Asymptotic Behavior of the Coupled Nonlinear Schrödinger Lattice System

and Applied Analysis 3 Taking the imaginary part of the inner product of the first equation and second equation of (4) with ?̃?k := η(|k|/M)uk and Ṽk := η(|k|/M)Vk in l , respectively, whereM ∈ Z is large enough, we have ε 2 d dt ∑ k∈Z η( |k| M ) 󵄨󵄨󵄨󵄨uk 󵄨󵄨󵄨󵄨 2 + σ1∑ k∈Z η ( |k| M ) 󵄨󵄨󵄨󵄨uk 󵄨󵄨󵄨󵄨 2 + V 2 Im ∑ k∈Z η ( |k| M ) Vkuk − Im ∑ k∈Z [(B?̃?) k (Bu)k] = Im ∑ k∈Z g 1 k η ( |k| M )uk, ε 2 d dt ∑ k∈Z η( |k| M ) 󵄨󵄨󵄨󵄨Vk 󵄨󵄨󵄨󵄨 2 + σ1∑ k∈Z η ( |k| M ) 󵄨󵄨󵄨󵄨Vk 󵄨󵄨󵄨󵄨 2 + V 2 Im ∑ k∈Z η ( |k| M )ukVk − Im ∑ k∈Z [(BṼ) k (BV)k] = Im ∑ k∈Z g 2 k η ( |k| M ) Vk. (16) Summing up (16), we get ε 2 d dt ∑ k∈Z η ( |k| M ) ( 󵄨󵄨󵄨󵄨uk 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨Vk 󵄨󵄨󵄨󵄨 2 ) + σ1∑ k∈Z η( |k| M ) 󵄨󵄨󵄨󵄨uk 󵄨󵄨󵄨󵄨 2 + σ2∑ k∈Z η ( |k| M ) 󵄨󵄨󵄨󵄨Vk 󵄨󵄨󵄨󵄨 2 = Im ∑ k∈Z [(B?̃?) k (Bu)k + (BṼ)k(BV)k] + Im ∑ k∈Z [g 1 k η ( |k| M )uk + g 2 k η ( |k| M ) Vk] . (17) Let σ = min{σ1/2, σ2/2} > 2. By Cauchy-Schwartz inequality and (17), we have d dt ∑ k∈Z η ( |k| M ) ( 󵄨󵄨󵄨󵄨uk 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨Vk 󵄨󵄨󵄨󵄨 2 ) + 2 (σ − 2) ε ∑ k∈Z η ( |k| M ) ( 󵄨󵄨󵄨󵄨uk 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨Vk 󵄨󵄨󵄨󵄨 2 )

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 2. The global attractor A  converges A in the sense of the Hausdorff semidistance related to ℓ 2 × ℓ 2 ; that is, where ( 1 ,  2 ) = sup ∈ 1 inf ∈ 2  X (, ), for any nonempty compact subsets  1 and  2 in a metric space X.
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.

Existence of the Global Attractor
This section shows the existence of compact attractor of system (1) in ℓ 2 × ℓ 2 .We denote a Hilbert space by ℓ 2 with the scalar product ⟨, V⟩ = ∑ ∈Z   V  , , V ∈ ℓ 2 .
Firstly, we prove the global existence of solution of system (1).For convenience, we take the scalar form of system (1): Then system (4) can be rewritten as (5) Then we get the mild solution of (5) as where the semigroup T() =  A generated by the operator A : ℓ 2 → ℓ 2 , and In order to prove the existence of compact attractor, we need the following proposition.
It is easy to verify that  1 (, V) and  2 (, V) satisfy the Lipschitz continuous property on any bounded set in ℓ 2 × ℓ 2 .Using the same method in [14], we obtain the following result.
Then there exists a bounded absorbing ball B of the semigroup {S()} ≥0 generated by system (4) in The radius of B is  > 0. Therefore, there exists  0 ≥ 0 depending on B such that Proof.Taking the imaginary part of the inner product of the first equation and second equation of ( 4) with () and V(), respectively, we have Summing up (10), Let  = min{ 1 /2,  2 /2}.By Young inequality, we have Applying Gronwall Lemma to (12), we have It implies that the semigroup {S()} ≥0 possesses a bounded absorbing ball B ⊂ ℓ 2 × ℓ 2 centered at 0 with radius Then, there exist () and () such that the solution ((), V()) of system (4) satisfies Proof.Define () ∈ C(R + ; [0, 1]) by where  0 is a positive constant number.
Taking the imaginary part of the inner product of the first equation and second equation of (4) with ũ := (||/)  and Ṽ := (||/)V  in ℓ 2 , respectively, where  ∈ Z is large enough, we have Summing up (16) where  0 and  are the time of entry of initial data bounded in ℓ 2 × ℓ 2 and radius of the absorbing ball in ℓ 2 × ℓ 2 .Since  1 ,  2 ∈ ℓ 2 , then, for any given  > 0 and  >  0 , there exist () and () such that where () denote a constant number depending on .
Proof.Define Γ = {S(  )(  , V  ) : (  , V  ) ∈ B,   → ∞ as  → ∞}.Our purpose is to prove that Γ has finite covering balls of radii .By Lemma 6, we know that, for all (  , V  ) ∈ Γ, there exist () and () such that We consider the set ; that is, there exists a family of balls of radii /2, which covers Γ.This together with (21) implies that the set Γ 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.

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).( Proof of Theorem 2. We denote global attractors generated by semigroups S() and S  () by A and A  , respectively.By Lemma 7, we can get that B := B ⋂ (C 2+1 × C 2+1 ) is also an absorbing set for S  ().Then, we have where O denotes an open neighborhood of absorbing ball B.