On the asymptotic behavior of solutions to the Benjamin-Ono equation

We prove that the limit infimum, as time $t$ goes to infinity, of any uniformly bounded in time $H^1\cap L^1$ solution to the Benjamin-Ono equation converge to zero locally in an increasing-in-time region of space of order $t\mathrm{/}\log t$. Also for a solution with a mild $L^1$-norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending of the rate of growth of its $L^1$-norm. In particular, we discard the existence of breathers and other solutions for the BO model moving with a speed \lq\lq slower" than a soliton.