The "good" Boussinesq equation on the half-line: a Riemann-Hilbert approach

TL;DR

This paper develops a Riemann-Hilbert framework to solve the 'good' Boussinesq equation on the half-line, linking solutions to initial and boundary data through a 3x3 matrix problem with a specific jump contour.

Contribution

It introduces a novel Riemann-Hilbert problem formulation for the 'good' Boussinesq equation on the half-line, enabling solution recovery from initial and boundary conditions.

Findings

Solution can be reconstructed from a 3x3 Riemann-Hilbert problem.

The jump contour consists of twelve half-lines.

The approach depends only on initial and boundary data.

Abstract

We consider the ``good" Boussinesq equation on the half-line. Assuming existence of the solution, we prove that it can be recovered from the solution of a $3\times 3$ Riemann-Hilbert problem that depends only on the initial and boundary values, and whose jump contour consists of twelve half-lines.