# On the Wave Turbulence Theory for the Nonlinear Schr\"odinger Equation with Random Potentials

## TL;DR

This paper derives new kinetic and porous medium equations from the nonlinear Schrödinger equation with random potentials, establishing connections to wave turbulence theory and demonstrating solutions that spread infinitely, confirming aspects of weak turbulence.

## Contribution

It introduces a novel kinetic equation similar to wave turbulence theory and constructs self-similar solutions for the porous medium equation in this context.

## Key findings

- Derived a new kinetic equation resembling 4-wave turbulence
- Constructed self-similar solutions that spread infinitely over time
- Confirmed the 'weak turbulence' behavior for the nonlinear Schrödinger equation with randomness

## Abstract

We derive a new kinetic and a porous medium equations from the nonlinear Schr\"odinger 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 Schr\"odinger equation with random potentials positively. We also derive Ohm's law for the porous medium equation.