On the energy transfer to high frequencies in the damped/driven nonlinear schro ̀ˆdinger equation

Guan HUANG YMSC,Tsinghua University Sergei Kuksin Universite PARIS VII

Dynamical Systems mathscidoc:2205.11003

Stoch PDE: Anal Comp, 2021.1
We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n\in\mathbb{N}$, 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}, $ uniformy 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 $\mathcal{O}(\frac{1}{\nu})$.
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@inproceedings{guan2021on,
  title={ON THE ENERGY TRANSFER TO HIGH FREQUENCIES IN THE DAMPED/DRIVEN NONLINEAR SCHRO ̀ˆDINGER EQUATION},
  author={Guan HUANG, and Sergei Kuksin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517133426144744199},
  booktitle={Stoch PDE: Anal Comp},
  year={2021},
}
Guan HUANG, and Sergei Kuksin. ON THE ENERGY TRANSFER TO HIGH FREQUENCIES IN THE DAMPED/DRIVEN NONLINEAR SCHRO ̀ˆDINGER EQUATION. 2021. In Stoch PDE: Anal Comp. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220517133426144744199.
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