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

We prove the asymptotic stability in H¹ (R) of the family of solitary waves for the Benjamin-Bona-Mahony equation, (1-²ₓ) 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.