Stochastic NLS Research Articles

Long Time Behavior of Stochastic NLS with a Small Multiplicative Noise

Long Time Behavior of Stochastic NLS with a Small Multiplicative Noise

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

We construct a martingale solution of the stochastic nonlinear Schrodinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $$H^1$$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood–Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of cadlag functions and Jakubowski’s generalization of the Skorohod-Theorem to nonmetric spaces.

The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates

We consider the stochastic NLS with nonlinear Stratonovich noise for initial values in $${L^2({\mathbb {R}^d})}$$ and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic and stochastic Strichartz estimates. In the subcritical case we prove that the solution is global, if we impose an additional assumption on the nonlinear noise.

Numerical Analysis on Ergodic Limit of Approximations for Stochastic NLS Equation via Multi-symplectic Scheme

We consider a finite dimensional approximation of the stochastic nonlinear Schr\"odinger equation driven by multiplicative noise, which is derived by applying a symplectic method to the original equation in spatial direction. Both the unique ergodicity and the charge conservation law for this finite dimensional approximation are obtained on the unit sphere. To simulate the ergodic limit over long time for the finite dimensional approximation, we discretize it further in temporal direction to obtain a fully discrete scheme, which inherits not only the stochastic multi-symplecticity and charge conservation law of the original equation but also the unique ergodicity of the finite dimensional approximation. The temporal average of the fully discrete numerical solution is proved to converge to the ergodic limit with order one with respect to the time step for a fixed spatial step. Numerical experiments verify our theoretical results on charge conservation, ergodicity and weak convergence.

Derivation of a wave kinetic equation from the resonant-averaged stochastic NLS equation

We suggest a new derivation of a wave kinetic equation for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kinetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analysed with full rigour in [8], and show how the second limit leads to a kinetic equation for the spectrum, if some further hypotheses (commonly employed in the weak turbulence theory) are accepted. Finally we show how to derive from these equations the Kolmogorov–Zakharov spectra.

Approach to Equilibrium for the Stochastic NLS

We study the approach to equilibrium, described by a Gibbs measure, for a system on a $d$-dimensional torus evolving according to a stochastic nonlinear Schr\"odinger equation (SNLS) with a high frequency truncation. We prove exponential approach to the truncated Gibbs measure both for the focusing and defocusing cases when the dynamics is constrained via suitable boundary conditions to regions of the Fourier space where the Hamiltonian is convex. Our method is based on establishing a spectral gap for the non self-adjoint Fokker-Planck operator governing the time evolution of the measure, which is {\it uniform} in the frequency truncation $N$. The limit $N\to\infty$ is discussed.

Modulation analysis for a stochastic NLS equation arising in Bose–Einstein condensation

We study the asymptotic behavior of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude e tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of e −2 , the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as e goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale.

Numerical study of two‐dimensional stochastic NLS equations

AbstractIn this article, we numerically solve the two‐dimensional stochastic nonlinear Schrödinger equation in the case of multiplicative and additive white noises. The aim is to investigate their influence on well‐known deterministic solutions: stationary states and blowing‐up solutions. In the first case, we find that a multiplicative noise has a damping effect very similar to diffusion. However, for small amplitudes of the noise, the structure of solitary state is still localized. In the second case, a local refinement algorithm is used to overcome the difficulty arising for the computation of singular solutions. Our experiments show that multiplicative white noise stops the deterministic blow‐up that occurs in the critical case. This extends the results of Debussche and Di Menza (Physica D, 162(3–4) 2002, 131–154) in the one‐dimensional case. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential, 2005.