# Formal expansions in stochastic model for wave turbulence 1: kinetic limit

## TL;DR

This paper investigates the behavior of solutions to a damped/driven cubic NLS equation on large tori, showing that the energy spectrum of a second order truncated series approaches the wave kinetic equation as amplitude diminishes and domain size increases.

## Contribution

It introduces a formal series expansion approach for the stochastic NLS and proves the convergence of the second order truncation to the wave kinetic equation in the limit.

## Key findings

- Energy spectrum converges to the wave kinetic equation.
- Second order truncation accurately models the kinetic limit.
- Discussion of higher order truncations included.

## Abstract

We consider the damped/driver (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove that when the amplitude goes to zero and the torus' size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.