  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.
