On the origin of heavy tail statistics in equations of the nonlinear Schrödinger type: an exact relation

Davide Proment
University of East Anglia

We study the formation of extreme events in incoherent systems described by the nonlinear Schrödinger type of equations. We derive an identity that relates the evolution of the kurtosis (a measure of the relevance of the tails in the probability density function) of the wave amplitude to the rate of change of the width of the Fourier spectrum of the wave field. The result is exact for all dispersive systems characterised by a nonlinear term of the form of the one contained in the nonlinear Schrödinger equation. Numerical simulations are also performed to confirm our findings. Our work sheds some light on the origin of rogue waves in incoherent dispersive nonlinear media ruled by local cubic nonlinearity. (joint work with Miguel Onorato, Gennady El, Stéphane Randoux, and Pierre Suret)