On the Wave Turbulence Theory for the Nonlinear Schr\"odinger Equation with Random Potentials
Mathematical Physics
2019-05-16 v1 Statistical Mechanics
Analysis of PDEs
math.MP
Chaotic Dynamics
Applied Physics
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.
Keywords
Cite
@article{arxiv.1905.06323,
title = {On the Wave Turbulence Theory for the Nonlinear Schr\"odinger Equation with Random Potentials},
author = {Sergey Nazarenko and Avy Soffer and Minh-Binh Tran},
journal= {arXiv preprint arXiv:1905.06323},
year = {2019}
}