Error estimates for Gaussian beam superpositions

Abstract:

Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrödinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength $\varepsilon$. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrödinger equations subject to highly oscillatory initial data of the form $Ae^{i\Phi /\varepsilon }$. Through a careful estimate of an oscillatory integral operator, we prove that the $k$-th order Gaussian beam superposition converges to the original wave field at a rate proportional to $\varepsilon ^{k/2}$ in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, $\varepsilon$-scaled, energy norm and for the Schrödinger equation in the $L^2$ norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in $\mathbb {R}^2$ to analyze the sharpness of the theoretical results.