Response solution to complex Ginzburg–Landau equation with quasi-periodic forcing of Liouvillean frequency

In this paper, the existence of a response solution with the Liouvillean frequency vector to the quasi-periodically forced complex Ginzburg–Landau equation, whose linearized system is elliptic–hyperbolic, is obtained. The proof is based on constructing a modified KAM theorem for an infinite-dimensional dissipative system with Liouvillean forcing frequency.


Introduction and main result
The complex Ginzburg-Landau equation u t = ru + (b + iν)∂ xx u + m∂ x u -(1 + iμ)|u| 2 u (1.1) is extensively studied in the physics community. Here, the real parameter m depicts the group velocity, and the real parameters ν and μ characterize linear and nonlinear dispersion, and b is real as is the control parameter r. It results from nonlinear stability theory and describes the evolution of complex amplitude coefficient u = u(t, x) of a neutral plane wave. See [1][2][3][4] and the references therein for more details and physical and mathematical background.
The existence and stability of periodic or quasi-periodic solutions to (1.1) have been extensively investigated in many papers, for example [2,[5][6][7][8]. When x ∈ T d := (R/2πZ) d , there are some papers concerning the existence of KAM-type tori for (1.1). More concretely, Chung and Yuan [9] and Cong, Liu and Yuan [10] proved the existence of quasiperiodic solutions which are not traveling waves for d = 1 and d ≥ 2 respectively in the case of the group velocity m = 0 by KAM-type theorems. See also [11][12][13].
In this paper, we assume the forcing frequency is ω = (1, α), with α ∈ (0, 1) being any irrational numbers. Since we do not impose arithmetic condition (i.e., Diophantine or Brjuno condition) on ω, it can also be Liouvillean. There are some works addressing this frequency. More concretely, Avila, Fayad and Krikorian [15] developed a new KAM scheme for discrete SL(2, R) co-cycles with one Liouvillean frequency by using the technique of CD bridge. Further, Hou and You [16] studied the reducibility problems for continuous two-dimensional quasi-periodic linear systems. For the nonlinear system, Wang, You and Zhou [17] and Lou and Geng [18] investigated the existence of response solutions for the quasi-periodically forced nonlinear harmonic oscillators in Hamiltonian and reversible case respectively.
Note that the above work is all about the systems that possess a Hamiltonian or reversible structure. The question is that whether the systems without such structures still possess the response solution or not. Motivated by this question, in this paper, we consider quasi-periodically forced complex Ginzburg-Landau equation (1.2), which is a dis-sipative system with the forcing frequency ω = (1, α). The linearized equation of (1.2) is given by u t = ru + (b + iν)∂ xx u + m∂ x u, x ∈ T, and the linear operator r + (b + iν)∂ xx + m∂ x possesses the eigenvalues λ n = rbn 2 + i mnνn 2 , n ∈ Z.
For any given j ∈ N \ {0}, we can choose suitable r, b ∈ R such that Re λ ±j = rbj 2 = 0 and Re λ l = rbl 2 = 0 for |l| = j. In this case, there are eigenvalues which are pure imaginary. Moreover, we assume the basic frequency ω is Liouvillean. Thus, the method in [9,10,14] cannot be directly applied since in these papers the frequency is Diophantine and the linear system is pure hyperbolic i.e., the real parts of all frequencies are not zero. In a Hamiltonian case like [17], one constructed the symplectic transformation by using the time-1-map of an auxiliary Hamiltonian flow to preserve the Hamiltonian structure in each KAM step. However, we deal with the infinite-dimensional dissipative system in this paper. So we directly construct the nearly identical coordinate transformation, which needs a more complicated computation. For the hyperbolic part, we only eliminate the terms depending only on the angle variables since Re λ n = Re λ -n for |n| = j.
For Eq. (1.2), we always assume: (H) f (ωt, x) and h(ωt, x) are quasi-periodic in t with frequency vector ω. Moreover, the functions f (θ , x) and h(θ , x) are analytic in (θ , x) ∈ T 2 × T with the following Fourier expansions: Now we state the main result of this paper.
Our paper is organized as follows. In Sect. 2, we give some definitions and notations on vector field and continued fraction expansion. In Sect. 3, a modified infinite-dimensional KAM theorem for our dissipative equation with Liouvillean frequency is presented. In Sects. 4 and 5, we prove the KAM theorem, Theorem 3.1. In Sect. 6, we apply our KAM theorem 3.1 to the quasi-periodically forced complex Ginzburg-Landau equation (1.2) and prove Theorem 1.1.

Functional setting
Let T 2 c = C 2 /(2πZ 2 ) be the two-dimensional complex torus. For δ > 0, we denote the complex neighborhood of 2-torus T 2 by where | · | is the supremum norm of the complex vector.
Suppose O ⊆ R 2 is a compact set. For a C 1 W (C 1 smooth in the sense of Whitney) function f : O → C, we define its norm as Given a function f : where k, θ = k 1 θ 1 + k 2 θ 2 and |k| = |k 1 | + |k 2 |.
For K > 0 and an analytic function f on D(δ) × O, we define the truncation operator T K and projection operator R K as The average [f (θ ; ξ )] θ of f (θ ; ξ ) over T 2 is defined as We denote the index sets by Then we define the space a,p := {q = (. . . , q j , . . .) n∈J 2 : q j ∈ C} of complex sequences equipped with the following norm: where j := max{1, |j|} and a ≥ 0, p > 1 2 are constants such that the Banach algebra property holds in this space. Lemma 2.1 ([21]) For w, z ∈ a,p , the convolution w * z is defined by (w * z) j = k∈Z w j-k z k . For a ≥ 0, p > 1 2 , then w * z a,p ≤ c w a,p z a,p with a constant c depending only on p.
For δ, s > 0, we introduce a complex neighborhood of Denote a = (a 1 , . . . , a d ), α = (. . . , α j , . . .) j∈J 2 , with finitely many non-zero components a j , α j ∈ N. Given a function P : D(δ, s) × O → C, which is analytic in (θ , ρ, z) ∈ D(δ, s) and C 1 W in ξ ∈ O and has Taylor-Fourier series expansion Consider the dynamical systeṁ where we have the vector field For the vector field X : D(δ, s) × O → P a,p , which is analytic in (θ , ρ, z) ∈ D(δ, s) and depends C 1 W smoothly on parameter ξ ∈ O, the weighted norm of X is defined as
Let p 0 = 0, p 1 = 1, q 0 = 1, q 1 = a 1 , and inductively Then {q n } is the sequence of denominators of the best rational approximations for α. It satisfies In the sequence {q n }, we will fix a special subsequence {q n k }. For simplicity, we denote the subsequences {q n k } and {q n k +1 } by {Q k } and {Q k }, respectively. Next, we introduce the concept of CD bridge which was first given in [15]. Definition 1 (CD bridge, [15]) Let 0 < A ≤ B ≤ C. We say that the pair of denominators (q m , q n ) forms a CD(A,

A modified KAM theorem
To prove Theorem 1.1, we give an abstract modified KAM theorem, which can be applied to the quasi-periodically forced complex Ginzburg-Landau equation (1.2). The proof of the KAM theorem will be finished by an iterative procedure in Sect. 5. Each step of the iterative procedure is set up by a finite Newton iteration. Consider the following system: Here, Ω(ξ ) = diag(Ω 1 (ξ ), . . . , Ω d (ξ )) with Ω j (ξ ) ∈ R, and Λ(ξ ) = diag(. . . , λ j (ξ ), . . .) j∈J 2 . We also identify Ω(ξ ) and Λ(ξ ) as vectors When p = g ≡ 0, the system (3.1) admits an invariant torus Our goal is to show that if the perturbations p, g are small enough, the system (3.1) still admits invariant torus with Liouvillean frequency ω = (1, α) for most of parameter ξ ∈ O (in Lebesgue measure sense) provided that Ω, Λ satisfy some non-degeneracy conditions. Now we state our KAM theorem.
then, for every sufficiently small γ > 0, there exists ε 0 > 0 depending on δ, δ * , s, , 1 , 2 , τ , d but not on α, such that whenever where p * , g * are at least of order 2 with response to ρ, z, and B * is a diagonal matrix.

Homological equation and its solution
The main idea of proving the KAM theorem, Theorem 3.1, is to construct a series of coordinate transformations {Φ l } ∞ l=0 such that the perturbation of transformed system is smaller and smaller. Because the system is dissipative, we construct the nearly identical transformation directly, which makes the proof more complicated. In this procedure, we need to solve a series of homological equations to construct the desired transformations. The idea of iterative procedure is detailed in Sect. 5.

Derivation of homological equation
and have the following form: We will construct a transformation Φ defined on a smaller domain such that the system (4.1) is transformed into where the norm of perturbation p + and g + in (4.2) on the small domain is smaller than that in (4.1) (see Lemma 5.3 for details).
For simplicity, we drop the parameter ξ in this section. Suppose that the coordinate transformation Φ has the following form: and the other expressions are similar. Let Eqs. (4.4)-(4.6) and (4.8)-(4.9) be equal to 0, then we obtain the following homological equations: If we find the solutions H j (j = 1, 2, 3) and F j (j = 1, 2) of the homological equations (4.11)-(4.15), we will obtain a new system with another perturbation, which will be smaller on a small domain.

Solution to homological equation
In this subsection, we consider the homological equations with variable coefficients (4.11)-(4.15) and find their solutions. We only give the solutions to homological equations (4.13) and (4.15) in detail while omitting the other solutions, since the other equations can be dealt in the same way.
In the following, we assume that for any given τ > 2, 0 < γ < 1 and K > 0, where Moreover, we let A := τ + 3, M := A 4 2 and {Q n } be the selected sequence of α in Lemma 2.2 with respect to A. In the process of solving the homological equations, we also use the following notations: where 0 <c < 1 is a constant which will be defined later and c 0 : The solutions to the homological equations (4.13) and (4.15) with estimates are given in Proposition 4.3 and 4.4, respectively. For the homological equation (4.13), we find an approximate solution with suitable small error term using idea in [16,17]. We remove the non-resonance terms of B(θ ; ξ ) to eliminate relatively large B(θ ; ξ ) by solving the equation Due to the lack of Diophantine condition on ω, we will use the technique of the CD bridge introduced in Sect. 2.2 to obtain a good estimate of solution B(θ; ξ ) for (4.17) (see Lemma 4.1).
The following lemma is about the estimate of small divisors.
, there exists a positive constant c 2 (τ ) such that, for |k| < K , and 0 < |l| ≤ 2, Now we solve the homological equation (4.13) in the following proposition.
For δ * > 0, 0 <c < 1, there exist positive constants c 3 (τ ) and 1 then each equation in (4.18) has an approximate solution which can be estimated as follows: Moreover, the error term R e lj satisfies In the case of B = 0, the equation has an approximate solution H 2lj satisfying and the error term R e lj satisfies Proof We only prove the case of B = 0. The case of B = 0 is similar and easier. Consider Eq. (4.18) for the unknown function H 2lj for any 1 ≤ l, j ≤ d, where Ω lj (ξ ) = Ω l (ξ ) -Ω j (ξ ) and similarly for B lj (θ ; ξ ) and b lj (θ ; ξ ). For 1 ≤ l = j ≤ d, H 2lj = 0, by assumption R lj = 0. Thus, we consider the case 1 ≤ l = j ≤ d in the following. Let Then, by Lemma 4.1 and assumption (4.19), we have Taking 0 < 1 < c 0 (τ ,c) 960c 1 (δ * ,τ ,c) , together with the definition of E in (4.16), we obtain Then Eq. (4.18) becomes where We first solve the truncated equation of (4.21), We write By comparing the Fourier coefficients of Eq. (4.22), for |k| < K ,we have This can be viewed as a vector equation: where Let E δ = diag . . . , e |k| δ , . . . |k|<K .
Then Eq. (4.23) is equivalent to It follows from Ω + [B] θ ∈ DC ω (γ , τ , K, O) and Lemma 4.2 that where the matrix norm is defined by we have (i.e., assumption (4.20) holds), we get This implies that T + E δ SE -1 δ has a bounded inverse and Therefore, Going back to H 2lj (θ ; ξ ) = e iB(θ;ξ ) H 2lj (θ ; ξ ), we get For the error term R e lj , i.e., R e lj = e iB(θ;ξ ) R K e -iB(θ;ξ ) ib lj (θ ; ξ )H 2lj (θ ; ξ ) + R lj (θ ; ξ ) , we have the following estimate: Note that the case B(θ ; ξ ) = 0 means that there is no need to define H 2lj or R lj . In this case, we directly deal with the equations for H 2lj and R lj to obtain the estimates for H 2lj and R e lj .
In the next proposition, we study the homological equation (4.15) using the nondegeneracy condition (3.2). Since the real part of λ j (ξ ) satisfies | Re λ j (ξ )| > 1 , there is no small divisor.
where ⊗ is the tensor product of two matrices (see [22] for details) and where W lj is the (l, j)th element of matrix W .
By comparing the Fourier coefficients of Eq. (4.25), we obtain can be viewed as the vector equation It follows from the non-degeneracy condition for Λ(ξ ) in (3.2) that Moreover, If we take 2 ≤ 1 c 4 , by (4.24), then we get This implies T + E δ SE -1 δ has a bounded inverse. Therefore, As a conclusion, we get Similarly, we can deal with other homological equations and obtain the approximate or exact solutions with estimates respectively. After this, we can get a new system with a new perturbation as follows: p + = (4.7) + R (pe) , g + = (4.10), (4.26) where R (pe) is the error term from solving the homological equations. In the proof of Proposition 5.1, we will prove that the above perturbation is smaller on a small domain.

Proof of Theorem 3.1
In this section, we will give the proof of Theorem 3.1. We will give one KAM step in detail, which needs finite Newton iteration. After one step of Newton iteration, the perturbation is smaller than that in previous step. Via finite steps of transformation, the perturbation is small enough to meet the KAM iterative requirement. So we can set up one cycle of KAM scheme. This is the essential difference from the classical KAM iteration with the Diophantine or Brjuno conditions. For simplifying our notations, we drop the subscript n and write the symbol "+" for (n + 1). Suppose at the nth step of the KAM scheme, we have the system Our goal is to find an analytic transformation Φ : D(δ + , s + ) × O → D(δ, s) × O such that the transformed system of (5.1) is of the form ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩θ = ω ρ + = i(Ω(ξ ) + B + (θ ; ξ ))ρ + + p + (θ , ρ + , z + ; ξ ), z + = (Λ(ξ ) + W + (θ ; ξ ))z + + g + (θ , ρ + , z + ; ξ ), (5.2) where the norm of new perturbation p + and g + in (5.2) on a small domain D(δ + , s + ) is smaller than (5.1) (see Proposition 5.1).

Moreover, the transformation Φ satisfies
Φid s + ,D(δ + ,s + )×O ≤ 4ε The above proposition plays an important role to run one KAM step. In the following, we use a Newtonian iterative procedure consisting of finite steps to prove Proposition 5.1.

Firstly, let
Then, for 1 ≤ j ≤ L, we define the following sequences: where In the following, we give some inequalities for the above sequences, which we use frequently in the proof.

Lemma 5.2 There exist positive constants J
then we have Now consider the inequality (5.12). When j ≥ j 0 , it is obvious by the definition of σ j in (5.8). When 0 ≤ j < j 0 , we have due to the smallness of ε 0 ≥ ε j and the choice of j 0 . Moreover, by the definition of K in (4.16) and Q n+1 ≥ T 0 γ -A/2 , we can get for sufficiently large T 0 . By the definition of ε j , we have It follows from (5.14) and (5.15) that Consider (5.13). When j ≥ j 0 , it is obvious from K ε 1 20 j < 1 by the choice of j 0 . In the case of j < j 0 , by (5.10) and (5.11), there exists 0 < 4 = 4 (τ , δ * ) such that if 0 ≤ min{ 3 , 4 }, then

A finite inductive lemma
We now give the following iterative lemma for a finite induction, which is used to prove Proposition 5.1.
In the following, we will denote by c a constant depending only on τ , d, but not on the iterative step number j.

Lemma 5.3 Suppose that ε 0 satisfies assumptions in Lemma 5.2 and the system
satisfies the conditions in Proposition 5.1 for B, W and Then there is an analytic transformation such that the transformed system of (5.16) is the system defined on D( δ j+1 , s j+1 ) × O, and satisfies the same assumption (5.17) with (j + 1) in place of j for 0 ≤ j ≤ L -1.
and the corresponding error term R Thus, we can obtain Similarly, we can get where p 0 j := p j (θ , 0, 0; ξ ). Moreover, the estimates of the error terms are By Proposition 4.4, we can get exact solutions for H 3 and F i (i = 1, 2) with estimates F 1 a,p, δ j ,O ≤ c g 0 a,p, δ j ,O , g 0 := g(θ , 0, 0; ξ ).
Since P j s j ,D( δ j , s j )×O ≤ ε j , by the weighted norm of vector field, we have Thus by Cauchy's estimate and the inequalities in Lemma 5.2, we have For F 1 , F 2 , F 3 , we can obtain Thus, we have That implies Φ j (D( δ j+1 , s j+1 )) ⊆ D( δ j , s j ). And we obtain Similarly, by Cauchy's estimate, we can also obtain the tangent map (3) Finally, we give the estimate of the new perturbation in detail. From the (4.26) in Sect. 4.2, we can get the new perturbation: We mainly focus on the term p j+1 since all others can be dealt with in the same way. For the form of p j+1 , we decompose it into five parts: where I 1 = p j • Φ jp j (θ , ρ + , z + ; ξ ), Now we need to verify δ + ≤ δ L for δ + = δ 0 (1η). It is sufficient to show that L-1 m=0 σ m ≤ η/2. By the definition of σ j in (5.8), we obtain And if L > j 0 , As a consequence, Proof of Proposition 5.1 The proof of Proposition 5.1 is an immediate result of Lemma 5.3. By applying Lemma 5.3 for L times inductively, we get a sequence of transformations Then, via transformation Φ, we get the new system (5.2) with B + = B + b L , W + = W + w L , satisfying and the perturbation P + = P L satisfying P + s + ,D(δ + ,s + )×O ≤ P L s L ,D( δ L , s L )×O ≤ ε L ≤ ε + .
Next, we verify that the transformation Φ satisfies (5.5) and (5.6). By the chain rule and (5.19), one has Then, by the mean value theorem and (5.18), As a consequence, Similarly,

Iterative lemma for KAM scheme
We define infinitely many successive steps of KAM iteration using Proposition 5.1. For By the discussion in [17], for T defined above, we can choose n 0 ∈ N such that Q n 0 ≥ T. Then we choose sufficiently small ε depending on the constants δ, δ * , τ , γ , such that We define the iterative sequences depending on ε, δ, s, γ by That is, For given 1 ≥ s > 0, we also define Firstly, the sequence {s n } n≥0 is decreasing and goes to 0 as n → ∞. For the sequences {δ n } n≥0 defined in (5.27), we show that δ n > δ * for every n ≥ 0. Indeed, by According to the analysis in Sect. 5.2, we can conclude to the following iterative lemma.
Secondly, we need to show B n and W n are small enough. From (5.36) and M = A 4 2 , we see that B n satisfies From (5.29) and W 0 = 0, we get Finally, we prove P n s n ,D(δ n ,s n )×O n-1 ≤ ε n ≤ 0 γ J n E n .
The definition of ε n in (5.27) and condition (5.26) show that When n = 0, it suffices to take ε 0 satisfying (5.26). Therefore, by applying Proposition 5.1, there exists an analytic transformation Φ n which is of the form (5.32) such that the transformed system (5.33) has the same properties as the system (5.28) at the nth KAM step. Moreover, the transformation Φ n satisfies the estimates (5.34) and (5.35) by Proposition 5.1 again.

Convergence and measure estimates
We begin with the system Since | Re λ j (ξ )| > 1 for some positive constant 1 , we do not encounter a small divisor when solving the homological equations with respond to z. Therefore, one does not need any non-resonant condition for Λ(ξ ).
It follows from the estimates (5.29) of B n and W n that B n and W n converge uniformly to limits B * and W * on domain D(δ * , s 2 ) × O γ with W n -W n-1 δ n ,p,p,O n-1 ≤ 4ε 0 .
Measure estimates: During the procedure of KAM iteration, we obtain a decreasing sequence of closed subsets O 0 ⊇ O 1 ⊇ · · · . It is crucial to prove that the Lebesgue measure of their intersection O γ is positive in KAM theory for small enough γ > 0.
Then using the non-degeneracy condition (3.2) and the analysis in Sect. 4.2 of [17], we have the following lemma for the measure of the parameter set O γ .

Lemma 5.5
For τ > 2 and sufficiently small γ > 0, we have As a conclusion, we complete the proof of Theorem 3.1.