A proof of the soliton resolution conjecture for the Benjamin--Ono equation

TL;DR

This paper proves the soliton resolution conjecture for the Benjamin--Ono equation, showing solutions decompose into solitons plus radiation over time, using spectral theory and Fourier analysis techniques.

Contribution

It provides the first rigorous proof of the soliton resolution conjecture for the Benjamin--Ono equation, linking spectral data to solution decomposition.

Findings

Solutions decompose into finite solitons and radiation asymptotically.

Spectral theory of the Lax operator characterizes the solution components.

New analytical methods improve understanding of long-time behavior.

Abstract

We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator.