Short Pulses Approximations in Dispersive Media

TL;DR

This paper rigorously analyzes various approximations for nonlinear hyperbolic systems with rapidly oscillating initial data, providing error estimates, validity conditions, and numerical validation for models like SVEA and modified Schrödinger equations.

Contribution

It offers a comprehensive theoretical framework for the validity of short pulse approximations in dispersive media, including error bounds and extensions to large spectrum waves.

Findings

Error estimates for envelope and Schrödinger approximations

Validation of the SVEA practical rule

Numerical confirmation of theoretical predictions

Abstract

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 Schr\"odinger 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 Schr\"odinger equation with improved frequency dispersion. Numerical computations are then presented, which confirm the theoretical predictions.