TY  - JOUR
CY  - Orlando, Fla
ID  - publications50730
UR  - http://tse-fr.eu/pub/130482
IS  - N° 538
A1  - Bégout, Pascal
A1  - Diaz, Jesus Ildefonso
Y1  - 2024/10/01/
N2  - We consider the damped nonlinear Schr¨odinger 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 iμu(t, x)/|u(t, x)| for a given parameter μ > 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´ym 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∞-norm of the forcing term f (t, x) becomes less than μ after a finite time. This presents some analogies with the so called feedback bang-bang controls v (here v = −iμu/|u| + f ).
PB  - Academic Press
JF  - Journal of Mathematical Analysis and Applications
VL  - Vol. 1
KW  - Damped Schrödinger equation
KW  - Saturated section, Stabilization
KW  - Finite time extinction
KW  - Maximal monotone operators
KW  - Existence and regularity of weak solutions
SN  - 1096-0813
TI  - Strong stabilization of damped nonlinear Schrödinger equation with saturation on unbounded domains
AV  - public
ER  -