We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by adding a linear damping cu and a stochastic force whose intensity is proportional to the square root of c, and then letting c -> 0. We prove indeed that the family of stationary solutions of the perturbed equation possesses an accumulation point for any vanishing sequence 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.
Stationary solutions for the nonlinear Schrödinger equation
Margherita Zanella
;Benedetta Ferrario
2025-01-01
Abstract
We construct stationary statistical solutions of a deterministic unforced nonlinear Schrödinger equation, by perturbing it by adding a linear damping cu and a stochastic force whose intensity is proportional to the square root of c, and then letting c -> 0. We prove indeed that the family of stationary solutions of the perturbed equation possesses an accumulation point for any vanishing sequence 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
