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

01 Mar 2023-
TL;DR: In this article , the authors considered the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves and provided an exact expression for the soliton-radiation interaction to leading order.
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.

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