Abstract

Integrable turbulence has proven to be a successful framework to describe random nonlinear waves in integrable dynamics such as the focusing nonlinear Schrödinger (fNLS) equation. The soliton gas, which can be seen as a stochastic ensemble of interacting solitons, constitutes one of the fundamental cases of integrable turbulence. We investigate in this work the likelihood of extreme events for a soliton gas of the fNLS equation, by computing analytically the kurtosis of the solution wave amplitude.The theory is illustrated by physically relevant examples of dense soliton gas. We first consider the asymptotic development of the noise induced modulational instability of the fNLS equation, which can be modeled by a bound state soliton gas. We then extend the result to the more general problem of the evolution of partially coherent waves, corresponding to stochastic initial conditions with slowly varying amplitude. For each example, the analytical formula of the kurtosis successfully compares to the value obtained with the numerical implementation of the problem.This is a joint work with G. El, S. Randoux, G. Roberti, P. Suret and A. Tovbis.