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Abstract and Applied Analysis

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

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

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

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.

#### References

- I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation.,”
*Reviews of Modern Physics*, vol. 74, no. 1, pp. 99–143, 2002. - B. A. Malomed, “Complex Ginzburg-Landau equation,” in
*Encyclopedia of Nonlinear Science*, A. Scott, Ed., pp. 157–160, Routledge, New York, NY, USA, 2005. - A. Eden, C. Foias, B. Nicolaenko, and K. Temam,
*Exponential Attractors for Dissipative Evolution Equations*, vol. 37, John Wiley, New York, NY, USA, 1994. - Z. Dai and B. Guo, “Inertial fractal sets for dissipative Zakharov system,”
*Acta Mathematicae Applicatae Sinica*, vol. 13, no. 3, pp. 279–288, 1997. View at Publisher · View at Google Scholar - Z. Dai and B. Guo,
*Inertial ManiFolds and Approximate Inertial Mani-Folds*, Science Press, Beijing, China, 2000. - Z. Dai, Y. Huang, and X. Sun, “Long-time behavior of solution for coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities,”
*Journal of Mathematical Analysis and Applications*, vol. 362, no. 1, pp. 125–139, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Sakaguchi and B. A. Malomed, “Stable solitons in coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities,”
*Physica D*, vol. 183, no. 3-4, pp. 282–292, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet