Short Pulses Approximations in Dispersive Media

We derive various approximations for the solutions of nonlinear hyperbolic systems with fastly oscillating initial data. We first provide error estimates for the so-called slowly varying envelope, full dispersion, and Schrödinger approximations in a Wiener algebra; this functional framework allows u...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 2009-01, Vol.41 (2), p.708-732
Main Authors: Colin, Mathieu, Lannes, David
Format: Article
Language:English
Subjects:
Approximation
Dispersions
Envelopes
Fourier transforms
Mathematical analysis
Mathematical models
Optics
Permissible error
Propagation
Schroedinger equation
Short pulses
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Summary:We derive various approximations for the solutions of nonlinear hyperbolic systems with fastly oscillating initial data. We first provide error estimates for the so-called slowly varying envelope, full dispersion, and Schrödinger approximations in a Wiener algebra; this functional framework allows us to give precise conditions on the validity of these models; we give in particular a rigorous proof of the "practical rule" which serves as a criterion for the use of the slowly varying envelope approximation (SVEA). We also discuss the extension of these models to short pulses and more generally to large spectrum waves, such as chirped pulses. We then derive and justify rigorously a modified Schrödinger equation with improved frequency dispersion. Numerical computations are then presented, which confirm the theoretical predictions.
ISSN:0036-1410
1095-7154
DOI:10.1137/070711724