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Annales Henri Poincaré, 2023, Volume 24, Pages 3685–3739
DOI: https://doi.org/10.1007/s00023-023-01366-2
(Mi annhp5)
 

This article is cited in 1 scientific paper (total in 1 paper)

The Large-Period Limit for Equations of Discrete Turbulence

Andrey Dymovabc, Sergei Kuksinadb, Alberto Maiocchie, Sergei Vlăduţfg

a Steklov Mathematical Institute of RAS, Moscow, Russia
b Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia
c Skolkovo Institute of Science and Technology, Skolkovo, Russia
d Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris
e Dipartimento di Matematica e Applicazioni - Edificio U5, Università degli Studi di Milano-Bicocca, via Roberto Cozzi, 55, 20125 Milan, Italy
f IITP RAS, 19 B. Karetnyi, Moscow, Russia
g CNRS, Centrale Marseille, I2M UMR 7373, Aix Marseille Université, 13453 Marseille, France
Citations (1)
Abstract: We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $\lambda$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first $\lambda\to0$ and then $L\to\infty$. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit $L\to\infty$ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second-order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.
Funding agency Grant number
Russian Science Foundation 23-11-00150
Ministry of Science and Higher Education of the Russian Federation 075-15-2022-1115
The work of AD on Sects. (1, 2, 3) was supported by the Russian Science Foundation under Grant no.23-11-00150. The work of AD on the other sections and work of AM and SK on the whole paper was supported by the Ministry of Science and Higher Education of the Russian Federation (megagrant No. 075-15-2022-1115).
Received: 13.01.2022
Accepted: 28.08.2023
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Document Type: Article
Language: English
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