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

Authors: Gui Mu; Jun Liu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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(1)iut+γ2Δu+iγu+(σ1+iσ2|u|2)|u|2u+v=0,ivt+γ2Δu+(iΓ-χ)v+u=0,
with the periodic boundary conditions
(2)u(x,t)=u(x+D,t),v(x,t)=v(x+D,t),u(x,t)=u(x+D,t)x∈R,t>0,
and initial value
(3)u(x,0)=u0(x),v(x,0)=v0(x),x∈R.
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, u and v are amplitudes of electromagnetic waves in two cores of the system, the evolutional variable t is either time or propagation distance in the dual-core optical fiber, and x 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 Section2, 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 V2 type exponential attractor is proved. In Section 4, we give some conclusions for this paper.


## 2. Preliminaries

LetV1,V2 be two Hilbert spaces, and let V2 be dense in V1 and compactly imbedded into V1. Let S(t)t≥0 be a continuous map from V1, V2 into itself. We study
(4)dudt+Au+g(u)=f(x),t>0,x∈Ω,(5)u(x,0)=u0(x),(6)Dirichletproblemorperiodicboundaryproblem,
where Ω is a bounded open set in Rn, ∂Ω is smooth, and A is a positive self-adjoint operator with a compact inverse. Letting {wi}i=1∞ denote the complete set of eigenvectors of A, the corresponding eigenvalues are
(7)0≤λ1<λ1⋯λi<⋯→+∞.
We assume that the nonlinear semigroup S(t)t≥0 defined in (4)–(6) possesses a compact attractor B of (V2,V1)-type; namely, there exists a compact set B in V1, and B attracts all bounded subsets in V2 and is invariant under the action of S(t)t≥0.LetC be a compact subset of V2. S(t)t≥0 leaves the set C invariant and set
(8)B=⋂s≥0‍⋃t≥s‍S(t)t≥0C¯,
that is, for S(t)t≥0 on C, B is the global attractor.Definition 1.
A compact setM is called an exponential attractors for S(t)t≥0,C if(i)
B⊆M⊆C;(ii)
S(t)M⊆M, for every t≥0;(iii)
M has finite fractal dimension dF<∞;(iv)
There exist constantsc1 and c2 such that
(9)distv2(S(t)C,M)≤c1exp(-c2t),∀t>0,where
(10)distv2(A,D)=supx∈Ainfy∈D∥x-y∥V2.Definition 2.
If there exists a bounded functionl(t) independent u and v such that
(11)∥S(t)u-S(t)v∥V2≤l(t)∥u-v∥V2,
for every u,v∈C, then we say S(t) is Lipschitz continuous in C and l(t) is Lipschitz constant for S(t) in C.Definition 3.
A continuous semigroupS(t)t≥0 is said to satisfy the squeezing property on C if there exists t*>0 such that S(t*) satisfies the following.
For everyδ∈(0,(1/8)), there exists an orthogonal projection PN0 of rank equal to N0 such that for every u and v in C if
(12)∥PN0(S(t*)u-S(t*)v)∥v2≤∥QN0(S(t*)u-S(t*)v)∥v2
holds, then we also have
(13)∥S(t*)u-S(t*)v∥v2≤δ∥u-v∥v2,
where QN0=I-PN0.Theorem 4 (see [3]).
Suppose (4)–(6) satisfy the following conditions.(1)
There exist nonlinear semigroupS(t)t≥0 and a compact attractor B.(2)
There exists a compact setC in V2 which is positively invariant for S(t)t≥0.(3)
S(t)t≥0 is Lipschitz continuous and is squeezing in C.Then (4)–(6) admit an exponential attractor M in V2 for S(t)t≥0 and
(14)M=⋃0≤t≤t*‍S(t)M*,
where
(15)M*=B⋃‍(⋃j=1∞‍⋃k=1∞‍S(t*)j(E(k))).
Moreover,
(16)dF(M)≤1+CN0,distV2(S(t)B,M)≤C0exp(-C1t),
where N0, E(k) are defined as in [4], C, C0, C1 are the constants independent of B, and t* is a positive constant.Proposition 5.
There existst0(B0) such that
(17)B*=⋃0≤t≤t0‍S(t)B0¯
is a compact positively invariant set in V1 and is absorbing set for all bounded subsets in V2, where B0 is a closed absorbing set in V2 for S(t)t≥0.Proposition 6.
LetB0, B1 be bounded and closed absorbing sets for (4)–(6) in (V2,V1), respectively. Then there exists a compact attractor A* of (V2,V1)-type. For the proof of Proposition 5 and Proposition 6, we refer the reader to [5].Denoting by|·|Lp the norm in Lp(0,L), 1≤p≤∞, for simplicity, we denote by |·|0 and |·|∞ the norm in the case p=2 and p=∞, respectively. Suppose that H=L2(0,L), Ei=Hi(0,L)×Hi(0,L) (i=1,2), where Hi(0,L) is a Hilbert space for the scalar product
(18)((·,·))Hi=(·,·)+∑j=1i‍(Dj·,Dj·),D=∂∂x.
The norm of Ei is defined by ∥(u,v)∥Ei2=∥u∥Hi2+∥v∥Hi2.We now establish some time-uniform a priori estimates on(u,v) in E1 and E2, respectively.Lemma 7.
Assume that(u0,v0)∈E1; then
(19)∥(u,v)∥E12≤c∥(u0,v0)∥E12e-δ1t+c1.
Thus there exists t1=t1(R)>0 such that
(20)∥(u,v)∥E12≤c2,t≥t1,
whenever ∥(u0,v0)∥E1≤R.Lemma 8.
Assume that(u0,v0)∈E2; then
(21)∥(u,v)∥E22≤c∥(u0,v0)∥E22e-δ2t+c3.
Thus there exists t2=t2(R)>0 such that
(22)∥(u,v)∥E22≤c4,t≥t2,
whenever ∥(u0,v0)∥E2≤R.Theorem 9.
Assume that all the parameters of (1) are positive. For (u0,v0) given in Ei (i=1,2), there exists a unique solution
(23)(u,v)∈L∞(R+,Ei).
And also
(24)(u,v)∈𝒞(R+,E1),∀(u0,v0)∈E1.
Furthermore, the solution operator of the system is a continuous semigroup S(t) on E1 which possesses bounded absorbing sets Bi⊂Ei, for i=1,2.Thus, we observe that Lemmas7 and 8 show that there exists constant k depending only on the data that the balls
(25)B1={(u,v)∈E1,∥u∥H1+∥v∥H1≤k},B2={(u,v)∈E2,∥u∥H2+∥v∥H2≤k}
are bounded absorbing sets for S(t) in E1 and E2, respectively:Let(26)V1=E1,V2=E2,B=⋃t≥0‍S(t)B2¯,
then B is a compact invariant subset in V2; we know that semigroup S(t) defined by problem (31)–(34) possesses a V2-type compact attractor. According to Theorem 4, we need only to show the Lipschitz continuity and the squeezing property of the dynamical system S(t) in B, respectively. That is what we proceed to do in the following sections.


## 3. Exponential Attractor inV2 for Problem (1)-(2)

In this section, we show the existence of the exponential attractor inV2 for problem (1)-(2). In order to prove the Lipschitz continuity and the squeezing property, we need to extend Hölder inequality
(27)∫Ω‍|u(x)u2(x)⋯uk(x)|dx≤∏j=1k∥uj∥Lpj,
where ∑j=1k‍(1/pj)=1, pj>1 and Gagliardo-Nirenberg (G-N) inequality
(28)∥∇ju∥p≤c∥∇mu∥ra∥u∥q1-a,
where
(29)1p=jn+a(1r-mn)+1-aq,1≤q,r≤∞,0≤j<m,jm≤a≤1,
and the Young's inequality
(30)ab≤εpap+1qε(-q/b)bq,a,b,ε>0,1<p,q<∞,1p+1q=1.Theorem 10.
Assumew1(t)=(u1(t),v1(t)), and w2(t)=(u2(t),v2(t)) are two solutions of problem (1)-(2) with initial values w10=(u10,v10), w20=(u20,v20)∈B=H2×H2; then one has
(31)∥w1(t)-w2(t)∥V2≤exp(2C0t)∥w10-w20∥V2.Proof.
Lettingh(t)=u1(t)-u2(t), g(t)=v1(t)-v2(t), from (1)-(2), we have
(32)iht+γ2Δh+iγh+f(u1,u2)+g=0,(33)igt+γ2Δg+(iΓ-χ)g+h=0,
with periodic initial value
(34)h(x,t)=h(x+D,t),g(x,t)=g(x+D,t),h(x,t)=h(x+D,t)x∈R,t>0,(35)h(x,0)=u10(x)-u20(x),g(x,0)=v10(x)-v20(x),h(x,0)=u10(x)-u20(x)x∈R,
where
(36)f(u1,u2)=σ1(|u1|2u1-|u2|2u2)+iσ2(|u1|4u1-|u2|4u2).
Taking ϕ1(u)=|u|2 and ϕ2(u)=|u|4, then we get
(37)ϕ1′(ξ)h=|u1|2-|u2|2,(38)ϕ2′(η)h=|u1|4-|u2|4.
Substituting (37) and (38) into (36), we get
(39)f(u1,u2)=σ1(|u1|2u1-|u1|2u2+|u1|2u2-|u2|2u2)+iσ2(|u1|4u1-|u1|4u1+|u1|4u2-|u2|4u2)=σ1h(ϕ1(u1)+u2ϕ1′(ξ))+iσ2h(ϕ2(u1)+u2ϕ2′(η)).
Substituting (39) into (32), we obtain
(40)iht+γ2Δh+iγh+σ1h(ϕ1(u1)+u2ϕ1′(ξ))+iσ2h(ϕ2(u1)+u2ϕ2′(η))+g=0,(41)igt+γ2Δg+(iΓ-χ)g+h=0.
To prove the Theorem 4, we take the following four steps.
Step1. Taking the inner product of (40) with h¯ and (41) with g¯, respectively, we have
(42)(iht,h¯)+(γ2Δh,h¯)+(iγh,h¯)+(σ1h(ϕ1(u1)+u2ϕ1′(ξ)),h¯)+(iσ2h(ϕ2(u1)+u2ϕ2′(η)),h¯)+(g,h¯)=0,(43)(igt,g¯)+(γ2Δg,g¯)+((iΓ-χ)g,g¯)+(h,g¯)=0,
using
(44)ddt∫Ω‍|u|2dx=ddt∫Ω‍uu¯dx=∫Ω‍(utu¯+uut¯)dx=2Re∫Ω‍utu¯dx.
Thus,
(45)Im(i∫Ω‍utu¯dx)=12ddt∫Ω‍|u|2dx,(γ2Δh,h¯)=-γ2∥hx∥2,(iγh,h¯)=iγ∥h∥2,
then taking the imaginary part of (42) and (43), respectively,
(46)12ddt∥h∥2+γ∥h∥2+σ1Im∫Ω‍u2ϕ1′(ξ)|h|2dx+σ2Im∫Ω‍ϕ2(u1)|h|2dx+σ2Re∫Ω‍ϕ2′(η)|h|2dx+Im∫Ω‍gh¯dx=0,(47)12ddt∥g∥2+Γ∥g∥2+Im∫Ω‍hg¯dx=0,
by using the extend Hölder inequality, we can obtain
(48)|Im∫Ω‍gh¯dx|≤12(∥g∥2+∥h∥2),|σ1Im∫Ω‍u2ϕ1′(ξ)|h|2dx|≤|σ1|∫Ω‍|u2|ϕ1′(ξ)|h|2dx≤|σ1|∥h∥2∥u2∥∞∥ϕ1′(ξ)∥∞≤C∥h∥2,|σ2Re∫Ω‍u2ϕ2′(η)|h|2dx|≤|σ2|∫Ω‍|u2|ϕ2′(ξ)|h|2dx≤|σ2|∥h∥2∥u2∥∞∥ϕ2′(η)∥∞≤C∥h∥2.
Combining (46) and (47), then we infer that
(49)12ddt(∥h∥2+∥g∥2)+γ∥h∥2+γ∥g∥2+σ2∫Ω‍ϕ2(u1)|h|2dx≤C∥h∥2+∥g∥2.Step2. Taking the inner product of (40) with -hxx¯ and (41) with -gxx¯, respectively, we have
(50)(iht,hxx¯)+(γ2Δh,-hxx¯)+(iγh,-hxx¯)+(σ1h(ϕ1(u1)+u2ϕ1′(ξ)),-hxx¯)+(iσ2h(ϕ2(u1)+u2ϕ2′(η)),-hxx¯)+(g,-hxx¯)=0,(51)(igt,-gxx¯)+(γ2Δg,-gxx¯)+((iΓ-χ)g,-gxx¯)+(h,-gxx¯)=0.
Note that
(52)(iht,-hxx¯)=i∫Ω‍hxthx¯dx,(g,-hxx¯)=∫Ω‍gxhx¯dx,(γ2Δh,-hxx¯)=∥Δh∥2,(iγh,-hxx¯)=iγ∥hx∥2,(σ1h(ϕ1(u1)+u2ϕ1′(ξ)),-hxx¯)=σ1∫Ω‍[|hx|2(ϕ1(u1)+u2ϕ1′(ξ))+hhx¯(ϕ1(u1)+u2ϕ1′(ξ))x]dx,(iσ2h(ϕ2(u1)+u2ϕ2′(η)),-hxx¯)=iσ2∫Ω[|hx|2(ϕ2(u1)+u2ϕ2′(η))‍+hhx¯(ϕ2(u1)+u2ϕ2′(η))x]dx,
then taking the imaginary part of (50) and (51), respectively,
(53)12ddt∥hx∥2+γ∥hx∥2+σ1Im∫Ω‍(u2ϕ1′(ξ)|hx|2σ1Im∫+hhx¯(ϕ1(u1)+u2ϕ1′(ξ))x)dx+σ2Im∫Ω‍ϕ2(u1)|hx|2dx+σ2Re∫Ω‍(u2ϕ2′(η)|hx|2σ2Re∫+hhx¯(ϕ2(u1)+u2ϕ2′(η))x)dx+Im∫Ω‍gxhx¯dx=0,(54)12ddt∥g∥2+Γ∥g∥2+Im∫Ω‍hg¯dx=0.
Note the following inequalities:
(55)|σ1Im∫Ω‍(u2ϕ1′(ξ)|hx|2+hhx¯(ϕ1(u1)+u2ϕ1′(ξ))x)dx|≤|σ1|Im∫Ω‍(|u2||ϕ1′(ξ)||hx|2|σ1|Im∫+|h||hx¯|(|ϕ1(u1)x|+|u2x||ϕ1′(ξ)||σ1|Im∫+|h||hx¯|n+|u2||ϕ1′(ξ)x|))dx≤C∥hx∥2+|σ1|∥h∥∥hx¯∥(∥ϕ1(u1)x∥∞+∥u2x∥∞∥ϕ1′(ξ)∥∞C∥hx∥2+|σ1|∥h∥∥hx¯∥nnn+∥u2∥∞∥ϕ1′(ξ)x∥∞),≤C∥hx∥2+c∥h∥2,σ2Re∫Ω‍(u2ϕ2′(η)|hx|2+hhx¯(ϕ2(u1)+u2ϕ2′(η))x)dx≤C∥hx∥2+c∥h∥2,
Combining (53) and (54), one can obtain
(56)12ddt(∥hx∥2+∥gx∥2)+γ∥hx∥2+γ∥gx∥2+σ2∫Ω‍ϕ2(u1)|hx|2dx≤C∥hx∥2+∥gx∥2+c∥h∥2.Step3. Taking the inner product of (40) with hxxxx¯ and (41) with gxxxx¯, respectively, we have
(57)(iht,hxxxx¯)+(γ2Δh,hxxxx¯)+(iγh,hxxxx¯)+(σ1h(ϕ1(u1)+u2ϕ1′(ξ)),hxxxx¯)+(iσ2h(ϕ2(u1)+u2ϕ2′(η)),hxxxx¯)+(g,hxxxx¯)=0,(igt,gxxxx¯)+(γ2Δg,gxxxx¯)+((iΓ-χ)g,gxxxx¯)+(h,gxxxx¯)=0,
using
(58)(iht,hxxxx¯)=i∫Ω‍hxxthxx¯dx,(g,hxxxx¯)=∫Ω‍gxxhxx¯dx,(γ2Δh,hxxxx¯)=∥hxxx∥2,(iγh,hxxxx¯)=iγ∥hxx∥2,(σ1h(ϕ1(u1)+u2ϕ1′(ξ)),hxxxx¯)=σ1((h(ϕ1(u1)+u2ϕ1′(ξ)))xx,hxx¯)=σ1(hxxϕ1(u1)+ψ1,hxx¯),(iσ2h(ϕ2(u1)+u2ϕ2′(η)),hxxxx¯)=iσ2((h(ϕ2(u1)+u2ϕ2′(η)))xx,hxx¯)=iσ2(hxxϕ2(u1)+ψ2,hxx¯),
where
(59)ψ1=hxxu2ϕ1′(ξ)+2hx(ϕ1(u1)x+u2xϕ1′(ξ)+u2ϕ1′(ξ)x)+h(ϕ1(u1)xx+u2xxϕ1′(ξ)+2u2xϕ1′(ξ)x+u2ϕ1′(ξ)xx),ψ2=hxxu2ϕ2′(η)+2hx(ϕ2(u1)x+u2xϕ2′(η)+u2ϕ2′(η)x)+h(ϕ2(u1)xx+u2xxϕ2′(η)+2u2xϕ2′(η)x+u2ϕ2′(η)xx),
then taking the imaginary part of (50) and (51), respectively,
(60)12ddt∥hxx∥2+γ∥hxx∥2+σ1Im(ψ1,hxx¯)+σ2(hxxϕ2(u1),hxx¯)+σ2Re(ψ2,hxx¯)+Im∫Ω‍gxxhxx¯dx=0,(61)12ddt∥gxx∥2+Γ∥gxx∥2+Im∫Ω‍hxxgxx¯dx=0.
Note the following inequalities:
(62)|Im(ψ1,hxx¯)|≤C∥h∥H22,|σ2Re(ψ2,hxx¯)|≤C∥h∥H22.
Combining (60) and (61), one can obtain
(63)12ddt(∥hxx∥2+∥gxx∥2)+γ∥hxx∥2+Γ∥gxx∥2+σ2∫Ω‍ϕ2(u1)|hxx|2dx≤C∥h∥H22+∥gxx∥2.Step4. Combining (49), (56) and (63), we get
(64)12ddt(∥h∥H22+∥g∥H22)+γ∥h∥H22+γ∥g∥H22+σ2∫Ω‍ϕ2(u1)(|h|2+|hx|2+|hxx|2)dx≤C(∥h∥2+∥hx∥2+∥h∥H22)+∥g∥2+∥gx∥2+∥gxx∥2.
Taking μ=min(Γ,γ),C0=max(C,1) and noting that
(65)σ2∫Ω‍(|h|2+|hx|2+|hxx|2)dx≥0,
so (64) can be reduced to
(66)12ddt(∥h∥H22+∥g∥H22)+μ(∥h∥H22+∥g∥H22)≤C0(∥h∥H22+∥g∥H22).
By Gronwall's inequality
(67)∥h∥H22+∥g∥H22≤exp(2C0t)(∥h(0)∥H22+∥g(0)∥H22),
that is,
(68)∥w1(t)-w2(t)∥V2≤exp(2C0t)∥w10-w20∥V2.
Meanwhile, it indicates that the Lipschitz constant l(t)≤exp(2C0t). This completes the proof.
Now, we intend to show the squeezing property for semigroupS(t). To this end, we introduce the operator A=-(∂/∂x2) from D(A) to H with domain
(69)D(A)={u∈H2(Ω)}.
Obviously, A is an unbounded self-adjoint positive operator and the inverse A-1 is compact. Thus, there exists an orthonormal basis {wi}i=1∞i=1 of H consisting of eigenvectors of A such that
(70)Awi=λiwi,0≤λ1<λ1⋯λi<⋯→+∞,wheni→∞.
For all N denote by P=Pn:H→span{w1,w2,…,wn} the projector Q=QN=I-PN. In the following, we will use
(71)∥A(1/2)u∥=∥∂u∂x∥,∥A(1/2)u∥≥λN+1(1/2),u∈QNH,∥QNu∥≤∥u∥,u∈H,∥AQNu∥=∥QNAu∥≤∥Au∥,u∈D(A).
Decompose h,g as
(72)h=Ph+Qh,g=Pg+Qg.
Applying Q to (32) and (33) we find that
(73)iQht+γ2ΔQh+iγh+Qf(u1,u2)+Qg=0,(74)iQgt+γ2ΔQg+(iΓ-χ)Qg+Qh=0.
Take the inner product of (73) with Qh¯ and (74) with Qg¯, respectively. Then like Step 1, we can get
(75)12ddt(∥Qh∥2+∥Qg∥2)+γ∥Qh∥2+Γ∥Qg∥2+σ2∫Ω‍Qϕ2(u1)|Qh|2dx≤C∥Qh∥2+∥Qg∥2.
Take the inner product of (73) with -Qhxx¯ and (74) with -Qgxx¯, respectively. Then like Step 2, we can get
(76)12ddt(∥Qhx∥2+∥Qgx∥2)+γ∥Qhx∥2+Γ∥Qgx∥2+σ2∫Ω‍Qϕ2(u1)|Qhx|2dx≤C∥Qhx∥2+∥Qgx∥2+c∥Qh∥2.
Take the inner product of (73) with Qhxxxx¯ and (74) with Qgxxxx¯, respectively. Then like Step 3, we can get
(77)12ddt(∥Qhxx∥2+∥Qgxx∥2)+γ∥Qhxx∥2+Γ∥Qgxx∥2+σ2∫Ω‍Qϕ2(u1)|Qhxx|2dx≤C∥Qh∥H22+∥Qgxx∥2.
Combining (75), (76), and (77), we get
(78)12ddt(∥Qh∥H22+∥Qg∥H22)+μ(∥Qh∥H22+∥Qg∥H22)≤C0(∥Qh∥H22+∥Qg∥H22).
Using the G-N inequality
(79)∥ux∥2≤∥u∥∥uxx∥≤12(∥u∥2+∥uxx∥2),
from (78), we have
(80)12ddt(∥Qh∥H22+∥Qg∥H22)+μ(∥Qh∥H22+∥Qg∥H22)≤3C02(∥Qh∥+∥Qhxx∥+∥Qg∥+∥Qgxx∥)≤3C02λN+1-1(∥Qhxx∥+∥Qgxx∥)≤3C02λN+1-1(∥Qh∥H22+∥Qg∥H22)≤3C02λN+1-1exp(2C0t)(∥h(0)∥H22+∥g(0)∥H22).
By Gronwall lemma we get
(81)∥Qh∥H22+∥Qg∥H22≤(∥h(0)∥H22+∥g(0)∥H22)exp(-2μt)+C¯λN+1-1exp(2C0t)(∥h(0)∥H22+∥g(0)∥H22)≤[exp(-2μt)+C¯λN+1-1exp(2C0t)]×(∥h(0)∥H22+∥g(0)∥H22).
Letting t*>0 be fixed we take w(t)=w1(t)-w2(t)=(h(t),g(t)) and assume that
(82)exp(-2μt*)≤1256.
Then we choose N large enough so that
(83)C¯λN+1-1exp(2C0t)≤1256,
that is,
(84)λN+1≥256C¯exp(2C0t).
From (82) and (84), we obtain
(85)∥Qh∥H22+∥Qg∥H22≤1128(∥h(0)∥H22+∥g(0)∥H22).
This shows that when t*>0 is fixed, Lipschitz constant for S(t) in B is equal to exp(2C0t*) and N satisfies
(86)λN+1≥256C¯exp(2C0t*).
We have
(87)∥Qw∥V2≤∥Qw(0)∥V2.
So when
(88)∥Qw(t*)∥V2>∥Pw(t*)∥V2,∥w(t*)∥V2=∥Qw(t*)∥V2+∥Pw(t*)∥V2<2∥Qw(t*)∥V2≤164∥Qw(0)∥V2≤164∥w(0)∥V2.
This completes the proof of Theorem 4.Theorem 11.
The semigroupS(t) associated with problem (1)-(2) is squeezing in B. Now we conclude this section by giving our main result.Theorem 12.
Suppose that problem (1)-(2) satisfies Theorem 9; there exist t*≥(1/2μ)ln(256) and N large enough such that
(89)λN+1≥256C¯exp(2C0t*).
Then for the nonlinear semigroup S(t) defined in (4) and (5), S(t)t≤0; B admits an exponential attractor M in V2 and
(90)dF(M)≤1+CN0,distV2(S(t)B,M)≤C0exp(-C1t),
where C0, C1, C 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,N is only large enough such that
(91)λN+1≥256C¯exp(2C0t*).



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*Source: 390476-2013-04-07.xml*