Stationary solutions for the nonlinear Schrödinger equation

We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by a linear damping $\gamma u$ and a stochastic force whose intensity is proportional to $\sqrt{\gamma }$, and then letting $\gamma \to 0^{+}$. We prove indeed that the family of stationary solutions $\{U_{\gamma }{\}}_{\gamma >0}$ of the perturbed equation possesses an accumulation point for any vanishing sequence ${\gamma }_j\to 0^{+}$ 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.