On the numerical accuracy of the method of multiple scales for nonlinear dispersive wave equations

TL;DR

This paper evaluates the numerical accuracy of the method of multiple scales (MMS) for nonlinear dispersive wave equations, highlighting its robustness even when the amplitude equation is ill-posed.

Contribution

It provides a detailed numerical analysis of MMS applied to dispersive wave equations, including cases with ill-posed amplitude equations, demonstrating its validity as an approximation.

Findings

MMS remains valid even for ill-posed amplitude equations

Numerical tests confirm the accuracy of MMS for different dispersion models

The amplitude equation's linearity and complex frequency are key features

Abstract

In this paper we study dispersive wave equation using the method of multiple scales (MMS) and perform several numerical tests to investigate its accuracy. The key feature of our MMS solution is the linearity of the amplitude equation and the complex nature of the time-frequency. The MMS is tested as an initial value problem using three choices of the dispersion model, one toy and two Lorentz models. Depending on the parameters of the problem, the amplitude equation can be both well- or ill-posed. Despite the ill-posedness, the MMS solution remains a valid approximation of the solution to the original nonlinear model.