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

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 $u(0,t)\le 0$ and $u(0,t)\ge 0$. 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 $u(0,t)\le 0$, the solution is determined by the initial data alone (assuming an appropriate decay as $x\to \mathrm{\infty }$), whereas for $u(0,t)\ge 0$, compatible boundary data are also required for the construction.