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We are concerned with the existence as well as the exponential stability in H1-level for the damped defocusing Schrödinger equation posed in a two-dimensional exterior domain Ω with smooth boundary ∂Ω. The proofs of the existence are based on the properties of pseudo-differential operators introduced in Dehman et al. [Math. Z. 254, 729–749 (2006)] and a Strichartz estimate proved by Anton [Bull. Soc. Math. Fr. 136, 27–65 (2008)], while the exponential stability is achieved by combining arguments firstly considered by Zuazua [J. Math. Pures Appl. 9, 513–529 (1991)] for the wave equation adapted to the present context and a global uniqueness theorem. In addition, we proved propagation results for the linear Schrödinger equation for any dimensional and for any (reasonable) boundary conditions employing microlocal analysis. [ABSTRACT FROM AUTHOR]

In this work, we study at the L 2 – level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrödinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and efficient code to study the energy decay of the higher order nonlinear Schrödinger equation and we prove its convergence and exponential stability of the discrete energy. [ABSTRACT FROM AUTHOR]

In this paper, we study the defocusing nonlinear Schrödinger 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 L2-sense by using the multiplier technique and a unique continuation property. Then, we prove the global existence as well as the L2-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. [ABSTRACT FROM AUTHOR]

In this work we present a finite difference scheme used to solve a High order Nonlinear Schrödinger Equation. These equations can model the propagation of solitons travelling in fiber optics ([3], [11]). The scheme is designed to preserve the numerical energy at L2 level, and control the energy at H1 level for a suitable choose on the equation's parameters. We show numerical results displaying conservation properties of the schemes using solitons as initial conditions. [ABSTRACT FROM AUTHOR]

Inspired by the highly cited work due to Lasiecka and Tataru (1993) [20] , a semilinear model of the wave equation in an inhomogeneous medium with simultaneous interior and boundary feedbacks is considered. While for a homogeneous medium Ω and for certain geometries nonlinear feedback is strong enough to derive uniform decay rates of the energy associated with the model, in contrast, in an inhomogeneous medium, and for general geometry of Ω, the problem requires a new approach. In the first case, the rays of the geometric optics are straight lines and the geometry is favorable. However, in the second one, more damping is required to well-known obtain a similar result. For this purpose, arguments of microlocal analysis are used instead of the multipliers technique. [ABSTRACT FROM AUTHOR]