# Boussinesq's equation for water waves: the soliton resolution conjecture for Sector IV

## TL;DR

This paper derives an asymptotic formula for the Boussinesq equation in Sector IV with solitons, providing insights into soliton-radiation interactions and supporting the soliton resolution conjecture for water waves.

## Contribution

It extends previous work by deriving the leading asymptotic behavior in Sector IV with solitons, confirming the soliton resolution conjecture for this case.

## Key findings

- Exact expression for soliton-radiation interaction
- Verification of the soliton resolution conjecture in Sector IV
- Asymptotic behavior characterized by specific x/t ratio

## Abstract

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by $\frac{x}{t}\in (\frac{1}{\sqrt{3}},1)$, in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.