In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of [Martel, Y. and Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), 219-254], [Martel, Y. and Merle, F.: Asymptotic stability of solitons of the gKdV equations with a general nonlinearity. Math. Ann. 341 (2008), 391-427]. As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.
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Carlos E. Kenig, Yvan Martel, Asymptotic stability of solitons for the Benjamin-Ono equation. Rev. Mat. Iberoam. 25 (2009), no. 3, pp. 909–970DOI 10.4171/RMI/586