Behavior of solutions to the 1D focusing stochastic L^2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

05/28/2020
by   Annie Millet, et al.
0

We study the focusing stochastic nonlinear Schrödinger equation in 1D in the L^2-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting, nor is the mass (or the L^2-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that we study the influence of noise on the global behavior of solutions. In particular, we show that the noise may induce blow-up, thus, ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow-up happens, 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 a random variable normally distributed.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 29

page 30

page 31

06/18/2020

Behavior of solutions to the 1D focusing stochastic nonlinear Schrödinger equation with spatially correlated noise

We study the focusing stochastic nonlinear Schrödinger equation in one s...
01/13/2021

Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises

We study the asymptotic properties of the stochastic Cahn-Hilliard equat...
03/29/2021

Higher dimensional generalization of the Benjamin-Ono equation: 2D case

We consider a higher-dimensional version of the Benjamin-Ono (HBO) equat...
02/14/2020

Stable blow-up dynamics in the L^2-critical and L^2-supercritical generalized Hartree equation

We study stable blow-up dynamics in the generalized Hartree equation wit...
08/12/2019

High-frequency analysis of parabolic stochastic PDEs with multiplicative noise: Part I

We consider the stochastic heat equation driven by a multiplicative Gaus...
03/17/2021

The least favorable noise

Suppose that a random variable X of interest is observed perturbed by in...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.