On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials

Abstract

We derive a new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form with the 4-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread infinitely as time goes to infinity and this fact answers the "weak turbulence" question for the nonlinear Schrödinger equation with random potentials positively. We also derive Ohm's law for the porous medium equation.

Weak Langmuir turbulence in disordered multimode optical fibers

“…Such a generalized theory can also shed new light on the recent experiments of supercontinuum generation that can be characterized by spatial beam cleaning effects [71][72][73][74]. From a broader perspective, this would contribute to the development of a wave turbulence theory that accounts for the presence of a structural disorder of the nonlinear medium [28,29,[75][76][77].…”

Interplay of thermalization and strong disorder: Wave turbulence theory, numerical simulations, and experiments in multimode optical fibers

“…On the other hand, a structural disorder of the nonlinear medium is known to deeply affect the coherence properties of the waves. Understanding the interplay of nonlinearity and disorder is a fundamental problem, in relation with the paradigm of statistical light-mode dynamics, glassy behaviors and complexity science [34][35][36][37][38][39][40][41][42]. Disorder is also known to impact light propagation in MMFs, a feature relevant to endoscopic imaging [43,44], or to study completely integrable Manakov systems [45][46][47][48].…”

“…Our work paves the way for the development of a systematic method to tackle the impact of a time-dependent disorder in wave turbulence -our methodology substantially differs from that developed for a time-independent disorder [38][39][40][41][42]. More generally this work contributes to the understanding of spontaneous organization of coherent states in nonlinear disordered systems [34][35][36][37][38].…”

“…The parameters α and γ denote the linear and nonlinear coefficients. The disorder being ('time') z−dependent, our system is of different nature than those studying the interplay of thermalization and Anderson localization [38][39][40][41][42].…”