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 Schr\"odinger 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 regimes of fNLSE integrable turbulence. The results have broad applicability to physical systems including water waves, nonlinear optics, and superfluids.