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

Abstract

We prove the asymptotic stability in H1 (ℝ) of the family of solitary waves for the Benjamin-Bona-Mahony equation, (1 - ∂ x2)ut + (u + u2)x = 0. We prove that a solution initially close to a solitary wave, once conveniently translated, converges weakly in H1(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.