Rogue waves, self-similar statistics, and self-similar intermediate asymptotics

Abstract

We explore extreme event emergence in statistical nonlinear wave systems with self-similar intermediate asymptotics. We show, within the framework of a generic (1 + 1)D nonlinear Schrödinger equation with linear gain, that the rogue waves in weakly nonlinear, statistical open systems emerge as parabolic-shape giant fluctuations in the self-similar asymptotic propagation regime. We analytically derive the non-Gaussian statistics of such rogue waves, validate our results with numerical simulations, and demonstrate the self-similar structure of the emergent rogue wave statistics. Our results shed new light on generic statistical features of rogue waves in nonlinear open systems with self-similar intermediate asymptotics.Rogue waves (RW), extremely rare, giant-amplitude waves obeying non-Gaussian statistics, were originally discussed in the oceanographic context [1-3]. The concept has been quickly recognized as germane to generic wave supporting physics settings and RWs have been discovered, among others, in supercontinuum generating optical fibers [4,5], optical cavities [6], Bose-Einstein condensates [7], Raman fiber amplifiers [8,9], fiber lasers [10,11], laser filamentation [12], plasmas [13], stimulated Raman scattering [14,15], discrete nonlinear lattices [16], and even in the multimode optical fibers and microwave transport in the linear propagation regime [17,18].