The soliton resolution conjecture for the Boussinesq equation

TL;DR

This paper advances understanding of the Boussinesq equation by providing asymptotic formulas in the presence of solitons, supporting the soliton resolution conjecture across most of the space-time plane.

Contribution

It extends previous work by deriving a formula in the sector with solitons, validating the soliton resolution conjecture for the Boussinesq equation in nearly all regions.

Findings

Asymptotic formula valid in sector with solitons

Validation of the soliton resolution conjecture

Results applicable to large positive M

Abstract

We analyze the Boussinesq equation on the line with Schwartz initial data belonging to the physically relevant class of global solutions. In a recent paper, we determined ten main asymptotic sectors describing the large $(x,t)$-behavior of the solution, and for each of these sectors we provided the leading order asymptotics in the case when no solitons are present. In this paper, we give a formula valid in the asymptotic sector $x/t \in (1,M]$, where $M$ is a large positive constant, in the case when solitons are present. Combined with earlier results, this validates the soliton resolution conjecture for the Boussinesq equation everywhere in the $(x,t)$-plane except in a number of small transition zones.