Hidden solitons in the Zabusky-Kruskal experiment: Analysis using the periodic, inverse scattering transform

TL;DR

This paper uses nonlinear Fourier analysis to confirm the existence and quantify hidden solitons in the Zabusky-Kruskal experiment, revealing detailed properties of solitons and other oscillation modes across dispersion parameters.

Contribution

It applies Osborne's periodic inverse scattering transform to accurately identify and analyze hidden solitons, providing exact counts, amplitudes, and reference levels, and explores their behavior with varying dispersion.

Findings

Confirmed the existence of hidden solitons using nonlinear Fourier analysis.

Computed the exact number, amplitudes, and reference levels of solitons.

Found non-soliton oscillation modes with nontrivial energy contributions.

Abstract

Recent numerical work on the Zabusky--Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne's nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the \emph{exact} number of solitons, their amplitudes and their reference level is computed. Other "less nonlinear" oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions.