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

We advance a statistical theory of extreme event emergence in random nonlinear wave systems with self-similar intermediate asymptotics. We show, within the framework of a generic (1 + 1)D nonlinear Schrodinger equation with linear gain, that extreme events and even rogue waves in weakly nonlinear, statistical open systems emerge as parabolic-shape giant fluctuations in the self-similar asymptotic propagation regime. We demonstrate analytically the self-similar structure of the non-Gaussian statistics of emergent rogue waves, and we validate our results with numerical simulations. Our results shed new light on the generic statistical features of extreme events in nonlinear open systems with self-similar intermediate asymptotics.