General superpositions of Gaussian beams and propagation errors

Hailiang Liu; James Ralston; Peimeng Yin

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arXiv · 1905.09090

General superpositions of Gaussian beams and propagation errors

Hailiang Liu·James Ralston·Peimeng Yin

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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 $k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.

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Hailiang Liu, James Ralston, Peimeng Yin. 2019-05-22. General superpositions of Gaussian beams and propagation errors. https://arxiv.org/abs/1905.09090

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