Abstract and Applied Analysis

Volume 2013 (2013), Article ID 390476, 8 pages

http://dx.doi.org/10.1155/2013/390476

Research Article

Exponential Attractor for Coupled Ginzburg-Landau Equations Describing Bose-Einstein Condensates and Nonlinear Optical Waveguides and Cavities

Gui Mu and Jun Liu

College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China

Received 4 February 2013; Accepted 6 March 2013

Academic Editor: de Dai

Copyright © 2013 Gui Mu and Jun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The existence of the exponential attractors for coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities with periodic initial boundary is obtained by showing Lipschitz continuity and the squeezing property.

1. Introduction

Inertial set was introduced (see [1–5]) in order to overcome some of the theoretical difficulties that are associated with inertial manifolds. An inertial set, by definition, contains the global attractor and attracts all trajectories at a uniform exponential rate. Consequently, it contains the slow transients as well as the global attractor. In the theory of dynamical systems the slow transients correspond to slowly converging stable manifolds that are in some sense close to central manifolds. Numerical simulations of infinite dimensional dynamical systems often capture both slow transients and parts of the attractor. After a large but finite time the state of the system obtained from the numerical calculation may often be at a finite distance from the global attractor but at an infinitesimal distance to the inertial set. In this sense, we propose to call the inertial set an exponential attractor to be consistent with the physical intuition [5].

An exponential attractor is an exponentially attracting compact set with finite fractal dimension that is positively invariant under the forward semiflow. The notion of exponential attractors was introduced by Eden et al. [3] and has been shown to be one of the very important notions in the study of long time behavior of solutions to nonlinear diffusion equations [6]. The easiest way of obtaining an exponential attractor is by taking the intersection of an absorbing set with an inertial manifold.

In the area of hyperbolic evolutionary equations, the existence of exponential attractors has been proved for many equations. In this paper, we will prove the existence of exponential attractor for coupled Ginzburg-Landau equations

with the periodic boundary conditions

and initial value

Its physical realizations include systems from nonlinear optics and a double-cigar-shaped Bose-Einstein condensate with a negative scattering length. In particular, in the case of the optical systems, and are amplitudes of electromagnetic waves in two cores of the system, the evolutional variable is either time or propagation distance in the dual-core optical fiber, and is the transverse coordinate in the cavity or the reduced time in the application to the fibers [7].

This paper is organized as follows. In Section 2, we give a description of preliminaries with existence of exponential attractor and the properties of solutions and bounded absorbing sets of (1). In Section 3, the existence of the exponential attractor in type exponential attractor is proved. In Section 4, we give some conclusions for this paper.

2. Preliminaries

Let be two Hilbert spaces, and let be dense in and compactly imbedded into . Let be a continuous map from , into itself. We study

where is a bounded open set in , is smooth, and is a positive self-adjoint operator with a compact inverse. Letting denote the complete set of eigenvectors of , the corresponding eigenvalues are

We assume that the nonlinear semigroup defined in (4)–(6) possesses a compact attractor of -type; namely, there exists a compact set in , and attracts all bounded subsets in and is invariant under the action of .

Let be a compact subset of . leaves the set invariant and set

that is, for on , is the global attractor.

Definition 1. A compact set is called an exponential attractors for if

(i);(ii), for every ;(iii) has finite fractal dimension ;(iv)There exist constants and such that

where

Definition 2. If there exists a bounded function independent and such that

for every , then we say is Lipschitz continuous in and is Lipschitz constant for in .

Definition 3. A continuous semigroup is said to satisfy the squeezing property on if there exists such that satisfies the following.

For every , there exists an orthogonal projection of rank equal to such that for every and in if

holds, then we also have

where .

Theorem 4 (see [3]). Suppose (4)–(6) satisfy the following conditions.

(1)There exist nonlinear semigroup and a compact attractor .(2)There exists a compact set in which is positively invariant for .(3) is Lipschitz continuous and is squeezing in .

Then (4)–(6) admit an exponential attractor in for and

where

Moreover,

where , are defined as in [4], , , are the constants independent of , and is a positive constant.

Proposition 5. There exists such that

is a compact positively invariant set in and is absorbing set for all bounded subsets in , where is a closed absorbing set in for .

Proposition 6. Let , be bounded and closed absorbing sets for (4)–(6) in , respectively. Then there exists a compact attractor of -type. For the proof of Proposition 5 and Proposition 6, we refer the reader to [5].

Denoting by the norm in , , for simplicity, we denote by and the norm in the case and , respectively. Suppose that , (), where is a Hilbert space for the scalar product

The norm of is defined by .

We now establish some time-uniform a priori estimates on in and , respectively.

Lemma 7. Assume that ; then

Thus there exists such that

whenever .

Lemma 8. Assume that ; then

Thus there exists such that

whenever .

Theorem 9. Assume that all the parameters of (1) are positive. For given in (), there exists a unique solution

And also

Furthermore, the solution operator of the system is a continuous semigroup on which possesses bounded absorbing sets , for .

Thus, we observe that Lemmas 7 and 8 show that there exists constant depending only on the data that the balls

are bounded absorbing sets for in and , respectively:

Let

then is a compact invariant subset in ; we know that semigroup defined by problem (31)–(34) possesses a -type compact attractor. According to Theorem 4, we need only to show the Lipschitz continuity and the squeezing property of the dynamical system in , respectively. That is what we proceed to do in the following sections.

3. Exponential Attractor in for Problem (1)-(2)

In this section, we show the existence of the exponential attractor in for problem (1)-(2). In order to prove the Lipschitz continuity and the squeezing property, we need to extend Hölder inequality

where , and Gagliardo-Nirenberg (G-N) inequality

where

and the Young's inequality

Theorem 10. Assume , and are two solutions of problem (1)-(2) with initial values , ; then one has

Proof. Letting , , from (1)-(2), we have

with periodic initial value

where

Taking and , then we get

Substituting (37) and (38) into (36), we get

Substituting (39) into (32), we obtain

To prove the Theorem 4, we take the following four steps.

Step 1. Taking the inner product of (40) with and (41) with , respectively, we have

using

Thus,

then taking the imaginary part of (42) and (43), respectively,

by using the extend Hölder inequality, we can obtain

Combining (46) and (47), then we infer that

Step 2. Taking the inner product of (40) with and (41) with , respectively, we have

Note that

then taking the imaginary part of (50) and (51), respectively,

Note the following inequalities:

Combining (53) and (54), one can obtain

Step 3. Taking the inner product of (40) with and (41) with , respectively, we have

using

where

then taking the imaginary part of (50) and (51), respectively,

Note the following inequalities:

Combining (60) and (61), one can obtain

Step 4. Combining (49), (56) and (63), we get

Taking , and noting that

so (64) can be reduced to

By Gronwall's inequality

that is,

Meanwhile, it indicates that the Lipschitz constant . This completes the proof.

Now, we intend to show the squeezing property for semigroup . To this end, we introduce the operator from to with domain

Obviously, is an unbounded self-adjoint positive operator and the inverse is compact. Thus, there exists an orthonormal basis of consisting of eigenvectors of such that

For all denote by the projector . In the following, we will use

Decompose as

Applying to (32) and (33) we find that

Take the inner product of (73) with and (74) with , respectively. Then like Step , we can get

Take the inner product of (73) with and (74) with , respectively. Then like Step 2, we can get

Take the inner product of (73) with and (74) with , respectively. Then like Step 3, we can get

Combining (75), (76), and (77), we get

Using the G-N inequality

from (78), we have

By Gronwall lemma we get

Letting be fixed we take and assume that

Then we choose large enough so that

that is,

From (82) and (84), we obtain

This shows that when is fixed, Lipschitz constant for in is equal to and satisfies

We have

So when

This completes the proof of Theorem 4.

Theorem 11. The semigroup associated with problem (1)-(2) is squeezing in . Now we conclude this section by giving our main result.

Theorem 12. Suppose that problem (1)-(2) satisfies Theorem 9; there exist and N large enough such that

Then for the nonlinear semigroup defined in (4) and (5), ; admits an exponential attractor in and

where , , are constants independent of the solution of the equation.

4. Conclusions

In this paper, we have studied the coupled Ginzburg-Landau equations which describe Bose-Einstein condensates and nonlinear optical waveguides and cavities with periodic initial boundary; the existence of the exponential attractors is obtained by showing Lipschitz continuity and the squeezing property. For exponential attractor, is only large enough such that

Acknowledgment

This work was supported by Chinese Natural Science Foundation Grant no. 11061028.

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