Error estimates for Gaussian beams at a fold caustic

Abstract: In this work we show an error estimate for a first order Gaussian beam at a fold caustic, approximating time-harmonic waves governed by the Helmholtz equation. For the caustic that we study the exact solution can be constructed using Airy functions and there are explicit formulae for the Gaussian beam parameters. Via precise comparisons we show that the pointwise error on the caustic is of the order $O(k^{-5/6})$ where $k$ is the wave number in Helmholtz.