Soliton gas resolution and statistics of random wave fields in semiclassical integrable turbulence
Abstract
We develop a general analytical framework for determining the probability distribution of random nonlinear wave fields governed by the focusing nonlinear Schrödinger equation (fNLSE) in regimes where typical realizations are dominated by solitons. We formulate the soliton gas resolution conjecture for the long-time evolution of slowly varying ("semiclassical") random initial states and implement a stochastic analogue of the inverse scattering transform by establishing a relationship between the spectral density of states of the underlying bound state soliton gas and the probability density function (PDF) of the intensity of the resulting turbulent wave field. The derived explicit integral representation for the PDF is shown to be in excellent agreement with direct numerical simulations across several representative examples of fNLSE integrable turbulence. The analytical results of the paper are exact; they do not involve any closure assumptions, in contrast to classical turbulence theories. The results have broad applicability to physical systems including water waves, nonlinear optics, and superfluids, where integrable equations serve as important asymptotic models.
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BibTeX
@misc{congy2026soliton,
title = {{Soliton gas resolution and statistics of random wave fields in semiclassical integrable turbulence}},
author = {T. Congy and G. A. El},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.06306},
eprint = {2605.06306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06306},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Soliton gas resolution and statistics of random wave fields in semiclassical integrable turbulence
AU - T. Congy
AU - G. A. El
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2605.06306
UR - https://arxiv.org/abs/2605.06306
AB - We develop a general analytical framework for determining the probability distribution of random nonlinear wave fields governed by the focusing nonlinear Schrödinger equation (fNLSE) in regimes where typical realizations are dominated by solitons. We formulate the soliton gas resolution conjecture for the long-time evolution of slowly varying ("semiclassical") random initial states and implement a stochastic analogue of the inverse scattering transform by establishing a relationship between the spectral density of states of the underlying bound state soliton gas and the probability density function (PDF) of the intensity of the resulting turbulent wave field. The derived explicit integral representation for the PDF is shown to be in excellent agreement with direct numerical simulations across several representative examples of fNLSE integrable turbulence. The analytical results of the paper are exact; they do not involve any closure assumptions, in contrast to classical turbulence theories. The results have broad applicability to physical systems including water waves, nonlinear optics, and superfluids, where integrable equations serve as important asymptotic models.
ER -