A geometric interpretation of the complex tensor Riccati equation for Gaussian beams

DOI

10.2991/jnmp.2007.14.1.8How to use a DOI?

Abstract

We study the complex Riccati tensor equation DcG + GCG - R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space. When time moves forward, Gaussian beams move along geodesics, and the Riccati equation determines the Hessian of the phase function for the Gaussian beam. The imaginary part of a solution G describes how a Gaussian beam decays in different directions of space. The main result of the present work is that the real part of G is the shape operator of the phase front for the Gaussian beam. This result generalizes a known result for the Riccati equation in R3 . The idea of the proof is to express the Riccati equation in Fermi coordinates adapted to the underlying geodesic. In Euclidean geometry we also study when the phase front is contained in the area of influence, or light cone.

Copyright

© 2007, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - M.F. Dahl
PY  - 2007
DA  - 2007/02/01
TI  - A geometric interpretation of the complex tensor Riccati equation for Gaussian beams
JO  - Journal of Nonlinear Mathematical Physics
SP  - 95
EP  - 111
VL  - 14
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2007.14.1.8
DO  - 10.2991/jnmp.2007.14.1.8
ID  - Dahl2007
ER  -