Behavior of solutions to the 1D focusing stochastic nonlinear Schr\"odinger equation with spatially correlated noise

TL;DR

This paper investigates how spatially correlated noise affects the dynamics of solutions to the 1D focusing stochastic nonlinear Schrödinger equation, revealing that such noise influences the blow-up location but not the blow-up behavior itself.

Contribution

It provides a detailed analysis of the impact of spatially correlated noise on blow-up and scattering in the 1D focusing stochastic nonlinear Schrödinger equation, extending previous results with white noise.

Findings

Energy is affected by spatially correlated noise.

Noise shifts the blow-up center without altering blow-up dynamics.

The probability of blow-up versus scattering depends on correlation parameters.

Abstract

We study the focusing stochastic nonlinear Schr\"odinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the $L^2$-critical and supercritical cases. The mass ($L^2$-norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate how the energy is affected by various spatially correlated random perturbations. We then study the influence of the noise on the global dynamics measuring the probability of blow-up versus scattering behavior depending on various parameters of correlation kernels. Finally, we study the effect of the spatially correlated noise on the blow-up behavior, and conclude that such random perturbations do not influence the blow-up dynamics, except for shifting of the blow-up center location. This is similar to what we observed in [32] for a space-time white driving noise.