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Open AccessArticle

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

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Institut de Physique de Nice, Université Côte d’Azur, Centre national de la recherche scientifique (CNRS), Parc Valrose, 06108 Nice, France
2
Mathematics Department, Rutgers University, New Brunswick, NJ 08903 USA
3
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(9), 823; https://doi.org/10.3390/e21090823
Received: 25 July 2019 / Revised: 13 August 2019 / Accepted: 19 August 2019 / Published: 23 August 2019
(This article belongs to the Special Issue Statistical Mechanics and Mathematical Physics)
We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-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 with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation. View Full-Text
Keywords: wave turbulence theory; nonlinear schrödinger equation with random potentials; 4-wave kinetic turbulence equation; ohm’s law; porous medium equation. wave turbulence theory; nonlinear schrödinger equation with random potentials; 4-wave kinetic turbulence equation; ohm’s law; porous medium equation.
MDPI and ACS Style

Nazarenko, S.; Soffer, A.; Tran, M.-B. On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials. Entropy 2019, 21, 823.

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