On a time discretization of a weakly damped forced nonlinear Schrödinger equation

Abstract

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a damped forced nonlinear Schrödinger equation. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.

Finite‐dimensional global attractor for a semi‐discrete fractional nonlinear Schrödinger equation

“…Actually, we have obtained an upper bound on the H 2˛n orm of solutions in the attractor that depends on (see (74)). On the other hand, the fractal dimension of the global attractor depends on this bound (see the proof of Proposition 5.6 in [30] for the discretization of the classical NLS equation [18]. This is not satisfactory because the dimension of the continuous limit equation is finite.…”

“…These numerical schemes, first introduced in [16] (see also [17]) for numerical purposes, provide also discrete infinite-dimensional dynamical systems. This point of view for classical NLS equations appear in [18]. We do not address here alternate discretizations for NLS such as splitting methods (see [19] and the references therein) or relaxation methods introduced in [20].Fruitful results about Crank-Nicolson scheme appear in the infinite-dimensional dynamical system literature: For the case when the space variable x belongs to a finite interval, with periodic boundary conditions, the existence of a global attractor was proved in [18].…”

“…These numerical schemes, first introduced in [16] (see also [17]) for numerical purposes, provide also discrete infinite-dimensional dynamical systems. This point of view for classical NLS equations appear in [18]. We do not address here alternate discretizations for NLS such as splitting methods (see [19] and the references therein) or relaxation methods introduced in [20].…”

“…Fruitful results about Crank-Nicolson scheme appear in the infinite-dimensional dynamical system literature: For the case when the space variable x belongs to a finite interval, with periodic boundary conditions, the existence of a global attractor was proved in [18]. It turns out that this result was also obtained in the case where the space variable x belongs to T 2 in [21].…”

Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

“…One of the particularly hard issues in hydrodynamics is the modeling of damping phenomena: according to the physical situations, viscous (entire or fractional power of −∆), local or non local additional terms (half-time or half-space derivative) have been proposed to represent the damping and the fitting with real physical data still remains a challenge, we refer the reader, e.g., to [57,58]. The mathematical analysis of the long time behavior of the solutions of the resulting models is also essential to the understanding of the underlying physics [23,42,43,44,45,48]. Of course the derivation of appropriate and robust numerical schemes is crucial to capture the dynamics and also to point out mathematical properties that are difficult to establish, [12,21,30,40].…”