On the Zakharov–L’vov Stochastic Model
for Wave Turbulence
A. V. Dymova,* and S. B. Kuksinb,c,d,**
a Steklov Mathematical Institute, Russian Academy
of Sciences, Moscow, 119991 Russia
b Université Paris-Diderot (Paris 7), Paris, 75205 France
c School of Mathematics, Shandong University, Jinan, PRC
d St. Petersburg State University, St. Petersburg, Russia
Correspondence to: * e-mail: dymov@mi-ras.ru
Correspondence to: ** e-mail: Sergei.Kuksin@imj-prg.fr
Received 9 November, 2019
Abstract—In this paper we discuss a number of rigorous results in the stochastic model for wave turbulence due to Zakharov–L’vov. Namely, we consider the damped/driven (modified) cubic nonlinear Schrödinger equation on a large torus and decompose its solutions to formal series in the amplitude. We show that when the amplitude goes to zero and the torus’ size goes to infinity the energy spectrum of the quadratic truncation of this series converges to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.
Keywords: wave turbulence, energy spectrum, wave kinetic equation, kinetic limit, nonlinear Schrödinger equation, stochastic perturbation
DOI: 10.1134/S1064562420020106