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

Abstract

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation u t i. /˛u C ijuj 2 u C u D f for˛2 . 1 2 , 1/ considered in the the whole space R. We prove that such semi-discrete equation provides a discrete infinite-dimensional dynamical system in H˛.R/ that possesses a global attractor in H˛.R/. We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension.In this article, we address the discrete counterpart of these continuous dynamical systems. Actually, we consider a suitable semidiscrete fractional NLS equation introducing a suitable Crank-Nicolson scheme in time, keeping the space variable continuous. 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]. It is worth recalling that the discrete Crank-Nicolson scheme in both space and time variables for 2 was studied in [22] and [23] for weakly damped NLS equation. The discrete in time relaxation scheme for a nonlocal NLS equation was studied in [24]. The discrete dynamical system provided by the Crank-Nicolson scheme to 2 is new to our knowledge.The rest of the article is organized as follows. Section 2 is devoted to the derivation of the scheme and the main results. Section 3 deals with the mathematical framework, where we state and prove a nonstandard commutator estimate. In Section 4, we focus on the existence and uniqueness of the discrete solution. Finally, in Section 5, we prove the existence of a compact global attractor in the phase space H˛. We then prove, assuming, moreover, that the external force has some decay at the infinity, that this global attractor has a finite fractal dimension.Using the discrete Gronwall lemma, we then infer that there exists K 1 that depends on the data , f such that the set E D fu 2 E 0 , J.u/ Ä K 1 g is a bounded absorbing set in H˛that is positively invariant by S. In the remaining of the paper, we denote by M 1 the radius of the smallest ball that contains E.

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