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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 paper, we obtain very general decay rate estimates associated to a wave–wave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in Lasiecka and Tataru (1993) and we shall present explicit decay rate estimates as considered in Alabau-Boussouira (2005) and Cavalcanti et al. (2007). In addition, we implement a precise and efficient code to study the behavior of the transmission problem when k 1 ≠ k 2 and when one has a nonlinear frictional dissipation g ( u t ) . More precisely, we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence at the H 1 -level as well as the stability for the damped defocusing Schrödinger equation in R d . The considered damping coefficient is time-dependent and may vanish at infinity. To prove the existence, we employ the method devised by Özsarı, Kalantarov and Lasiecka [27] , which is based on monotone operators theory. In particular, when d = 1 or d = 2 , we obtain the uniqueness. Decay estimates for the L 2 -level and ( H 1 ∩ L p + 2 ) -level energies are established with the help of direct multipliers method, coupled with refined energy estimates and a lower semi-continuity argument. [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]

In this work we present a finite difference scheme used to solve a higher order nonlinear Schrödinger equation. These equations are related to models of propagation of solitons travelling in fiber optics. The scheme is designed to preserve the numerical L 2 norm, and control the energy for a suitable choose on the equation's parameters. We prove the convergence of the numerical solution for this equation in a boundary domain, and we show numerical results displaying conservation properties of the schemes using solitons as initial conditions. [ABSTRACT FROM AUTHOR]

We are concerned with the asymptotic behavior of two different cubic, defocusing and damped nonlinear Schrödinger equations on compact Riemannian manifolds without boundary. Two mechanisms of locally distributed damping are considered: a weak damping and a stronger one. In the first problem, we consider a two-dimensional case and prove that the corresponding energy functional goes to zero as time goes to infinity. The proof is based on a result of propagation of singularities due to Dehman et al. (Math Z 254(4):729-749, 2006) and Strichartz type inequalities due to Burq et al. (Am J Math 126(3):569-605, 2004), combined with new ingredients which come from the observability inequality associated with the linear problem. When a stronger damping is in place, we show that the energy functional decays exponentially to zero and for this purpose a forced smoothing effect due to Aloui (Collect Math 59(1):53-62, 2008) plays an essential role in the proof. [ABSTRACT FROM AUTHOR]