# Asymptotic behavior of solutions of the dispersive generalized Benjamin-Ono equation

## TL;DR

This paper investigates the long-term behavior of solutions to the dispersive generalized Benjamin-Ono equation, showing that under certain conditions, solutions tend to zero locally in expanding regions of space as time progresses.

## Contribution

It establishes the asymptotic decay of solutions in expanding spatial regions and rules out the existence of breathers or slow-moving solutions for the equation.

## Key findings

- Solutions tend to zero locally in expanding regions as time goes to infinity.
- The decay rate depends on the growth of the solution's L^1-norm.
- Breathers or slow-moving solutions are proven not to exist for this equation.

## Abstract

We show that for any uniformly bounded in time $H^1\cap L^1$ solution of the dispersive generalized Benjamin-Ono equation, the limit infimum, as time $t$ goes to infinity, converges to zero locally in an increasing-in-time region of space of order $t/\log t$. This result is in accordance with the one established by Mu\~noz and Ponce \cite{MP1} for solutions of the Benjamin-Ono equation. Similar to solutions of the Benjamin-Ono equation, for a solution of the dispersive generalized Benjamin-Ono equation, 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 on the rate of growth of its $L^1$-norm. As a consequence, the existence of breathers or any other solution for the dispersive generalized Benjamin-Ono equation moving with a speed "slower" than a soliton is discarded. In our analysis the use of commutators expansions is essential.