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

TL;DR

This paper investigates the long-term behavior of solutions to the Benjamin-Ono equation, showing they tend to zero locally in expanding regions of space, and rules out certain persistent localized solutions like breathers.

Contribution

It establishes the asymptotic decay of solutions in expanding spatial regions and excludes the existence of breathers for the Benjamin-Ono equation.

Findings

Solutions tend to zero locally in regions expanding as t/log t.

Solutions with mild L^1 growth also tend to zero in expanding regions.

Breathers and similar solutions moving slower than solitons do not exist.

Abstract

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 $\,t/\log t$. 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.