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

Abstract

We study the focusing stochastic nonlinear Schrödinger 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.

A note on log-log blow up solutions for stochastic nonlinear Schrödinger equations

“…Thus, asymptotically, Qe iW pt,λptqx`xptq will converge to Qe iW pTω,x ˚q in H 1 . This is consistent with Conjecture 1 in [30].…”

“…See, e.g., [10,13,12]. In particular, we refer to the recent works [30,31] for the study of noise effects on the log-log blow-up dynamics.…”

Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equation

“…Furthermore, in the stochastic case, a remarkable result proved by de Bouard and Debussche [21,23] is that, stochastic solutions can blow up at any short time with positive probability in the L 2 -supercritical case. Several numerical experiments have been also made to investigate the dynamics of stochastic blow-up solutions, see, e.g., [24,25,26,57,58].…”

“…One major challenge in the stochastic case is that, in contrast to NLS, the classical pseudoconformal symmetry is lost due to the input of noise. Moreover, the energy of solutions is no longer conserved, which makes it more difficult to understand the global behavior in the stochastic L 2 -supercritical case, see [57,58] for the numerical tracking of energy.…”

Multi solitary waves to stochastic nonlinear Schrödinger equations

“…We refer to [9, Section 2] for more physical interpretations. We also refer to [14,22,27,50,51] for the numerical experiments to investigate the dynamics of stochastic solutions.…”