The Gaussian semiclassical soliton ensemble and numerical methods for the focusing nonlinear Schr\"odinger equation

TL;DR

This paper investigates the semiclassical limit of the focusing nonlinear Schr"odinger equation through numerical experiments, validating schemes, and analyzing the evolution of perturbed initial data, revealing the propagation of small perturbations over time.

Contribution

It introduces an implicit finite difference scheme validated against inverse scattering, and studies the effect of small spectral perturbations in the semiclassical limit of the focusing NLS.

Findings

Modified initial data converges at an O(ε) rate to true data.

Perturbations propagate and persist after wave breaking.

Validated numerical schemes for the focusing NLS in the semiclassical regime.

Abstract

We report on a number of careful numerical experiments motivated by the semiclassical (zero-dispersion, \epsilon\downarrow 0) limit of the focusing nonlinear Schr\"odinger equation. Our experiments are designed to study the evolution of a particular family of perturbations of the initial data. These asymptotically small perturbations are precisely those that result from modifying the initial-data by using formal approximations to the spectrum of the associated spectral problem; such modified data has always been a standard part of the analysis of zero-dispersion limits of integrable systems. However, in the context of the focusing nonlinear Schr\"odinger equation, the ellipticity of the Whitham equations casts some doubt on the validity of this procedure. To carry out our experiments, we introduce an implicit finite difference scheme for the partial differential equation, and we validate both the proposed scheme and the standard split-step scheme against a numerical implementation of the inverse scattering transform for a special case in which the scattering data is known exactly. As part of this validation, we also investigate the use of the Krasny filter which is sometimes suggested as appropriate for nearly ill-posed problems such as we consider here. Our experiments show that that the O(\epsilon) rate of convergence of the modified data to the true data is propagated to positive times including times after wave breaking.