The "Nonlinear Waves and Coherent Structures Online Colloquium" is a monthly seminar that will have speakers from a diverse range of disciplines to speak on a broad range of topics connected to nonlinear waves and coherent structures. The organizers hope this series will help maintain a sense of community and provide a broad audience with access to research developments on the forefront of this field.
Title: From soliton gases to wave statistics in integrable turbulence
Abstract: The focusing nonlinear Schrödinger equation provides a canonical model for integrable turbulence, describing the evolution of random nonlinear waves in an integrable system, with applications to water waves, nonlinear optics and superfluids. In regimes dominated by solitons, these dynamics can be understood in terms of a soliton gas: a statistical ensemble of interacting, localised nonlinear waves. Such a gas is naturally characterised by the distribution of soliton amplitudes and velocities, which are encoded in the spectrum of the associated Zakharov-Shabat problem. Relating this spectral distribution (the density of states) to quantities measured in experiments, such as the probability of observing a wave of a given height, remains a central challenge.
This talk presents an analytical framework for determining the probability distribution (PDF) of wave intensity in these (soliton-dominated) regimes. We formulate a soliton gas resolution conjecture for the long-time evolution of slowly varying, or “semiclassical”, random initial states. Extending the classical soliton resolution picture, the conjecture, supported by numerical simulations and experimental data, asserts that such states evolve at long times into a soliton gas whose statistical properties are determined by the initial data. A stochastic analogue of the inverse scattering transform then relates the spectral density of states, which encodes the distribution of soliton parameters in this gas, to the probability density function of the wave intensity.
The resulting explicit integral representation for the intensity PDF is in excellent agreement with direct numerical simulations for a representative examples of integrable turbulence. Within the proposed framework, the analytical results are exact and require no statistical closure assumptions. These findings provide a direct link between the soliton description of integrable turbulence and the observable statistics of random nonlinear waves.
Based on joint work with Gennady El (arXiv:arXiv:2605.06306 [nlin.PS]).
How to join: An email containing the zoom link and password for the seminar will be sent out by 9:30am (Eastern Time) the day of the talk. To join the email list please register here
Speakers: To see the full list of upcoming speakers and recordings of prior talks, click here
Organizers:
Stathis Charalampidis, SDSU, San Diego
Sathyanarayanan Chandramouli, Okinawa Institute of Science and Technology
Panos Kevrekidis, UMass Amherst