Stationary solutions for the nonlinear Schrödinger equation

Abstract We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by adding a linear damping $$\gamma u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> and a stochastic force whose intensity is proportional to $$\sqrt{\gamma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>γ</mml:mi> </mml:msqrt> </mml:math> , and then letting $$\gamma \rightarrow 0^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> . We prove indeed that the family of stationary solutions $$\{U_\gamma \}_{\gamma &gt;0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> of the perturbed equation possesses an accumulation point for any vanishing sequence $$\gamma _j\rightarrow 0^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> 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.