Solitons in a box-shaped wavefield with noise: perturbation theory and statistics

TL;DR

This paper develops a linear perturbation theory for the nonlinear Schrödinger equation to analyze how noise affects soliton parameters, providing analytical expressions and probabilistic predictions validated by numerical simulations.

Contribution

It introduces a complete linear perturbation framework for scattering data corrections in the focusing NLSE, including statistical modeling of noise effects on solitons.

Findings

Analytical formulas for soliton parameter sensitivity to perturbations.

Statistical relations for soliton parameter fluctuations due to noise.

Numerical validation of probabilistic soliton emergence predictions.

Abstract

We investigate the fundamental problem of the nonlinear wavefield scattering data corrections in response to a perturbation of initial condition using inverse scattering transform theory. We present a complete theoretical linear perturbation framework to evaluate first-order corrections of the full set of the scattering data within the integrable one-dimensional focusing nonlinear Schr\"odinger (NLSE) equation. The general scattering data portrait reveals nonlinear coherent structures - solitons - playing the key role in the wavefield evolution. Applying the developed theory to a classic box-shaped wavefield we solve the derived equations analytically for a single Fourier mode acting as a perturbation to the initial condition, thus, leading to the sensitivity closed-form expressions for basic soliton characteristics, i.e. the amplitude, velocity, phase and its position. With the appropriate statistical averaging we model the soliton noise-induced effects resulting in compact relations for standard deviations of soliton parameters. Relying on a concept of a virtual soliton eigenvalue we derive the probability of a soliton emergence or the opposite due to noise and illustrate these theoretical predictions with direct numerical simulations of the NLSE evolution. The presented framework can be generalised to other integrable systems and wavefield patterns.