General superpositions of Gaussian beams and propagation errors

Abstract

Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension m in phase space, and show that the tools recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919-952, 2013] can be applied to obtain the propagation error of order k 1− N, where N is the order of beams and d is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.2000 Mathematics Subject Classification. Primary 35L05, 35A35, 41A60.