1. Exponential stability for the nonlinear Schrödinger equation with locally distributed damping
- Author
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Wellington J. Corrêa, Marcelo M. Cavalcanti, Mauricio Sepúlveda Cortese, Türker Özsarı, Rodrigo Véjar Aseme, and Maltepe Üniversitesi, İnsan ve Toplum Bilimleri Fakültesi
- Subjects
Finite volume method ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Schrödinger equation ,Space (mathematics) ,Exponential stability ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,symbols.namesake ,Locally distributed damping ,Bounded function ,symbols ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Nonlinear Schrödinger equation ,Analysis ,Mathematics - Abstract
In this paper, we study the defocusing nonlinear Schrodinger equation with a locally distributed damping on a smooth bounded domain as well as on the whole space and on an exterior domain. We first construct approximate solutions using the theory of monotone operators. We show that approximate solutions decay exponentially fast in the L-2-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the L-2-decay of solutions for the original model by passing to the limit and using a weak lower semicontinuity argument, respectively. The distinctive feature of the paper is the monotonicity approach, which makes the analysis independent from the commonly used Strichartz estimates and allows us to work without artificial smoothing terms inserted into the main equation. We in addition implement a precise and efficient algorithm for studying the exponential decay established in the first part of the paper numerically. Our simulations illustrate the efficacy of the proposed control design.
- Published
- 2020
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