Asymptotic Behavior of the Coupled Nonlinear Schrödinger Lattice System

Abstract

This paper studies asymptotic behavior of solutions for the coupled nonlinear Schrödinger lattice system. We obtain the existence and stability of compact attractor by means of tail estimates method and finite-dimensional approximations.

1. Introduction

In the present paper, we consider the following dissipative coupled Schrödinger lattice system: with initial condition where ; ,   denotes a bounded positive potential with ; , denote external forces.

Nonlinear coupled Schödinger lattice system (1) can be seen as a discretization model of the two-component system of time-dependent nonlinear Gross-Pitaevskii equations. Gross-Pitaevskii equation arises quite naturally in a binary mixture of Bose-Einstein condensates with two different hyperfine states [1]. There are many analytical and numerical results on solitary wave solutions for this system (see [2–11]).

The study of the existence of compact attractor for general infinite lattice system can date back to Bates et al. [12], who used the tail estimates method to prove the asymptotic compactness of dissipative lattice system and the existence of compact attractor. For more general results on the existence of compact attractor for infinite lattice system, one can see [13]. Karachalios and Yannacopoulos [14] studied the asymptotic behavior of single nonlinear discrete Schrödinger equations.

We state our main results in this paper.

Theorem 1. The semigroup generated by system (4) possesses a global attractor which is compact, connected, and maximal among the functional invariant sets in .

Theorem 2. The global attractor converges in the sense of the Hausdorff semidistance related to ; that is, where , for any nonempty compact subsets and in a metric space .

This paper is organized as follows. In the next section, we prove the global existence of the dissipative coupled Schrödinger lattice system (1). In Section 3, we show the stability of the global attractor.

2. Existence of the Global Attractor

This section shows the existence of compact attractor of system (1) in . We denote a Hilbert space by with the scalar product , .

Firstly, we prove the global existence of solution of system (1). For convenience, we take the scalar form of system (1): with initial condition , .

Let and . Then system (4) can be rewritten as Then we get the mild solution of (5) as where the semigroup generated by the operator , and

In order to prove the existence of compact attractor, we need the following proposition.

Proposition 3 (Hale [15], Temam [16]). Assume that is a metric space and is a semigroup of continuous operators in . If has an absorbing set and is asymptotically compact, then process a global attractor.

It is easy to verify that and satisfy the Lipschitz continuous property on any bounded set in . Using the same method in [14], we obtain the following result.

Theorem 4. Let . Then system (1) possesses a unique solution for some . If , then

Lemma 5. Let and . Assume that holds. Then there exists a bounded absorbing ball of the semigroup generated by system (4) in . The radius of is . Therefore, there exists depending on such that

Proof. Taking the imaginary part of the inner product of the first equation and second equation of (4) with and , respectively, we have Summing up (10), Let . By Young inequality, we have Applying Gronwall Lemma to (12), we have It implies that the semigroup possesses a bounded absorbing ball centered at with radius .

Lemma 6. Assume that , , and . Then, there exist and such that the solution of system (4) satisfies

Proof. Define by where is a positive constant number.
Taking the imaginary part of the inner product of the first equation and second equation of (4) with and in , respectively, where is large enough, we have Summing up (16), we get Let . By Cauchy-Schwartz inequality and (17), we have Applying Gronwall Lemma, we can obtain where and are the time of entry of initial data bounded in and radius of the absorbing ball in .
Since , then, for any given and , there exist and such that where denote a constant number depending on .

Lemma 7. The semigroup is asymptotically compact in ; that is, if sequences , are bounded in and , then is precompact in .

Proof. Define . Our purpose is to prove that has finite covering balls of radii .
By Lemma 6, we know that, for all , there exist and such that We consider the set in . Note that is bounded in , so it is precompact in ; that is, there exists a family of balls of radii , which covers . This together with (21) implies that the set has finite covering balls of radii . This completes the proof.

Therefore, by Lemmas 6 and 7 and Proposition 3, we conclude that Theorem 1 holds.

3. Finite Approximation of the Global Attractor

In this section, we study the stability of global attractor of lattice dynamical system generated by (1)-(2) under its approximation by a global attractor of an appropriate infinite dimensional dynamical system.

We consider the following finite dimensional boundary value problem: In similar process with infinite dimensional problem (1)-(2), we have the following well-posedness and asymptotic behavior of finite dimensional system (22).

Lemma 8. Let , . Assume that holds. Then there exists a unique solution . The dynamical system generated by (22) possesses a bounded absorbing set and global attractor .

In order to verify the global attractor of semigroup being approximated by the global attractor of semigroup as , we extend the solution of (22) to infinite dimensional space , as

Proof of Theorem 2. We denote global attractors generated by semigroups and by and , respectively. By Lemma 7, we can get that is also an absorbing set for . Then, we have where denotes an open neighborhood of absorbing ball . In light of Theorem 6.1 in [14], for obtaining (3), we only need to verify that, for every compact interval of , Let be a solution of problem (22). Since , for every . If we denote as the radius of absorbing ball , by (23) and (22), we have which implies that, for every , there exists a subsequence of (still denoted by itself) such that Next, according to Theorem 6.1 in [14], for proving (25), it suffices to show that solution of system (22) converges to solution of system (1) in any compact interval of and in a bounded set of .
By (26), we can define differentiable functions Then, It follows from the mean-value theorem that there exist such that, for any fixed , Hence, there exists a constant (independent of ) such that which implies that the sequences are equicontinuous. This combines with (26) and Ascoli-Arzelá theorem; we get that the weak convergence in (27) is actually strong convergence.
We define where and are defined in (5). Similar to Lemma 5, we can get that are Lipschitz continuous on any bounded sets in ; that is, there exists constant such that Then, by (27), , we have By the formula in [17, page 59], for , we get Since is arbitrary, (36) holds for all . By (27) and (36), we obtain that is a bounded solution of (1) and , which implies that . If the convergence holds for any other subsequence satisfying (36), then it contradicts uniqueness of the solution. Therefore, we deduce that the convergence holds for original sequence , which implies that (25) holds. This completes the proof.

Remark 9. We have discussed the existence of compact attractor for the one dimensional coupled Schrödinger lattice. By the same method, we also can obtain the same result for spatial discretization of coupled Schrödinger system in higher dimensional , . It takes the following form: where is a small parameter, , , are constant potential , and is the discrete Laplacian operator defined as Define operators as Then the operator satisfies

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.