On spatial decay for coherent states of the Benjamin-Ono equation

We consider solutions to the Benjamin-Ono equation ∂ₜ u - H ∂ₓ² u = -∂ₓ(u²) that are localized in a reference frame moving to the right with constant speed. We show that any such solution that decays at least like x-1-ε for some $ε> 0$ in a comoving coordinate frame must in fact decay like x⁻². In view of the explicit soliton solutions, this decay rate is sharp. Our proof has two main ingredients. The first is microlocal dispersive estimates for the Benjamin-Ono equation in a moving frame, which allow us to prove spatial decay of the solution provided the nonlinearity has sufficient decay. The second is a careful normal form analysis, which allows us to obtain rapid decay of the nonlinearity for a transformed equation assuming only modest decay of the solution. Our arguments are entirely time dependent, and do not require the solution to be an exact traveling wave.