The large-period limit for equations of discrete turbulence

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 $\lambda \to 0$ 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.

References in corpus (6)

• Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
• Weakly nonlinear Schrödinger equation with random initial data
• On the derivation of the homogeneous kinetic wave equation
• Derivation of the homogeneous kinetic wave equation: longer time scales
• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
• Resonant averaging for small solutions of stochastic NLS equations

Cited by in corpus (4)

• Wave Turbulence and thermalization in one-dimensional chains
• Propagation of chaos and the higher order statistics in the wave kinetic theory
• Wave turbulence and collective behavior models for wave equations with short- and long-range interactions
• Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition (complete version)