# American Institute of Mathematical Sciences

April  2005, 13(3): 583-622. doi: 10.3934/dcds.2005.13.583

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

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

Received  August 2004 Revised  March 2005 Published  May 2005

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.

Citation: Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583
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