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

## TL;DR

This paper develops a statistical theory explaining how rogue waves and extreme events emerge as self-similar giant fluctuations in nonlinear wave systems, supported by analytical and numerical evidence.

## Contribution

It introduces a novel self-similar statistical framework for rogue wave emergence in nonlinear systems with gain, advancing understanding of extreme event formation.

## Key findings

- Rogue waves appear as parabolic giant fluctuations in self-similar regimes.
- The non-Gaussian statistics of rogue waves are analytically characterized.
- Numerical simulations validate the self-similar structure of extreme events.

## Abstract

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 analytically demonstrate the self-similar structure of the non-Gaussian statistics of emergent rogue waves and validate our results with numerical simulations. Our results shed new light on generic statistical features of extreme events in nonlinear open systems with self-similar intermediate asymptotics.