doi:10.3934/dcds.2005.13.583

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2005, Volume 13, Issue 3: 583-622. Doi: 10.3934/dcds.2005.13.583

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Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation

Khaled El Dika^1,

1.
Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay

Received: July 31, 2004

Revised: February 28, 2005

Published: March 2005

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Abstract

We prove the asymptotic stability in $H^1(\mathbb R)$ of the family of solitary waves for the Benjamin-Bona-Mahony equation,

$(1-\partial^2_x)u_t+(u+u^2)_x=0.$

We prove that a solution initially close to a solitary wave, once conveniently translated, converges weakly in $H^1(\mathbb R)$, as time goes to infinity, to a possibly different solitary wave. The proof is based on a Liouville type theorem for the flow close to the solitary waves, and makes an extensive use of a monotonicity property.

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Mathematics Subject Classification: 35B40, 35Q51, 35Q53.

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