Kinetic approximation for equations of discrete turbulence in the subcritical case

TL;DR

This paper provides a new proof for the kinetic approximation of the energy spectrum in a damped/driven cubic NLS equation on a torus, extending previous results to exact solutions without using Feynman diagrams.

Contribution

It introduces an alternative proof method based on inductive cumulant analysis, extending prior quasisolution results to exact solutions in the subcritical case.

Findings

Kinetic approximation holds for exact solutions in the subcritical regime.

The proof avoids Feynman diagrams, simplifying the analysis.

Results extend previous quasisolution findings to full solutions.

Abstract

We consider a damped/driven cubic NLS equation on a torus under the limit when first the amplitude of solutions goes to zero and then the period of the torus goes to infinity. We suggest another proof of the kinetic approximation for the energy spectrum under a subcritical scaling, extending to the exact solutions result obtained in [Dymov, Kuksin, Maiocchi, Vladuts '2023] for quasisolutions which were defined as the second order truncations of decompositions for the solutions in amplitude. The proof does not involve Feynman diagrams, instead relying on a robust inductive analysis of cumulants.