Boussinesq's equation for water waves: asymptotics in sector V

TL;DR

This paper proves the asymptotic formula for solutions of the Boussinesq equation in a specific sector, expanding understanding of water wave behavior for broad initial conditions without solitons.

Contribution

It provides a rigorous proof for the asymptotic behavior of Boussinesq solutions in a particular sector, complementing previous work on other sectors.

Findings

Exact asymptotic formula derived for sector x/t in (0, 1/√3)

Validation of asymptotic behavior for broad class of initial data

Extension of asymptotic analysis to new sector

Abstract

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data for which (i) no solitons are present, (ii) the spectral functions have generic behavior near $\pm 1$, and (iii) the solution exists globally. 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 this paper, we give a proof for the formula corresponding to the sector $\frac{x}{t}\in (0,\frac{1}{\sqrt{3}})$.