Approximation of high-frequency wave propagation in dispersive media

TL;DR

This paper develops an analytical approximation for high-frequency solutions of semilinear hyperbolic systems with dispersive effects, achieving higher accuracy than classical methods over long time intervals.

Contribution

It introduces a new approximation method that accurately captures nonlinear dispersive wave propagation with an error of order , surpassing traditional approaches.

Findings

Approximation error is  over time intervals of length 1/.

Classical nonlinear Schrf6dinger approximation has  error.

Method enables feasible long-time simulations of dispersive waves.

Abstract

We consider semilinear hyperbolic systems with a trilinear nonlinearity. Both the differential equation and the initial data contain the inverse of a small parameter $\varepsilon$, and typical solutions oscillate with frequency proportional to $1/\varepsilon$ in time and space. Moreover, solutions have to be computed on time intervals of length $1/\varepsilon$ in order to study nonlinear and diffractive effects. As a consequence, direct numerical simulations are extremely costly or even impossible. We propose an analytical approximation and prove that it approximates the exact solution up to an error of $\mathcal{O}(\varepsilon^2)$ on time intervals of length $1/\varepsilon$. This is a significant improvement over the classical nonlinear Schr\"odinger approximation, which only yields an accuracy of $\mathcal{O}(\varepsilon)$.