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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study the focusing stochastic nonlinear Schr\"odinger equation in 1D inthe $L^2$-critical and supercritical cases with an additive or multiplicativeperturbation driven by space-time white noise. Unlike the deterministic case,the Hamiltonian (or energy) is not conserved in the stochastic setting, nor isthe mass (or the $L^2$-norm) conserved in the additive case. Therefore, weinvestigate the time evolution of these quantities. After that we study theinfluence of noise on the global behavior of solutions. In particular, we showthat the noise may induce blow-up, thus, ceasing the global existence of thesolution, which otherwise would be global in the deterministic setting.Furthermore, we study the effect of the noise on the blow-up dynamics in bothmultiplicative and additive noise settings and obtain profiles and rates of theblow-up solutions. Our findings conclude that the blow-up parameters (rate andprofile) are insensitive to the type or strength of the noise: if blow-uphappens, it has the same dynamics as in the deterministic setting, however,there is a (random) shift of the blow-up center, which can be described as arandom variable normally distributed.

Behavior of solutions to the 1D focusing stochastic $L^2$-critical and supercritical nonlinear Schr\"odinger equation with space-time white noise