English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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