The Short Wave equation on the half-line by the Unified Transform Method

Abstract

We study the initial-boundary value problem for the Short Wave equation on the half-line $x\ge 0$. A distinctive feature of this problem is that the boundary $x=0$ cannot be characterized \emph{a priori} as an inflow or outflow boundary: its character is determined dynamically by the sign of the unknown trace $u(0,t)$. This leads to different analyticity properties of the associated eigenfunctions and, consequently, to different spectral formulations in the regimes $u(0,t)\le0$ and $u(0,t)\ge0$. Using the Unified Transform Method (aka the Fokas method), we formulate the solution in terms of matrix Riemann--Hilbert problems. We construct the associated spectral functions, derive the global relations, and show how the solution is reconstructed from the corresponding Riemann--Hilbert problem. In the case $u(0,t)\le0$, the solution is determined by the initial data alone (assuming an appropriate decay as $x\to \infty$), whereas for $u(0,t)\ge0$, compatible boundary data are also required for the construction.