Work

Stationary solutions for the nonlinear Schrödinger equation

Year: 2025

Type: article

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

Source: Stochastic Partial Differential Equations Analysis and Computations

DOI: 10.1007/s40072-025-00350-7

Authors: Benedetta Ferrario, Margherita Zanella

Institutions: University of Pavia, Politecnico di Milano

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", Università degli Studi di Pavia, Politecnico di Milano