Stationary solutions for the nonlinear Schrödinger equation

We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by adding a linear damping γu and a stochastic force whose intensity is proportional to γ, and then letting γ→0+. We prove indeed that the family of stationary solutions {Uγ}γ>0 of the perturbed equation possesses an accumulation point for any vanishing sequence γj→0+ and this stationary limit solves the deterministic unforced nonlinear Schrödinger equation and is not a trivial process. This technique has been introduced in Kuksin and Shirikyan (J Phys A: Math Gen 37:1–18, 2004), 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.