Asymptotic stability of Benjamin--Ono multisolitons in L²(mathbb R)

We prove the following dichotomy result for $L^2(\mathbb R)$ solutions to the Benjamin--Ono equation: On windows traveling at any speed, the solution either converges to zero or to a soliton dictated by the spectral properties of the Lax operator associated to the initial data. As an application of this result, we prove asymptotic stability of Benjamin--Ono multisolitons in $L^2(\mathbb R)$. Specifically, we show that solutions to the Benjamin--Ono equation emanating from small $L^2(\mathbb R)$ perturbations of multisolitons evolve towards a series of separating one-solitons when viewed in windows traveling with these solitons.