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

TL;DR

This paper proves that solutions to the Benjamin-Ono equation, localized in a moving frame, must decay at a specific sharp rate, using microlocal dispersive estimates and normal form analysis, without requiring exact traveling waves.

Contribution

It establishes the optimal spatial decay rate for localized solutions of the Benjamin-Ono equation in a moving frame, employing novel microlocal and normal form techniques.

Findings

Solutions decay at least like ⟨x⟩^{-2} in a moving frame.

Decay rate of ⟨x⟩^{-2} is sharp, matching explicit soliton solutions.

The method does not require solutions to be exact traveling waves.

Abstract

We consider solutions to the Benjamin-Ono equation $$\partial_t u - H \partial_x^2 u = -\partial_x(u^2)$$ 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 $\langle x\rangle^{-1-\epsilon}$ for some $\epsilon > 0$ in a comoving coordinate frame must in fact decay like $\langle x\rangle^{-2}$. 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.