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

2005; American Institute of Mathematical Sciences; Volume: 13; Issue: 3 Linguagem: Inglês

10.3934/dcds.2005.13.583

ISSN

1553-5231

Autores

Khaled El Dika,

Tópico(s)

Stability and Controllability of Differential Equations

Resumo

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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