Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains

Pascal Bégout (1) , Jesús Ildefonso Díaz

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider the damped nonlinear Schr\"{o}dinger equation with saturation: i.e., the complex evolution equation contains in its left hand side, besides the potential term $V(x)u,$ a nonlinear term of the form $\mathrm{i}\mu u(t,x)/|u(t,x)|$ for a given parameter $\mu >0$ (arising in optical applications on non-Kerr-like fibers). In the right hand side we assume a given forcing term $f(t,x).$ The important new difficulty, in contrast to previous results in the literature, comes from the fact that the spatial domain is assumed to be unbounded. We start by proving the existence and uniqueness of weak and strong solutions according the regularity of the data of the problem. The existence of solutions with a lower regularity is also obtained by working with a sequence of spaces verifying the Radon-Nikod\'{y}m property. Concerning the asymptotic behavior for large times we prove a strong stabilization result. For instance, in the one dimensional case we prove that there is extinction in finite time of the solutions under the mere assumption that the $L^\infty$-norm of the forcing term $f(t,x)$ becomes less than $\mu$ after a finite time. This presents some analogies with the so called feedback \textit{bang-bang controls} $v$ (here $v=-\mathrm{i}\mu u/|u|+f).$

Keywords: Damped Schrödinger equation; Saturated section; Stabilization; Finite time extinction; Maximal monotone operators; Existence and regularity of weak solutions; 2020 Mathematics Subject Classification: 35Q55 (35A01 35A02 35B40 35D30 28A20 28B05 46B22) Damped Schrödinger equation Saturated section Stabilization Finite time extinction Maximal monotone operators Existence and regularity of weak solutions 1; 2020 Mathematics Subject Classification: 35Q55 (35A01; 35A02; 35B40; 35D30; 28A20; 28B05; 46B22) Damped Schrödinger equation; Existence and regularity of weak solutions 1

Domains: Analysis of PDEs [math.AP]

Pascal Bégout, Jesús Ildefonso Díaz. Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains. Journal of Mathematical Analysis and Applications, 2024, 538 (1), pp.128329. ⟨10.1016/j.jmaa.2024.128329⟩. ⟨hal-04538659⟩