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

Annie Millet, Alex D Rodriguez+2 more|arXiv (Cornell University)|Jun 18, 2020

Advanced Mathematical Physics Problems45 references6 citations

TL;DR

This paper investigates the 1D focusing stochastic nonlinear Schrödinger equation with multiplicative noise driven by spatially correlated Wiener processes. Using Stratonovich integration, mass is conserved while energy evolves stochastically; numerical simulations show that spatially correlated noise primarily shifts the blow-up center location without altering blow-up profiles or rates, and reduces blow-up probability in critical and supercritical regimes, especially with stronger or more localized noise.

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.

Motivation & Objective

• To analyze the impact of spatially correlated multiplicative noise on the energy and global dynamics of the 1D focusing stochastic NLS.
• To determine how noise correlation structure influences the probability of finite-time blow-up versus scattering behavior.
• To investigate whether spatially correlated noise alters blow-up profiles, rates, or center location compared to the deterministic case.
• To compare results with previous findings on space-time white noise and extend understanding of stochastic NLS in L2-critical and supercritical regimes.

Proposed method

• Uses Stratonovich integral to ensure L2-norm (mass) conservation in the stochastic NLS equation.
• Models noise via Q-Wiener processes with trace-class covariance (Gaussian or polynomial decay) and homogeneous Wiener processes with Riesz or exponential kernels.
• Employs mass-conservative numerical schemes to track discrete mass and energy evolution over time.
• Performs extensive numerical simulations (1000–3000 trials) to estimate blow-up probability and distribution of blow-up center locations.
• Analyzes blow-up dynamics via profile convergence and rate tracking, comparing to deterministic ground states.
• Computes and compares energy bounds and asymptotic energy levels across different noise correlation types and parameters.

Experimental results

Research questions

• RQ1 How does spatially correlated noise affect the time evolution of energy in the 1D stochastic NLS equation?
• RQ2 What is the influence of noise correlation structure (e.g., Riesz kernel vs. exponential decay) on the probability of finite-time blow-up?
• RQ3 Does spatially correlated noise alter the blow-up profile or rate compared to the deterministic case?
• RQ4 How does the location of the blow-up center distribute under different noise realizations, and how does its variance depend on noise parameters?
• RQ5 Can spatially correlated noise induce blow-up in initial data that would otherwise scatter globally in the deterministic setting?

Key findings

• Energy initially increases and asymptotically approaches a horizontal level close to that observed under space-time white noise, indicating a stable long-term energy state.
• Larger noise strength and greater spatial concentration near the origin reduce the probability of blow-up in both L2-critical and supercritical cases, with a more pronounced effect in the critical case.
• Spatially correlated noise can drive globally existing deterministic solutions to blow up, particularly in the supercritical regime, by perturbing energy and mass thresholds.
• Blow-up profiles and rates remain statistically identical to the deterministic case; only the blow-up center location is stochastically shifted.
• The blow-up center location follows a normal distribution whose variance increases with noise intensity β, especially in Examples 3 and 4 (Riesz and exponential kernels).
• The mean blow-up center location remains near zero, but the variance grows significantly with β, indicating increased uncertainty in blow-up position under stronger noise.