On The Energy Transfer To High Frequencies In The Damped/Driven Nonlinear Schr\"odinger Equation (Extended Version)

TL;DR

This paper investigates how energy transfers to high frequencies in a damped/driven nonlinear Schr"odinger equation, showing that solutions can develop large Sobolev norms over time, indicating energy cascade phenomena.

Contribution

It proves that for small damping, solutions with arbitrary initial data can develop large high-frequency components with high probability within a specific timescale.

Findings

Sobolev norms grow at least to order ^{-\u03ba_{n,m}} with high probability

Energy transfer to high frequencies occurs over timescales of order 1/

Large Sobolev norms are achieved despite damping and randomness

Abstract

We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions \[ u_t-\nu\Delta u+i|u|^2u=\sqrt{\nu}\eta(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad \nu>0, \] where $\eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le C\nu^{-m}, $ uniformly in $t\ge0$ and $\nu>0$. In this work we prove that for small $\nu>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ become large at least to the order of $\nu^{-\kappa_{n,m}}$ with $\kappa_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu})$.