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

Definition 1. A compact set M 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 d_F < ∞;

• (iv)

  There exist constants c₁ and c₂ such that

  $$ ()

where

Definition 2. If there exists a bounded function l(t) independent u and v such that

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 semigroup S(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 every δ ∈ (0, (1/8)), there exists an orthogonal projection $$ of rank equal to N₀ such that for every u and v in C if

holds, then we also have where $$.

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

• (1)

  There exist nonlinear semigroup S(t) _t≥0 and a compact attractor B.

• (2)

  There exists a compact set C 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 where Moreover, 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 exists t₀(B₀) such that

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. Let B₀, 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].

Lemma 7. Assume that (u₀, v₀) ∈ E₁; then

Thus there exists t₁ = t₁(R) > 0 such that whenever $$.

Lemma 8. Assume that (u₀, v₀) ∈ E₂; then

Thus there exists t₂ = t₂(R) > 0 such that whenever $$.

Theorem 9. Assume that all the parameters of (1) are positive. For (u₀, v₀) given in E_i (i = 1,2), there exists a unique solution

And also Furthermore, the solution operator of the system is a continuous semigroup S(t) on E₁ which possesses bounded absorbing sets B_i ⊂ E_i, for i = 1,2.

Theorem 10. Assume w₁(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

Proof. Letting h(t) = u₁(t) − u₂(t), g(t) = v₁(t) − v₂(t), from (1)-(2), we have

with periodic initial value where Taking ϕ₁(u) = |u|² and ϕ₂(u) = |u|⁴, 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 μ = min (Γ, γ),C₀ = max (C, 1) and noting that so (64) can be reduced to By Gronwall′s inequality that is, Meanwhile, it indicates that the Lipschitz constant l(t) ≤ exp (2C₀t). This completes the proof.

Now, we intend to show the squeezing property for semigroup S(t). To this end, we introduce the operator A = −(∂/∂x²) from D(A) to H with domain

Obviously, A is an unbounded self-adjoint positive operator and the inverse A⁻¹ is compact. Thus, there exists an orthonormal basis $$ of H consisting of eigenvectors of A such that For all N denote by P = P_n : H → span {w₁, w₂, …, w_n} the projector Q = Q_N = I − P_N. In the following, we will use Decompose h, g as Applying Q to (32) and (33) we find that Take the inner product of (73) with $$ and (74) with $$, respectively. Then like Step 1, we can get Take the inner product of (73) with $$ 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 t_* > 0 be fixed we take w(t) = w₁(t) − w₂(t) = (h(t), g(t)) and assume that Then we choose N large enough so that that is, From (82) and (84), we obtain This shows that when t_* > 0 is fixed, Lipschitz constant for S(t) in B is equal to exp (2C₀t_*) and N satisfies We have So when This completes the proof of Theorem 4.

Theorem 11. The semigroup S(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

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 where C₀, C₁, C are constants independent of the solution of the equation.