Stationary solutions for the nonlinear Schrödinger equation
Benedetta Ferrario, Margherita Zanella
TL;DR
This work develops a framework to obtain stationary statistical solutions for the nonlinear Schrödinger equation by perturbing the deterministic flow with a damping term $\gamma u$ and a white-in-time stochastic forcing of intensity $\sqrt{\gamma}$, then sending $\gamma\to 0^+$. Using a Faedo–Galerkin approximation and Krylov–Bogoliubov methods, the authors prove existence of stationary martingale solutions to the damped stochastic NLS in general domains and for a broad range of nonlinearities, including focusing and defocusing cases. In the inviscid limit, any accumulation point $\hat U$ of the perturbed stationary solutions is a nontrivial stationary solution of the deterministic unforced NLS, with its mass determined by the noise covariance; under extra regularity assumptions, the limiting process enjoys stronger regularity and energy conservation. The results extend previous Kuksin–Shirikyan-type approaches to zero-order damping, larger spatial dimensions, fractional Laplacians, and more general boundary/domain settings, providing a robust pathway to stationary statistics for deterministic NLS in subcritical regimes.
Abstract
We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by a linear damping $γu$ and a stochastic force whose intensity is proportional to $\sqrt γ$, and then letting $γ\to 0^+$. We prove indeed that the family of stationary solutions $\{U_γ\}_{γ>0}$ of the perturbed equation possesses an accumulation point for any vanishing sequence $γ_j\to 0^+$ and this stationary limit solves the deterministic unforced nonlinear Schrödinger equation and is not the trivial zero solution. This technique has been introduced in [KS04], using a different dissipation. However considering a linear damping of zero order and weaker solutions we can deal with larger ranges of the nonlinearity and of the spatial dimension; moreover we consider the focusing equation and the defocusing equation as well.