# Bound state soliton gas dynamics underlying the noise-induced modulational instability

## TL;DR

This paper models the long-term behavior of noise-induced modulational instability in nonlinear waves using multi-soliton solutions of the nonlinear Schrödinger equation, explaining experimental observations through integrable turbulence theory.

## Contribution

It introduces a novel model based on N-soliton solutions to explain the statistical properties of noise-induced MI, bridging experiments and integrable turbulence theory.

## Key findings

- Complete spectral and statistical agreement with experiments
- Construction of large ensembles of multi-soliton solutions
- Generalization potential to other integrable turbulence problems

## Abstract

We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IST), we propose a model of the asymptotic stage of the noise-induced MI based on $N$-soliton solutions ($N$-SS) of the integrable focusing one-dimensional nonlinear Schr\"odinger equation (1D-NLSE). These $N$-SS are bound states of strongly interacting solitons having a specific distribution of the IST eigenvalues together with random phases. We use a special approach to construct ensembles of multi-soliton solutions with statistically large number of solitons $N\sim100$. Our investigation demonstrates complete agreement in spectral (Fourier) and statistical properties between the long-term evolution of the condensate perturbed by noise and the constructed multi-soliton bound states. Our results can be generalised to a broad class of integrable turbulence problems in the cases when the wave field dynamics is strongly nonlinear and driven by solitons.