Quasi-periodic solutions for a completely resonant beam equation with a nonlinear term depending on the time and space variables

Nonlinear Analysis 189 (2019) 111585

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Quasi-periodic solutions for a completely resonant beam equation with a nonlinear term depending on the time and space variables✩ Yi WANG School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, PR China

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Article history: Received 6 April 2019 Accepted 20 July 2019 Communicated by Enrico Valdinoci MSC: 37K55 70K43 70K45 70K40

abstract This article is devoted to the study of a completely resonant beam equation with an x-periodic and t-quasi-periodic nonlinear term. It is proved that the equation admits small amplitude, linearly stable and quasi-periodic solutions for most values of the frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some Birkhoff normal form. We show an infinite dimensional KAM theorem for non-autonomous beam equations, which is applied to prove the existence of quasi-periodic solutions. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Completely resonant beam equation Quasi-periodically forced x-dependent Quasi-periodic solution Normal form

1. Introduction In this paper, a completely resonant beam equation with an x-periodic and t-quasiperiodic nonlinear term utt + uxxxx + εg(ωt, x)u3 = 0,

x ∈ [0, π]

(1.1)

u(t, 0) = uxx (t, 0) = u(t, π) = uxx (t, π) = 0

(1.2)

under the hinged boundary conditions

is considered, where ε is a small positive parameter; ω = (ω1 , ω2 , · · · , ωm ) ∈ [ϱ, 2ϱ]m (ϱ > 0) is a frequency vector; and the function g(ωt, x) = g(ϑ, x), (ϑ, x) ∈ Tm ×[0, π], is real analytic in (ϑ, x) and quasi-periodic in t. Beam equations are important models in structural mechanics. We aim to explore whether the boundary value problem (1.1) with (1.2) has analytic and linearly stable quasi-periodic solutions. ✩ This work was supported by National Natural Science Foundation of China (Grant No. 11601270). E-mail addresses: [email protected], [email protected].

https://doi.org/10.1016/j.na.2019.111585 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

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Y. WANG / Nonlinear Analysis 189 (2019) 111585

We study this equation as an infinite dimensional Hamiltonian system. The infinite-dimensional KAM theory is one of main approaches to construct the periodic and quasi-periodic solutions. The periodic and quasi-periodic solutions constructed via the KAM method can provide some dynamics information, such as linear stability and zero Lyapunov exponents. In this paper, we apply the KAM theory as well as Birkhoff normal forms to attain quasi-periodic solutions for (1.1) with (1.2). About the KAM method and its applications, readers refer to [10,12,15,18,21] and references therein. In [6–8], Geng and You studied one-dimensional and multi-dimensional nonlinear beam equations for quasi-periodic solutions via KAM method. Their works treat equations with constant-coefficient nonlinearities. When a nonlinearity depends on t, a variant-coefficient transformation is necessary in the normal form step, while in a constant-coefficient nonlinearity case, a constant-coefficient transformation is sufficient. Tuo and Si [19] proved the existence of quasi-periodic solutions for the one-dimensional nonlinear beam equations with quasi-periodic quintic nonlinearities utt + uxxxx + (B + εϕ(t))u5 = 0

(1.3)

where B is a positive constant, ε is a small positive parameter, and ϕ(t) is a real analytic quasi-periodic function. They changed the equation to a non-autonomous one and reduced a Hill’s equation to a constantcoefficient equation. Eq. (1.3) is a completely resonant equation, namely in the linear part the frequencies are rational and all the bounded solutions are periodic. In order to solve the completely resonant problem, Tuo and Si [19] transformed (1.3) to a non-resonant beam equation by letting u = u0 + εv(x, t), where u0 is a non-zero quasi-periodic solution for an ordinary differential equation u ¨0 + (B + εϕ(t))u50 = 0. For our Eq. (1.1), the corresponding ordinary differential equation is u ¨0 + εg(ωt, x)u30 = 0, which cannot be solved. So we must look for other methods. The nonlinearities of the above equations cannot depend on the space variable x. The x-dependent nonlinearity implies that the equation is variant in space translations and the momentum is not conserved. After one expands the perturbation by infinite-dimensional coordinates, the coefficients are integrals of eigenfunctions product. With the assumption that the equations do not explicitly contain x, the normal form step is simplified in [3,6–9,14]. For instance, in [6], the coefficients with i ± j ± d ± l ̸= 0 are zero and the authors only discussed the divisors with i ± j ± d ± l = 0. In multi-dimensional equations, the x-dependent nonlinearity has the effect that the normal form is not diagonal in the purely elliptic directions. Eliasson, Grebert, and Kuksin [4] thought that [7,8] treat equations with a constant-coefficient nonlinearity g(x, u) = g(u), which is significantly easier than the general case. They [4] proved a KAM result for the nonlinear beam equation on the d-dimensional torus utt + ∆2 u + mu + g(x, u) = 0, t ∈ R, x ∈ Td ,

(1.4)

where g(x, u) = 4u3 + O(u4 ). They showed that, for generic m > 0, most of the small amplitude invariant finite dimensional tori of the linear equation persist as invariant tori of the nonlinear equation. It is worth mentioning that the nonlinearity u3 in (1.4) is still constant-coefficient and the nonlinearity of (1.3) cannot depend on x. For models with variant-coefficient nonlinearities, [1,2,5,21] can be referred. For Eq. (1.4), m is an important parameter. The number of parameters is reduced when m = 0, which is a quite different case and cannot be included in [4]. Liang and Geng [11] and Procesi [17] proved the existence and stability of quasi-periodic solutions for the completely resonant equations. The Hamiltonian system after the normal form step appears to have a very intricate behavior. In the case of finite-dimensional systems this may pose serious problems in proving the existence of invariant tori [17]. However, their equations are constant-coefficient and the normal form step is simplified. Our Eq. (1.1) is quite different from the above. It is quasi-periodically forced, completely resonant and has an x-dependent nonlinearity. This equation simultaneously has three characters, which is complex and difficult. To the best of our knowledge, there are no results regarding the quasi-periodic solutions for (1.1).

Y. WANG / Nonlinear Analysis 189 (2019) 111585

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We first transform the Hamiltonian into a Birkhoff normal form, selecting the coordinates of the non-resonant index set, namely the admissible index set, and adding the small divisor condition 1 ⏐ √ ⏐ √ √ √ ϱε 3 ⏐± λi ± λj ± λd ± λl + ⟨k, ω⟩⏐ ≥ m+2 (see details in Section 4). Fortunately, the growth of |k|

eigenvalues is quartic, which is crucial for measure estimates. However, we have to lose some regularity for estimating the gradient of perturbation. Second, we construct a KAM theorem for non-autonomous beam equations in order to obtain quasi-periodic solutions. The paper is organized as follows. In Section 2, the main result, notations, remarks and comments are given. We transform the equation to an infinite dimensional Hamiltonian system in Section 3. Section 4 is devoted to a Birkhoff normal form. In Section 5, we show an infinite-dimensional KAM theorem for our equation. Using this theorem, we prove our main theorem. A lemma is proved in the last section — the Appendix. 2. The main result and some notations For ε = 0, Eq. (1.1) becomes utt + uxxxx = 0.

(2.1)

d4 dx4

with hinged boundary conditions has eigenvalues {λj = j 4 } and the corresponding √ eigenfunctions ϕj (x) = π2 sin jx, j ∈ Z+ := {1, 2, . . .}. Every solution of the linear beam equation (2.1) can be written as a super-position of the basic modes ϕj , namely, for any subset J of Z, ∑ √ (2.2) u(t, x) = qj (t)ϕj (x), qj (t) = Ij cos( λj t + φ0j ) The operator A =

j∈J

with amplitudes Ij ≥ 0 and initial phases φ0j . We prove that the boundary value problem (1.1) and (1.2) admits quasi-periodic solutions which have similar form as (2.2). Theorem 2.1 is the main result of this paper. Before giving the theorem, we first make some assumptions and notations. Define D1 (σ1 ) := {ϑ| |Imϑ| < σ1 }, D2 (2a) := {x| |Imx| < 2a}, |g|σ1 ,2a :=

sup

|g(ϑ, x)|

(ϑ,x)∈D1 (σ1 )×D2 (2a)

and |g|2a := supx∈D2 (2a) |g(ϑ, x)| for every fixed ϑ ∈ D1 (σ1 ). We use “meas” to represent the Lebesgue measure. Throughout this paper, we suppose that: ∑ (H1 ) g(ϑ, x) = g0 + k∈Zm \{0} gk (x)ei⟨k,ϑ⟩ , 0 ̸= g0 ∈ R, where ⟨·, ·⟩ is the standard inner product in Cm . (H2 ) For some σ1 > σ ˜1 and a > 0, g analytically in ϑ, x extends to the domain D1 (σ1 ) × D2 (2a) and g is bounded in D1 (σ1 ) × D2 (2a) with finite norm |g|σ1 ,2a . (H3 ) ∂x2k+1 g(ϑ, 0) = 0,

∀k ∈ N.

One cannot find out quasi-periodic solutions for any index set due to complete resonance. To overcome this problem, the method of [11] is applied. It is proved that the quasi-periodic solutions exist for any admissible index set whose definition is as follows. For each index set I := {n1 < n2 < · · · < nb }, we define ∆k , k = 0, 1, 2, 3, where ∆k (k = 0, 1, 2) is the set of indices (i, j, d, l) such that there exactly 4 − k components in I, and ∆3 is the set of indices (i, j, d, l) such ⋂ that there exist at least three components not in I. Suppose that I0 = {(i, j, d, l) ≡ (i, j, i, j)}, N = I0 ∆0 ⋂ and M = I0 ∆2 . It was proved that, in [11], there exist infinitely many admissible index sets. Definition 2.1 (Definition 3.1 in [11]). The index set I is said to be admissible if and only if n1 , n2 , . . . , nb satisfy the following assumptions A–G.

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√ √ √ √ A If i ± j ± d ± l = 0, (i, j, d, l) ∈ ∆0 \ N , then λi + λj − λd − λl ̸= 0. √ √ √ √ B If i ± j ± d ± l = 0, (i, j, d, l) ∈ ∆1 , then λi + λj − λd − λl ̸= 0. √ √ √ √ C If i ± j ± d ± l = 0, (i, j, d, l) ∈ ∆2 \ M, then λi + λj − λd − λl ̸= 0. √ √ √ √ D If i ± j ± d ± l = 0, (i, j, d, l) ∈ ∆0 , then λi + λj + λd − λl ̸= 0. √ √ √ √ E If i ± j ± d ± l = 0, (i, j, d, l) ∈ ∆1 , then λi + λj + λd − λl ̸= 0. √ √ √ √ F If i ± j ± d ± l = 0, (i, j, d, l) ∈ ∆2 , then λi + λj + λd − λl ̸= 0. ∑ G For b > 1, (4b − 1) † ( 1≤i≤b n4i ), where x † y means that x is not a factor of y. Now we state the main result of this paper. Theorem 2.1. Let ϱ be a positive constant and b ≥ 2 be an integer with m + b ≥ 3. Assume that (H1 ), (H2 ) and (H3 ) hold. For each admissible index set I = {n1 < n2 < · · · < nb } ⊂ Z+ , there is a constant ε∗ such that for any 0 < ε < ε∗ , there exist a subset Ω ⊂ [ϱ, 2ϱ]m with meas Ω > 0 and a subset Σε ⊂ Σ := Ω × [0, 1]b with measΣε > 0 which satisfy that for any ξ = (ω, ςn1 , . . . , ςnb ) ∈ Σε , the boundary value problem (1.1) with (1.2) has a quasi-periodic solution of the form √ u(t, x) =

∑ j∈I

1

1 4(ςj + O(ε 4 )) sin jx cos ϖ∗j (ξ)t + O(ε 4 ), πj 2

where |ϖ∗j − j 2 | ≤ cε. From the above theorem, one can get that Eq. (1.1) has a large Cantor invariant m + b-dimensional tori carrying quasi-periodic solutions with frequency vector ω ˆ ∗ , where ω ˆ ∗ = (ω∗1 , ω∗2 , . . . , ω∗m , ϖ∗n1 , ϖ∗n2 , . . . , ϖ∗nb ), with ω∗i = ωi + O(ε), 1 ≤ i ≤ m, and ϖ∗nj = λnj + O(ε), 1 ≤ j ≤ b. Remark 2.1. (H1 ) ensures that g can be expanded to a Fourier series. (H3 ) makes us extend g into a complex domain. For the following even extension of g (see Section 3), we suppose (H3 ). We comment on the result as follows: 1. x-dependence. The nonlinearity g in (1.1) can explicitly contain the space variable x, which implies that the momentum is not conserved. The reference [2] introduced a weighted majorant norm of vector field to handle this. In this paper, we only assume g is bounded, i.e. (H2 ). The decay property plays an important role, e.g. (4.16). 2. quasi-periodical forcing. Since Eq. (1.1) is quasi-periodically forced, a quasi-periodic symplectic transform is required, which can transform the Hamiltonian into its normal form. 3. The frequency ω ˆ ∗ of the solution. Using the method of this paper we cannot expect an existence result of quasi-periodic solutions u(t, x) with the same frequency ω as g. That is because we need to add extra parameters while we use the KAM method. Extra parameters create solutions with additional frequencies. 4. The completely resonant beam equation. The completely resonant beam equation is more difficult than the non-resonant one, especially when the nonlinearity explicitly contains x. The admissible index set is applied to overcome the resonant problem.

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3. The Hamiltonian setting In this paper, C denotes a universal constant if we do not care its value. First of all, we rewrite the beam equation (1.1) as follows ∂t u = v, ∂t v + Au = −εg(ωt, x)u3 , (3.1) where A = d4 /dx4 . It is well known that Eq. (3.1) can be studied as an infinite dimensional Hamiltonian system with coordinates u and v = ∂t u. The Hamiltonian for (3.1) is then ∫ π 1 1 χ(u, x, ωt)dx, H = (v, v) + (Au, u) + ε 2 2 0 where χ(u, x, ωt) =

1 g(ωt, x)u4 , 4

and (·, ·) denotes the scalar product in L2 [0, π]. We introduce the coordinates q = (q1 , q2 , . . .) and p = (p1 , p2 , . . .) through the relations u(t, x) =

∑ qj (t) √ ϕj (x), 4 λj j≥1

v(t, x) =

∑√ 4 λj pj (t)ϕj (x). j≥1

The coordinates are taken from some Hilbert space la,s (a > 0, s > 21 ) of all real valued infinite sequences ∑ 2 la,s = la,s (R) := {q = (q1 , q2 , . . .), qi ∈ R, i ≥ 1 s.t. (∥q∥a,s )2 = |qi | i2s e2ai < ∞}. i≥1

Thus, we obtain the Hamiltonian H = Λ + εG, where 1 ∑√ λj (qj2 + p2j ) Λ= 2

and G =

j≥1

1 4

∫ 0

π

⎞4 ⎛ ∑ qj (t) √ ϕj (x)⎠ dx. g(ωt, x) ⎝ 4 λj j≥1

The corresponding equations of motion are √ √ ∂H ∂H ∂G = λj pj , p˙j = − = − λj qj − ε , ∂pj ∂qj ∂qj ∑ with respect to the symplectic structure dqi ∧ dpi on la,s × la,s . q˙j =

j≥1

(3.2)

Lemma 3.1. Let I ⊆ R be an interval and a curve I → (p(t), q(t)) be a real analytic solution of (3.2), then u(t, x) =

∑ qj (t) √ ϕj (x) 4 λj j≥1

(3.3)

is a classical solution of (1.1) that is real analytic on I × [0, π]. From the above lemma, u is a classical solution of (1.1). So we need to find a solution with form (3.3). The proof is common and we omit it. Details can be found in [6]. Introducing a pair of action–angle variables (J, ϑ) ∈ Rm × Tm (Tm := Rm /2πZm ), one can obtain an equivalent Hamiltonian that does not depend on the time variable. The autonomous formulation of our problem is reached as follows: ∫π ∂ χdx ∂H ∂H ∂G q˙j = = −ε 0 , , p˙j = − , j ≥ 1, ϑ˙ = ω, J˙ = −ε ∂pj ∂qj ∂ϑ ∂ϑ

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which is a Hamiltonian system with Hamiltonian 1 ∑√ λj (qj2 + p2j ) + εG(q, ϑ) (3.4) H = ⟨ω, J⟩ + 2 j≥1 ∑ and symplectic structure dϑ ∧ dJ + dqi ∧ dpi . To continue our investigation for the Hamiltonian (3.4), we need to establish the regularity of the nonlinear Hamiltonian vector field XG associated to G. Let lb2 and L2 , respectively, be the Hilbert spaces of all bi-infinite, square summated sequences with complex coefficients and all square-integrable complex-valued functions on [−π, π]. Let 1 ∑ F : lb2 → L2 , q ↦→ Fq = √ qj eijx 2π j be the inverse discrete Fourier transform, which defines an isometry between the two spaces. The subspaces 2 lba,s ⊂ lb2 consist, by definition, of all bi-infinite sequences with the finite form (∥q∥a,s )2 = |q0 | + ∑ 2 2s 2a|i| . Through F they define subspaces W a,s [−π, π] ⊂ L2 [−π, π] that are normed by setting i |qi | |i| e ∥Fq∥a,s = ∥q∥a,s . Lemma 3.2 is classical. See [6]. Using it, we can prove a regularity result– Lemma 3.3. For the proof, see [20]. Lemma 3.2. For a > 0 and s > 21 , the space lba,s is a Hilbert algebra with respect to convolution of the ∑ sequences (q ∗ p)j := k qj−k pk , and ∥q ∗ p∥a,s ≤ c∥q∥a,s ∥p∥a,s , where the constant c depends only on s. Consequently, W a,s is a Hilbert algebra with respect to multiplication of functions. a,s Lemma 3.3. The gradient ∂G into la,s , ∂q is real analytic as a map from some neighborhood of origin in l ∂G 3 with ∥ ∂q ∥a,s = O((∥q∥a,s ) ).

4. Partial Birkhoff normal form Since χ(u, x, ϑ) = 14 g(ϑ, x)u4 and u =

√qj j 4 λ ϕj , j

∑

we attain that

∫ π 1 ∑ 1 √ G(q, ϑ) = g(ϑ, x)ϕi ϕj ϕd ϕl dxqi qj qd ql . 4 4 λi λj λd λl 0 i,j,d,l From (H1 ), G(q, ϑ) =

∑ 1 ∑ Gijdl 1 Gk,ijdl √ √ qi qj qd ql + ei⟨k,ϑ⟩ qi qj qd ql , 4 4 4 4 λ λ λ λ λ λ λ λ i j d l i j d l i,j,d,l |k|≥1,i,j,d,l

where

∫ Gijdl = g0

(4.1)

π

ϕi ϕj ϕd ϕl dx

(4.2)

0

and

∫ Gk,ijdl =

π

gk (x)ϕi ϕj ϕd ϕl dx,

|k| ≥ 1.

(4.3)

0

An easy computation shows that Gijdl = 0 unless i ± j ± d ± l = 0 for at least one combination of plus and minus signs. In particular, we have g0 Gijij = (2 + δij ) (4.4) 2π { 1, i = j where δij = 0, i ̸= j.

Y. WANG / Nonlinear Analysis 189 (2019) 111585

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We switch to the complex coordinates zi =

qi + ipi √ , 2

z¯i =

qi − ipi √ 2

that live in the now complex Hilbert space la,s = la,s (C) := {z = (z1 , z2 , . . .), zi ∈ C, i ≥ 1 s.t. (∥z∥a,s )2 =

∑

2

|zi | i2s e2ai < ∞}.

i≥1

Then, a real analytic Hamiltonian H = Λ + εG, with symplectic structure dϑ ∧ dJ + i

(4.5)

∑

dzi ∧ d¯ zi is obtained, where ⎛ ⎞4 ∫ ∑ zj + z¯j ∑√ 1 π √ λj zj z¯j , G = g(ϑ, x) ⎝ ϕj (x)⎠ dx. Λ = ⟨ω, J⟩ + 4 4 4λ 0 j j i

j≥1

Moreover, from (4.1), (4.2) and (4.3), we have G=

+

1 16

1 16

∑ i,j,d,l,i±j±d±l=0

∑ |k|≥1,i,j,d,l

Gijdl (zi + z¯i )(zj + z¯j )(zd + z¯d )(zl + z¯l ) ijdl

Gk,ijdl i⟨k,ϑ⟩ e (zi + z¯i )(zj + z¯j )(zd + z¯d )(zl + z¯l ). ijdl

Next we transform the Hamiltonian (4.5) into some partial Birkhoff form of order four so that it may serve as a small perturbation of some nonlinear integrable system in a sufficiently small neighborhood of the origin. The following Lemmas are necessary. We will apply them to proving Proposition 4.1. The proof of Lemma 4.2 is shown in the Appendix. The proofs of Lemmas 4.1 and 4.3 can be found in [20]. Define ¯ := (∆ \ N ) ⋃ ∆ ⋃(∆ \ M). ¯ := ∆0 ⋃ ∆1 ⋃ ∆2 and ∆ ∆ 0 1 2 Lemma 4.1. Fix ϱ > 0. There exists ε such that for any 0 < ε < ε, the following holds true: there is a subset Ω ε ≡ Ω ⊂ [ϱ, 2ϱ]m such that every ω ∈ Ω satisfies that 1

|⟨k, ω⟩| ≥

ϱε 3

m+2 ,

|k|

f or all 0 ̸= k ∈ Zm

(4.6)

1

and measΩ ≥ (1 − C1 ε 3 )ϱm , where the constant C1 depends on m. ¯ and k ̸= 0. Then, there exists ε such that for any Lemma 4.2. Fix ϱ > 0. Assume that (i, j, d, l) ∈ ∆ m 0 < ε < ε, the following holds true: there is a subset Ω ε ≡ Ω ⊂ [ϱ, 2ϱ] satisfying that, for any ω ∈ Ω and √ √ √ √ ± λi ± λj ± λd ± λl ̸= 0, 1

⏐ √ ⏐ √ √ √ ⏐ ⏐ ⏐± λi ± λj ± λd ± λl + ⟨k, ω⟩⏐ ≥

ϱε 3 |k|

m+2 .

(4.7)

( ) 1 Moreover, measΩ ≥ ϱm 1 − C2 ε 3 , where C2 is a constant depending on m, ϱ, n1 and nb . ∑ 2 2m+1 Lemma 4.3. For xk ∈ C and k ∈ Zm , if the series |xk | converges, then the inequality k [k] ∑ ∑ 2 2 | k xk | ≤ c k [k]2m+1 |xk | holds, where [k] = max{|k|, 1}, |k| = |k1 | + |k2 | + · · · + |km | and c is a constant depending on m.

Y. WANG / Nonlinear Analysis 189 (2019) 111585

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Consider the Hamiltonian (4.5). For each admissible index set I, Proposition 4.1 holds. Proposition 4.1. When ε is small enough, there exists a subset Ω ⊂ [ϱ, 2ϱ]m with measΩ > 0 and for every ω ∈ Ω , there is a real analytic, symplectic change of coordinates Ψ which transforms the Hamiltonian (4.5) into its Birkhoff normal form, i.e. ¯ + εG ˆ + ε2 K, H ◦ Ψ = Λ + εG where

¯ z¯) = 1 G(z, 2

∑

¯ ij |zi |2 |zj |2 , G

(4.8)

i∈I or j∈I

⎧ ⎪ ⎪ ⎨

3g0 , if i ̸= j 2πi2 j 2 ¯ ij = G 9g0 ⎪ ⎪ , if i = j, ⎩ 8πi2 j 2 ∑ gijdl (zi + z¯i )(zj + z¯j )(zd + z¯d )(zl + z¯l )

ˆ= G

(i,j,d,l)∈∆3 , i±j±d±l=0

+

∑ |k|≥1

∑

ei⟨k,ϑ⟩

gk,ijdl (zi + z¯i )(zj + z¯j )(zd + z¯d )(zl + z¯l ),

(4.9)

(i,j,d,l)∈∆3

1

and ε 3 |K| = O((∥z∥a,s )6 ). Moreover, the transformation Ψ is defined in a complex neighborhood D1 ( σ21 ) := {ϑ||Imϑ| < σ21 } of the torus Tm and a neighborhood of the origin in la,s . Proof . For convenience, we introduce another coordinates (. . . , w−2 , w−1 , w0 w1 , w2 , . . .) in lba,s by letting zj = wj , z¯j = w−j (j ≥ 1) and w0 = 0. So we have ∑√ ∑ H = ⟨ω, J⟩ + λj wj w−j + ε

i,j,d,l, |i|±|j|±|d|±|l|=0

j≥1

+ε

∑ ∑ |k|≥1

i,j,d,l

gijdl wi wj wd wl

gk,ijdl ei⟨k,ϑ⟩ wi wj wd wl ,

where i, j, d, l run through all nonzero integers and gijdl :=

Gijdl , 16|ijdl|

gk,ijdl :=

Gk,ijdl . 16|ijdl|

We have gijdl = 0 unless |i| ± |j| ± |d| ± |l| = 0. Step 1. We construct the symplectic transformation. Consider a Hamiltonian function ∑ ∑ ∑ F = εF = ε Fijdl wi wj wd wl + ε Fk,ijdl ei⟨k,ϑ⟩ wi wj wd wl i,j,d,l

|k|≥1 i,j,d,l

with coefficients iFijdl

gijdl ¯ if |i| ± |j| ± |d| ± |l| = 0 and |i|, |j|, |d|, |l| ∈ ∆ ′ + λ′ + λ′ + λ′ , λ = i j d l ⎩ 0, otherwise, ⎧ ⎨

(4.10)

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and for k ̸= 0,

iFk,ijdl

⎧ g k,ijdl ⎪ , ⎪ ⎪ ⟨k, ω⟩ ⎪ ⎪ ⎪ ⎪ ⎨

|k| ≥ 1,

if

¯ |i|, |j|, |d|, |l| ∈ ∆,

λ′i + λ′j + λ′d + λ′l = 0, ¯ if |k| ≥ 1, |i|, |j|, |d|, |l| ∈ ∆,

gk,ijdl = ′ + λ′ + λ′ + λ′ + ⟨k, ω⟩ , ⎪ λ ⎪ j i d l ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0,

and and

λ′i + λ′j + λ′d + λ′l ̸= 0, otherwise,

2

where λ′i = sgni · |i| . Let Ψ = XF1 be the time-1 map of the vector field of the Hamiltonian F. Expanding at t = 0 and using Taylor’s formula we obtain 1

∫

(1 − t){{H, F}, F} ◦ XFt dt

H ◦ Ψ = H + {H, F} + 0

= Λ + εG + ε{Λ, F } ∫ 1 2 2 + ε {G, F } + ε (1 − t){{H, F }, F } ◦ XFt dt.

(4.11)

0

Now we compute G + {Λ, F }. We let [ ] Iijdl := gijdl − i(λ′i + λ′j + λ′d + λ′l )Fijdl wi wj wd wl and [ ] Ik,ijdl := gk,ijdl − i(λ′i + λ′j + λ′d + λ′l + ⟨k, ω⟩)Fk,ijdl ei⟨k,ϑ⟩ wi wj wd wl . In the second line of (4.11), we compute and obtain that ¯ + G, ˆ G + {Λ, F } = G

(4.12)

where ¯ := G

∑ (|i|,|j|,|d|,|l|)∈N or (|i|,|j|,|d|,|l|)∈M and |i|±|j|±|d|±|l|=0

gijdl wi wj wd wl

and ˆ := G

∑ (|i|,|j|,|d|,|l|)∈∆3

Iijdl +

∑

ei⟨k,ϑ⟩

∑

|k|≥1

(|i|,|j|,|d|,|l|)∈∆3

Ik,ijdl .

So, (4.8) and (4.9) hold true as well as ¯ + εG ˆ + ε2 {G, F } + ε2 H ◦ Ψ = Λ + εG

∫

1

(1 − t){{H, F }, F } ◦ XFt .

0

Step 2. We prove that Ψ is real analytic. CLAIM. The vectorfield of the Hamiltonian XF is real analytic in a complex neighborhood ϑ ∈ D1 ( σ21 ) of Tm and some neighborhood of the origin in lba,s . Furthermore, it satisfies    ∂F  C 3   (4.13)  ∂w  ≤ 13 (∥w∥a,s ) . ε a,s The vector-field of F is XF = (0, −

∂F ∂F −1 ∂F +∞ , {−i } , {i } ). ∂ϑ ∂wj −∞ ∂wj 1

Y. WANG / Nonlinear Analysis 189 (2019) 111585

10

a,s σ1 Firstly, we discuss − ∂F ∂ϑ . For ϑ ∈ D1 ( 2 ) and w ∈ lb , ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∑ ⏐ ⏐ ∑ i⟨k,ϑ⟩ ⏐ ∂F ⏐ ⏐ ⏐ ⏐ e ik Fk,ijdl wi wj wd wl ⏐⏐ =⏐ ⏐ ∂ϑ ⏐ σ1 ⏐ ⏐|k|≥1 D1 ( 2 ) ¯ (|i|,|j|,|d|,|l|)∈∆

σ D1 ( 21 )

≤

∑

e

|k|σ1 2

∑

|k|

|Fk,ijdl ||wi wj wd wl |.

(4.14)

¯ (|i|,|j|,|d|,|l|)∈∆

|k|≥1

However, with the even extension of g on x ∈ [−π, π], we obtain the Fourier Cosine expansion ∑ gk (x) = gkτ cos τ x, τ ≥0

which together with (4.3) and Lemma 4.2 yield the estimate of Fk,ijdl : |Fk,ijdl | ≤ C

m+2 ∑

|k|

ε

1 3

τ,τ +|i|±|j|±|d|±|l|=0

|gkτ |,

(4.15)

where C is dependent on ϱ and I. Note that 0 ≤ τ ∈ Z and i, j, d, l are non-zero integers. It follows from (4.14) and (4.15) that ⎛ ⎞ ⏐ ⏐ ∑ ∑ |k|σ1 m+3 ⏐ ∂F ⏐ C ⏐ ⏐ ⎝ |gkτ ||wi ||wj ||wd ||wl |⎠ . e 2 |k| ≤ 1 ⏐ ∂ϑ ⏐ σ ε 3 |k|≥1 D1 ( 21 ) ¯ τ +|i|±|j|±|d|±|l|=0,(|i|,|j|,|d|,|l|)∈∆ However, from Lemma A.1 in [16], we have |gkτ | < |g|σ1 ,2a e−|k|σ1 e−2aτ ,

k ̸= 0,

τ ≥ 0.

(4.16)

So ⏐ ⏐ ⏐ ∂F ⏐ C ∑ − |k|σ1 m+3 ⏐ ⏐ e 2 |k| ≤ 1 ⏐ ∂ϑ ⏐ σ ε 3 |k|≥1 D1 ( 21 ) ⎛ ⎞ ∑ ·⎝ e−2aτ |wi ||wj ||wd ||wl |⎠ .

(4.17)

¯ τ +|i|±|j|±|d|±|l|=0,(|i|,|j|,|d|,|l|)∈∆

Suppose that g˜ = (. . . , g˜−2 , g˜−1 , g˜0 , g˜1 , g˜2 , . . .)

(4.18)

and the component g˜τ = e−2aτ for τ ≥ 0, and g˜τ = 0 for τ ≤ −1, then ∑ ∑ ∑ 2 ∥˜ g ∥2a,s = |˜ gτ | [τ ]2s e2a|τ | = e−4aτ [τ ]2s e2aτ = e−2aτ [τ ]2s . τ ≥0

τ ∈Z

Since

∑

τ ≥0

e−2aτ [τ ]2s is convergent, g˜ ∈

lba,s .

τ ≥0

Assuming

w ˜j = |wj | + |w−j |,

w0 = 0

it follows from (4.17) that ⏐ ⏐ ⏐ ∂F ⏐ C ∑ − |k|σ1 m+3 ⏐ ⏐ ≤ 1 e 2 |k| (˜ g∗w ˜∗w ˜∗w ˜ ∗ w) ˜ 0. ⏐ ∂ϑ ⏐ σ ε 3 |k|≥1 D1 ( 21 )

(4.19)

(4.20)

Since ♯{k ∈ Zm : |k| = l} ≤ 2m lm−1 ,

l ∈ Z+ ,

(4.21)

Y. WANG / Nonlinear Analysis 189 (2019) 111585

11

we obtain, from (4.20) and Lemma 3.2, ⏐ ⏐ ⏐ ∂F ⏐ C ∑ 2m+2 − lσ1 ⏐ ⏐ l e 2 ≤ ∥˜ g∗w ˜∗w ˜∗w ˜ ∗ w∥ ˜ a,s 1 ⏐ ∂ϑ ⏐ σ ε3 D1 ( 1 ) l≥1

2

C

≤

ε

1 3

∥˜ g∗w ˜∗w ˜∗w ˜ ∗ w∥ ˜ a,s ≤

C ε

1 3

∥w∥ ˜ 4a,s ≤

C 1

ε3

∥w∥4a,s .

   Secondly, we discuss  ∂F ∂w a,s . Since   ∑ ⏐⏐ ∂F ⏐⏐2  ∂F  2 ⏐  ) = ⏐ 2a|l| |l|2s , (  ∂w  ⏐ ∂wl ⏐ e a,s

(4.22)

|l|≥1

⏐ ⏐ ⏐ ∂F ⏐2 we need to estimate ⏐ ∂w ⏐ . It is evident that l ⏐ ⏐ ⏐ ∂F ⏐2 2 2 ⏐ ⏐ ⏐ ∂wl ⏐ ≤ C(F0l ) + C(F1l ) , where

⏐∑ ⏐ F0l = ⏐⏐ |i|±|j|±|d|±|l|=0,

and

(4.23)

⏐ ⏐ Fijdl wi wj wd ⏐⏐ ¯ (|i|,|j|,|d|,|l|)∈∆

⏐ ⏐ ⏐∑ ⏐ ∑ ⏐ ⏐ ⏐ F1l = ⏐⏐ ei⟨k,ϑ⟩ F w w w k,ijdl i j d ⏐ . ¯ (|i|,|j|,|d|,|l|)∈∆ ⏐|k|≥1 ⏐

For Fijdl , according to the definition of admissible set, the divisor δ := λ′i + λ′j + λ′d + λ′l ̸= 0. ¯ Therefore, So, |δ| ≥ 1 holds for all (|i|, |j|, |d|, |l|) ∈ ∆. (∑ 2 F0l ≤ C ¯

(|i|,|j|,|d|,|l|)∈∆, ±|i|±|j|±|d|=l

)2 |wi wj wd | .

It follows from (4.19) that 2

2 F0l ≤ C ((w ˜∗w ˜ ∗ w) ˜ l) .

So, using Lemma 3.2, we have ∑

∑

2s

2 F0l |l| e2a|l| ≤ C

|l|≥1

2

2s

|(w ˜∗w ˜ ∗ w) ˜ l | |l| e2a|l|

|l|≥1

( )2 ≤ C (∥w ˜∗w ˜ ∗ w∥ ˜ a,s ) ≤ C ∥w∥3a,s , 2

(4.24)

where C depends on g and I. Supposing that Ω = Ω ε ∩ Ω ε , according to Lemmas 4.1 and 4.2, 1

measΩ ≥ ϱm (1 − Cε 3 ) holds and measΩ > 0 when ε is small enough. In the following, we assume ω ∈ Ω . Thus (4.6) and (4.7) are true. Using Lemma 4.3 and (4.15), ⎛ ⎞2 ∑ ∑ 2 2 |Fk,ijdl ||wi wj wd |⎠ F1l ≤C [k]2m+1 |ei⟨k,ϑ⟩ | ⎝ ¯ (|i|,|j|,|d|,|l|)∈∆

|k|≥1

⎞2

⎛ ≤

C ∑ 2

ε3

|k|≥1

[k]4m+5 |e

i⟨k,ϑ⟩ 2

| ⎝

∑ ¯ τ +|i|±|j|±|d|±|l|=0,(|i|,|j|,|d|,|l|)∈∆

|gkτ ||wi ||wj ||wd |⎠ .

Y. WANG / Nonlinear Analysis 189 (2019) 111585

12

From (4.16), we have 2 F1l ≤

C ∑ ε

2 3

2

[k]4m+5 |ei⟨k,ϑ⟩ | e−2|k|σ1

|k|≥1

⎞2

⎛ ∑

·⎝

e−2aτ |wi ||wj ||wd |⎠ .

¯ τ +|i|±|j|±|d|±|l|=0,(|i|,|j|,|d|,|l|)∈∆

It is derived from (4.18) and (4.19) that 2 F1l ≤

C ∑ ε

2 3

2

[k]4m+5 |ei⟨k,ϑ⟩ | e−2|k|σ1 ((˜ g∗w ˜∗w ˜ ∗ w) ˜ l )2 .

|k|≥1

For all (ϑ, x) ∈ D1 (σ1 /2) × D2 (2a), 2 F1l ≤

≤

C ∑ ε

2 3

C ε

2 3

2

[k]4m+5 eσ1 |k| e−2|k|σ1 ((˜ g∗w ˜∗w ˜ ∗ w) ˜ l)

|k|≥1

((˜ g∗w ˜∗w ˜ ∗ w) ˜ l )2

∑

[k]4m+5 e−|k|σ1 .

(4.25)

|k|≥1

Since (4.21) holds, we obtain, from (4.25), 2 F1l ≤

≤

C 2 3

ε C

2

ε3

((˜ g∗w ˜∗w ˜ ∗ w) ˜ l )2

∑

j 5m+4 e−j σ˜1

j≥1

((˜ g∗w ˜∗w ˜ ∗ w) ˜ l )2 .

Therefore, ∑

2s

2 F1l |l| e2a|l| ≤

|l|≥1

≤

C ∑ 2

ε3 C ε

2 3

2

2s

|(˜ g∗w ˜∗w ˜ ∗ w) ˜ l | |l| e2a|l|

|l|≥1 2

(∥˜ g∗w ˜∗w ˜ ∗ w∥ ˜ a,s ) ≤

C ( ε

2 3

∥w∥3a,s

)2

,

which together with (4.22)–(4.24) yield that   ∑  ∂F  2 )2 C ( 2s 2 2  ( (F0l + F1l )|l| e2a|l| ≤ 2 ∥w∥3a,s ,  ∂w  ) ≤ C ε3 a,s |l|≥1 for 0 < ε < 1, where the constant C depends on ϱ, g, σ ˜1 , a, m, I and s. The analyticity of XF follows from the analyticity of each component function and its local boundedness. The Claim is proved. ˆ = O(∥w∥4a,s ). Using Lemma 3.3, (4.13), |K| ≤ C1 (∥w∥a,s )6 Step 3. Similarly, we can attain that |G| holds for ∥w∥a,s ≤ 1. The detailed proof can be found in [6]. This completes the proof. □ We introduce the symplectic polar and complex coordinates by setting ⎧ √ ⎨ znj = ςj + Ij e−iθj , 1 ≤ j ≤ b ⎩

(4.26)

j ∈ Z1

zj = wj ,

where ςj ∈ [0, 1] and Z1 := Z+ \I. Then, the Hamiltonian is changed to ∑ ∑ ∑ ˆ l wl w H= ωi Ji + ϖj Ij + Ω ¯l + P, 1≤i≤m

ε3

1≤j≤b

l∈Z1

(4.27)

Y. WANG / Nonlinear Analysis 189 (2019) 111585

with symplectic structure

∑

˘=1 G 2

1≤i≤m

∑

dϑi ∧dJi +

∑

1≤j≤b

dθj ∧dIj +i

∑

l∈Z1

∑

¯ n n (Ii + ςi )(Ij + ςj ) + G i j

13

˘ G+ε ˆ 2 K, dwl ∧dw ¯l , where P = εG+ε

¯ ln (Ij + ςj )|wl |2 , G j

l∈Z1 , 1≤l≤b

1≤i,j≤b

√ √ √ √ ˜ Ω ˆ = ι + εBς, α = ( λn , λn , . . . , λn ), ι = ( λl )l∈Z , A˜ = (G ¯ n n )1≤i,j≤b and ϖ = α + εAς, 1 1 2 i j b ¯ ¯ ln )l∈Z , 1≤j≤b . For any ω− ∈ Ω fixed and ω ∈ Ω := {ω ∈ Ω | |ω − ω− | ≤ ε 32 }, we introduce B = (G 1 j new parameter ω ¯ as following 3 ¯, ω ¯ ∈ [0, 1]m . ω = ω− + ε 2 ω ¯ × [0, 1]b ⊂ Ω × [0, 1]b . Clearly, Ω Scaling the variables as follows

⎧ 1 ˜¯ , ⎪ ω ¯ = ε2 ω ⎪ ⎪ 1 ⎪ ⎪ ⎪ ς = ε 2 ς˜, ⎪ ⎪ 1 ⎨ w = ε 4 w, ˜ 1 ¯˜ ⎪ w 4 w, ¯ = ε ⎪ ⎪ ⎪ 1 ⎪ ˜ 2 ⎪ I = ε I, ⎪ ⎪ 1 ⎩ ˜ J = ε 2 J,

one can obtain a Hamiltonian system given by the rescaled Hamiltonian 1

1 ˜ = ε− 32 H(ϑ, ε 12 J, ˜ θ, ε 12 I, ˜ ε 14 w, ¯˜ ε 41 ω ˜¯ , ε 4 ς˜) H ˜ ε 4 w, ∑ ∑ ∑ ˜ ˆj w ¯˜j + P˜ , = ω ˜ i J˜i + ϖ ˜ l I˜l + Ω ˜j w

1≤i≤m

1≤l≤b

j∈Z1

where ˜ ˜¯ i , 1 ≤ i ≤ m, ω ˜ i (ω ¯ ) = ε−1 ω−i + εω ˜ˆ ˜ and Ω ϖ(ς) ˜ = ε−1 α + εAς, (ς) = ε−1 ι + εBς. ˜ ˆ = εΩ ˆ. Clearly, ω = ε˜ ω , ϖ = εϖ, ˜ Ω Next, we introduce some notations. Assume that ˜ θ, I, ˜ w, ˜ < r2 , ∥w∥ ¯ ¯˜ a,s < r} D(σ, r) := {(ϑ, J, ˜ w) ˜ : |Imϑ| < σ, |Imθ| < σ, |I| ˜ a,s < r, ∥w∥ ¯˜ = 0}, where | · | denotes the is a complex neighborhood of Tm × {J˜ = 0} × Tb × {I˜ = 0} × {w ˜ = 0} × {w sup-norm for complex vectors. For a p(p ≥ 1) order Whitney smooth function F (ξ), define { ⏐ p ⏐} ⏐∂ F ⏐ ∗ ∥F ∥Π = max sup |F | , . . . , sup ⏐⏐ p ⏐⏐ ξ∈Π ξ∈Π ∂ξ and

{

∥F ∥∗Π

⏐ ⏐ p ⏐} ⏐ ⏐ ∂F ⏐ ⏐ ⏐ ⏐ , . . . , sup ⏐ ∂ F ⏐ . = max sup ⏐⏐ ⏐ ⏐ p ⏐ ξ∈Π ∂ξ ξ∈Π ∂ξ ∗

If F (ξ) is a vector function from ξ to la,s (Rn ), which is p order Whitney smooth on ξ, define ∥F ∥a,s,Π = ∗ ∗ ∗ ∥(∥Fi (ξ)∥Π )i ∥a,s (∥F ∥Rn ,Π = maxi (∥Fi (ξ)∥Π )). If F (η, ξ) is a vector function from D(σ, r) × Π to la,s , ∗ ∗ define ∥F ∥a,s,D×Π = supη∈D(σ,r) ∥F ∥a,s,Π . We usually omit D for brevity. For functions F , a corresponding Hamiltonian vector field is defined as XF = (FJ˜, −Fϑ , FI˜, −Fθ , iFw¯˜ , −iFw˜ )T . Denote the weighted norm for XF by letting ∗

|XF |r,D(σ,r)×Π =

1 1 1 1 ∥Fϑ ∥∗Π + 2 ∥Fθ ∥∗Π + ∥FI˜∥∗Π + ∥Fw˜ ∥∗a,s,Π + ∥Fw¯˜ ∥∗a,s,Π . 2 r r r r

Y. WANG / Nonlinear Analysis 189 (2019) 111585

14

In the following, we suppose p = 1. Fix σ0 = σ21 and r = r0 , where 0 < r0 < 1 and r0 is fixed. On D(σ0 , r0 ) ˜ and for ξ˜ = (ω ¯ , ς˜) ∈ [β, 3β]m+b , from Proposition 4.1 we can get that ˜ ˜˘ ˆ + |εG| ˜ ≤ Cε 21 . |P˜ | ≤ |εG| + |ε2 K| Using Cauchy estimates, we have that ∗

|XP˜ |r,D(σ,r)×Π ≤ on D(σ, r) with σ = min{ σ20 , 12 }, r = we assume

r0 2 ,

C3 1 −1 ε2 β σ

(4.28)

Π = [β, 2β]m+b , and β will be denoted later. Since (¯ ω , ς) ∈ [0, 1]m+b , 1

3β < ε− 2 .

(4.29) ˜ ˜ by H, J˜ by J, I˜ by I, w ˆ by Ω ˆ, ω ¯˜ by w, ˜¯ by ω ¯ ω ˜ by ω, ϖ ˜ by ϖ, Ω ¯, For simplicity, we still denote H ˜ by w, w ˜ ς˜ by ς, and P by P . 1

Remark 4.1. Actually we eventually take β = ε− 3 as ε is small enough. The reason can be found in Section 5.1. 5. An infinite-dimensional KAM theorem Consider small perturbations of an infinite dimension Hamiltonian in the parameter dependent normal form ∑ ∑ ∑ ˆ j (ς)wj w N= ωi (¯ ω )Ji + ϖl (ς)Il + Ω ¯j 1≤i≤m

1≤l≤b

j∈Z1

on a phase space P a,s = Tm × Rm × Tb × Rb × la,s × la,s ∋ (ϑ, J, θ, I, w, w), ¯ where

2 l2 + · · · ω−i ˆ j = j + · · · + O(ς), + O(¯ ω ), ϖl = + O(ς), Ω ε ε ε a > 0, s > 1/2; the dots stand for finite lower order terms of l and j, respectively; ω− ∈ Ω is a constant vector; and O(¯ ω ) and O(ς) mean 1st order terms in ω ¯1, . . . , ω ¯ m and ς1 , . . . , ςb , respectively. Denote Π1 = [0, 1]m , ˆ j ∥∗,Π ≤ M2 , and Π2 = [0, 1]b , and Π := Π1 × Π2 . Suppose that ∥ω∥∗,Π1 ≤ M11 , ∥ϖ∥∗,Π2 ≤ M12 , ∥Ω 2 max{M11 , M12 } +M2 ≥ 1. Define M = max{M11 , M12 } + M2 . For the Hamiltonian H = N + P , there exists m + b-dimensional, linearly stable torus T0m+b = Tm × {0} × Tb × {0, 0, 0} with frequencies ω ˆ = (ω(¯ ω ), ϖ(ς)) when P = 0, where { ωi (¯ ω ), 1 ≤ i ≤ m ω ˆi = ϖi−m (ς), m + 1 ≤ i ≤ m + b.

ωi =

Our aim is to prove the persistence of a large portion of this family of linearly stable rotational tori under small perturbations. Suppose that the perturbation P is real analytic in the space variables, C 1 in (¯ ω , ς), and for each ξ = (¯ ω , ς) ∈ Π its Hamiltonian vector field XP = (PJ , −Pϑ , PI , −Pθ , iPw¯ , −iPw )T defines near T0m+b a real analytic map XP : P a,s → P a,s . Under the above assumptions, we have the following theorem. Theorem 5.1.

Suppose that H = N + P satisfies ∗

ϵ = |XP |r,D(σ,r)×Π ≤ γσ 2(1+µ) ,

(5.1)

where γ depends on m, b, τ and M , µ = 2τ +1+(m+b)/2. Then there exists a Cantor set Πϵ ⊂ Π , a Whitney smooth family of torus embedding Φ : Tm+b × Πϵ → P a,s and a Whitney smooth map ω ˆ ∗ = (ω∗ , ϖ∗ ) : Πϵ →

Y. WANG / Nonlinear Analysis 189 (2019) 111585

15

Rm+b , such that for each ξ ∈ Πϵ , the map Φ restricted to Tm+b × {ξ} is a real analytic embedding of a rotational torus with frequencies ω ˆ ∗ (ξ) for the Hamiltonian H at ξ. Each embedding is real analytic on |Imϑ| < σ2 and |Imθ| < σ2 , and 1

∗

∥ˆ ω∗ − ω ˆ ∥∗ ≤ cϵ

|Φ − Φ0 |r ≤ cϵ 2 ,

(5.2)

uniformly on that domain and Πϵ , where Φ0 is the trivial embedding Tm+b × Π → T0m+b . Remark 5.1. The parameter τ is denoted in Section 5.1. ′

Remark 5.2. The regularity of the vectorfield is XP : P a,s → P a,s with s′ = s. In this theorem, one cannot expect s′ > s, since the original equation explicitly contains x. Assume εβ < 1. Since

∂ω ∂ω ¯

=

(5.3)

εIdm , det( ∂ω ∂ω ¯) ⎛ ⎜ ⎜ ⎜ 3g 0 ⎜ ˜ ⎜ A= 2π ⎜ ⎜ ⎜ ⎝

= εm , where Idm is the m × m unit ⎞⎛ 1 3 0 · · · 0 ⎟⎜ 4 1 ··· n21 ⎟⎜ 1 3 0 ··· 0 ⎟ 1 ··· ⎟⎜ 2 n2 ⎟⎜ ⎜ . .4 ⎟ .. .. .. . . .. .. .. ⎟⎜ . . . . . ⎟⎜ ⎝ ⎠ 1 1 1 ··· 0 0 ··· n2b

It follows that det(

matrix. We can take M11 = 1. Clearly, 1 1 .. . 3 4

⎞⎛

1 n2 1

⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

0

···

0

···

0

.. .

0 1 n22 .. .

0

0

···

..

.

⎞

⎟ ⎟ ⎟ ⎟ ⎟. .. ⎟ . ⎟ 1 ⎠ n2b

3g0 1 1 ∂ϖ ) = ε det A˜ = ε (1 − 4b)(− )b . ∂ς 2π n41 n42 · · · n4b 4

ˆ

ˆ

∂Ω ¯ jn , G ¯ jn , . . . , G ¯ jn ). So ∥ ∂ Ωj ∥∗ ≤ 3|g0 | . We can take M12 = 3|g0 | (4b − 1)( 1 )b and However, ∂ςj = ε(G 1 2 b ∂ς 2π 2π 4 0| M2 = 3|g . So we take M = max{M , M + M }. Note that M does not depend on ε. 11 12 2 2π Using (4.28) and assuming C3 β −1 < 1, (5.4)

we have that

1

ε2 , =ϵ≤ σ as ε is small enough. According to the assumption, we need ∗ |XP |r,D(r,σ)

(5.5)

1

ε2 ≤ γσ 2(1+µ) . σ So, we can take ε ≤ (γσ 3+2µ )2 . According to the reference [11], one can take γ = 1 . c(M +1)26+4µ

γ06

2µ+3 2

γ06

γ0 ≤ Therefore, ε ≤ ( 20(1+µ)(cM )6 σ ) . 20 Suppose x = (ϑ, θ) and y = (J, I), where { { ϑi , 1 ≤ i ≤ m Ji , 1 ≤ i ≤ m xi = and yi = θi−m , m + 1 ≤ i ≤ m + b Ii−m , m + 1 ≤ i ≤ m + b. We take ξ ∈ Π . Then Hamiltonian can be written as ∑ ∑ ˆ j (ξ)wj w H =N +P = ω ˆ i (ξ)yi + Ω ¯j + P (x, y, w, w, ¯ ξ). 1≤i≤m+b

j∈Z1

6

202(1+µ)(cM )

, where

Y. WANG / Nonlinear Analysis 189 (2019) 111585

16

Using exactly the same KAM method with [11], we can prove this theorem. The proof is standard and the detailed steps can be found in [11,13]. In every KAM iteration step, some parameter sets are thrown ˜ ϵ is positive. In order to settle this problem, (see [11,13]). So, it is necessary to identify that the measure of Π one can estimate the measure of the thrown parameter sets, which are thrown in all the steps. We compute ˜ , is O(ε− 11 12 ). When ε is small enough, and attain that the total measure of these parameters sets, namely Π ˜ < ε−1 . The details can be found in [13]. For clarity, we show the measure estimate in we can have measΠ the first KAM iteration step and put the proof in Section 5.1. κ+b 1 We take β = ε− 3 in terms of (4.29), (5.3), (5.4), and (5.6). So measΠ = ε− 3 . If m + b ≥ 3, the ˜ ϵ > 0, where Σ ˜ ϵ := Π \Π ˜ . Therefore, there exists a Cantor set Σ ˜ ϵ ⊂ Π with positive measure, such measΣ κ+b ˜ that for each ξ ∈ Σϵ , the map Φ restricted to T × {ξ} is a real analytic embedding of a rotational torus ¯ ×[0, 1]b , namely Π , there ˜ ˜ ϵ to the subset in Ω with frequencies ω ˆ ∗ (ξ) for the Hamiltonian H at ξ. Returning Π ε ˜ˆ ∗ to the frequencies in exists a Cantor set Πε ⊂ Ω × [0, 1]b satisfying the result in Theorem 2.1. Returning ω the system with Hamiltonian (4.27), namely ω ˆ ∗ , we can get the frequency. Remark 5.3. By using the method of this paper, one cannot expect quasi-periodic solutions for m + b < 3. m+b ˜ < ε−1 . So measΠ ˜ ϵ may This is mainly because when m + b reduces, measΠ = ε− 3 reduces. But measΠ be not positive. 5.1. Measure estimates in the first step The thrown parameter sets in the first step are ( )⋃( ) ∪k̸=0 A1k1 ∪k (A1k2 ∪ A1k3 ∪ A1k4 ) , where

} 1 4 ϵ , A1k1 = ξ ∈ Π : |⟨k, ω ˆ ⟩| < Ak } { 1 ⋃ 1,1 ⋃ 4 ϵ ˆi| < A1k2 = Bki = , ξ ∈ Π : |⟨k, ω ˆ⟩ + Ω Ak i∈Z1 i∈Z1 { } 1 ⋃ ⋃ 4 (|i − j| + 1) ϵ 1,11 1 ˆi + Ω ˆj | < Ak3 = Bkij = ξ ∈ Π : |⟨k, ω ˆ⟩ + Ω , Ak i,j∈Z1 i,j∈Z1 { } 1 ⋃ ⋃ ϵ 4 (|i − j| + 1) 1,12 1 ˆ ˆ Ak4 = Bkij = ξ ∈ Π : |⟨k, ω ˆ ⟩ + Ωi − Ωj | < , Ak {

i,j∈Z1 ,i̸=j

τ

i,j∈Z1 ,i̸=j

meas(A102 ∪A103 ∪A104 )

where Ak = 1+|k| . Clearly, = 0. We show the measure estimate of the most complex 1 condition Ak4 . The others can be estimated in a similar way. Lemma 5.1.

1

11

For τ > m + b + 5, meas(∪k̸=0 A1k4 ) = O(ϵ 4 ε−1 ) = O(ε− 12 ).

Proof . Without loss of generality, we assume i ≥ j. When i ≥ c|k|, it is clearly that, for ς ∈ Π2 , ⏐ ⏐ b b b b ⏐ ∑ ⏐ ∑ ∑ ∑ ⏐ ¯ in ςk − ε ¯ jn ςk ⏐⏐ ≤ ε ¯ in ςk | + ε ¯ jn ςk | G G | G |G ⏐ε k k k k ⏐ ⏐ k=1

k=1

≤ε

k=1 b ∑

k=1

k=1

¯ in | + |G ¯ jn |)β, (|G k k

Y. WANG / Nonlinear Analysis 189 (2019) 111585

17

¯ jn ≤ C holds for any 1 ≤ k ≤ b and any j ∈ Z1 , where c is a constant large enough. Since G k ⏐ b ⏐ b ⏐ ∑ ⏐ ∑ ⏐ ⏐ ¯ ¯ Gink ςk − ε Gjnk ςk ⏐ ≤ εC1 β, ⏐ε ⏐ ⏐ k=1

k=1

where C1 depends on g0 and b. Assume εβ < 1.

(5.6)

So, √ √ ∑b ¯ in ςk − ε ∑b G ¯ ˆi − Ω ˆj | |ε−1 λi − ε−1 λj + ε k=1 G |Ω k=1 jnk ςk | k = 1 + |i − j| 1 + |i − j| √ √ √ √ −1 −1 λi − ε λj | |ε−1 λi − ε−1 λj | |ε βεC1 − ≥ − εC1 β ≥ 1 + |i − j| 1 + |i − j| 2|i − j| ε−1 ≥ c|k| − εC1 β 2

(5.7)

hold. However, 1 |⟨k, ω ˆ ⟩| ≤ |k||ˆ ω | ≤ C2 |k|( + εβ) 1 + |i − j| ε

(5.8)

is true, where C2 depends on I and g0 . Hence, from (5.7) and (5.8), we have ˆi − Ω ˆ j + ⟨k, ω ˆi − Ω ˆj | |Ω ˆ ⟩| |Ω |⟨k, ω ˆ ⟩| ≥ − 1 + |i − j| 1 + |i − j| 1 + |i − j| ε−1 C2 |k| ≥ c|k| − εC1 β − − ϵβC2 |k|. 2 ε It follows that, when c > 4C2 + 4C1 + 1 and ε < 1, ˆi − Ω ˆ j + ⟨k, ω |Ω ˆ ⟩| > 1. 1 + |i − j| When j ≥ c|k|, since i ≥ j, i ≥ c|k| also holds and we get the same result. We only need to deal with the case max{i, j} ≤ c|k|. (5.9) Suppose that ˆi − Ω ˆj . f (ξ) := ⟨k, ω ˆ⟩ + Ω 1,12 The set Bkij is equivalent to {

ω ¯ ∈ Π1 , ς ∈ Π2 : |⟨k

1

(1)

, ω(¯ ω )⟩ + ⟨k

(2)

ϵ 4 (|i − j| + 1) ˆ i (ς) − Ω ˆ j (ς)| < , ϖ(ς)⟩ + Ω 1 + (|k (1) | + |k (2) |)τ

} ,

where k (1) := (k11 , k12 , . . . , k1m ) ∈ Zm and k (2) := (k21 , k22 , . . . , k2b ) ∈ Zb . (ξ) ∂f (ξ) Case 1.When k (1) ̸= 0, there exists an t0 satisfying that ∂f ∂ω ¯ t0 = k1t0 ε ̸= 0. So, | ∂ ω ¯ t0 | ≥ ε. By using Lemma 10 of [13], from (5.9), we have that meas(∪k(1) ̸=0 A1k4 ) ≤

∑ ∑ 2ϵ 14 (|i − j| + 1) ∑ 2ϵ 14 3c|k| ≤ τ εAk ε|k| k̸=0

k̸=0 i,j≤c|k|

≤

∑ 6cϵ k̸=0

ε|k|

1 4

τ −1

When τ > (m + b) + 1, the last series

≤

1 4

1 4

∑ 6cϵ ϵ ∑ 1 2m+b lm+b−1 ≤ C . τ −1 τ −(m+b) εl ε l l̸=0 l̸=0

∑

l̸=0

1

lτ −(m+b) is convergent. Therefore, meas(∪k̸=0 A1k4 ) = O(ϵ 4 ε−1 ).

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( ) ¯n n , . . . , G ¯ n n for 1 ≤ l ≤ b, Bi = (G ¯ in , . . ., G ¯ in ), Bj = Case 2. If we suppose Al = G 1 1 l l b b ( ) ¯ ¯ Gjn1 , . . . , Gjnb for i, j ∈ Z1 , and δA,B,i,j := k21 A1 · ς + · · · + k2b Ab · ς + (Bi − Bj ) · ς, then, by using of (5.15), |δA,B,i,j | ≤ |k (2) ||A1 + · · · + Ab ||ς| + |Bi − Bj ||ς| ≤ C2 β, where C2 depends on I. Thus, when ε is small enough, ε |δA,B,i,j | < εC2 β < C2 . On the other hand, when ˆi − Ω ˆ j . Clearly, the 1st to mth elements of the k (1) = 0 and k (2) ̸= 0 holds, we have f (ξ) = ⟨k (2) , ϖ⟩ + Ω ∂f (ξ) vector ∂ξ are zeros. We only calculate the (m + 1)th to (m + b)th elements of the vector ∂f∂ξ(ξ) . It follows that 8πn41 n22 · · · n2b i2 j 2 ∂f · = 3k21 n22 n23 · · · n2b i2 j 2 + 4k22 n21 n23 · · · n2b i2 j 2 3g0 ε ∂ς1 + · · · + 4k2b n21 n22 · · · n2b−1 i2 j 2 + 4n21 n22 · · · n2b j 2 − 4n21 n22 · · · n2b i2 , 8πn21 n42 · · · n2b i2 j 2 ∂f · = 4k21 n22 n23 · · · n2b i2 j 2 + 3k22 n21 n23 · · · n2b i2 j 2 3g0 ε ∂ς2 (5.10) + · · · + 4k2b n21 n22 · · · n2b−1 i2 j 2 + 4n21 n22 · · · n2b j 2 − 4n21 n22 · · · n2b i2 , ··· 8πn21 n22 · · · n4b i2 j 2 ∂f · = 4k21 n22 n23 · · · n2b i2 j 2 + 4k22 n21 n23 · · · n2b i2 j 2 3g0 ε ∂ςb + · · · + 3k2b n21 n22 · · · n2b−1 i2 j 2 + 4n21 n22 · · · n2b j 2 − 4n21 n22 · · · n2b i2 . Then, there exists a t0 (1 ≤ t0 ≤ b) s.t. 8πn21 · · · n4t0 · · · n2b i2 j 2 ∂f · ̸= 0. 3g0 ε ∂ςt0

(5.11)

Otherwise, suppose that the equalities in (5.10) are all zeros and we can get that −4n2t (j 2 − i2 ) , t = 1, 2, . . . , b. (4b − 1)i2 j 2

k2t =

(5.12)

It follows that δn1 ,...,nb ,i,j : = k21 n21 + k22 n22 + · · · + k2b n2b + i2 − j 2 ) ( 4(n41 + n42 + · · · + n4b ) + 1 (i2 − j 2 ). = (4b − 1)i2 j 2

(5.13)

Hence, we have δn1 ,...,nb ,i,j ̸= 0 and |δn1 ,...,nb ,i,j | ≥ 1 hold for i ̸= j and i, j ∈ Z1 . When ε is small enough, ε−1 |δn1 ,...,nb ,i,j | > C2 + 1

(5.14)

is always true. In addition, from (5.12), |k (2) | =

b ∑

|k2t | ≤

t=1

⏐ ⏐ 4(n21 + n22 + · · · + n2b ) ⏐⏐ 1 1 ⏐⏐ 8bn2b − ≤ . ⏐ ⏐ 2 2 4b − 1 i j 4b − 1

It follows from (5.14) that |f (ξ)| ≥ ε−1 |δn1 ,...,nb ,i,j | − ε |δA,B,i,j | > 1, 1,11 as long as ε is small enough. However, for any ξ ∈ Bkij , from (5.9), the inequalities 1

|f (ξ)| <

1

1

ϵ 4 (|i − j| + 1) ϵ 4 3c|k| 3cϵ 4 < < τ −1 < 1 Ak Ak |k|

(5.15)

Y. WANG / Nonlinear Analysis 189 (2019) 111585

19

⏐ ⏐ ⏐ ⏐ hold with ϵ small enough, which is a contradiction. So, (5.11) is true. Furthermore, we can have ⏐ ∂ς∂ft ⏐ ≥ 0 Cε i21j 2 . It follows that, by using of Lemma 10 in [13], ⎞

⎛ ⋃

meas ⎝

⎛

A1k4 ⎠ ≤ meas ⎝

⎞ ⋃

⋃

1,12 ⎠ Bkij

k̸=0 i,j≤c|k|

k(1) =0, k(2) ̸=0

∑ ∑ ϵ 14 (|i − j| + 1) i2 j 2 · ≤ τ Cε 1 + |k| k̸=0 i,j≤c|k| 1

≤ Cϵ 4 ε−1

∑ |k|5 1 −1 ∑ 1 τ ≤ Cϵ 4 ε lτ −m−b−4 |k|

k̸=0 1 4

l̸=0

−1

= O(ϵ ε ). (⋃ (⋃ ) ) 11 1 −1 − 12 1 1 4 ). From (5.5), meas ) Therefore, when τ > m + b + 5, meas k̸=0 Ak4 = O(ε k̸=0 Ak4 = O(ϵ ε holds. This completes the proof. □ Similarly, we prove the same results for A1k1 , A1k2 , and A1k3 . So the following lemma holds. Lemma 5.2. true.

For τ > m + b + 5, meas

((

∪k̸=0 A1k1

)⋃(

)) 1 11 ∪k (A1k2 ∪ A1k3 ∪ A1k4 ) = O(ϵ 4 ε−1 ) = O(ε− 12 ) is

Remark 5.4. For meas(∪k̸=0 A1k3 ), similarly to (5.13), we can get the equality ( ) −4(n41 + n42 + · · · + n4b ) δn1 ,...,nb ,i,j = + 1 (i2 + j 2 ). (4b − 1)i2 j 2 By G of Definition 2.1, δn1 ,...,nb ,i,j ̸= 0. Acknowledgment The author would like to thank the referee for his/her valuable comments and detailed opinions which helped to improve the presentation of this paper. Appendix Proof of Lemma 4.2. Assume that √ √ √ √ fijdl,k = ± λi ± λj ± λd ± λl + ⟨k, ω⟩, { } 1 ϱε 3 m Rijdl,k = ω ∈ [ϱ, 2ϱ] : |fijdl,k | < , m+2 |k| ⋃ ⋃ ⋃ ⋃ ˜ = Ω Rijdl,k , Ω 0 = Rijdl,k , ¯ |k|≥1 (i,j,d,l)∈∆

Ω1 =

⋃

⋃

|k|≥1 (i,j,d,l)∈∆1

˜ = Ω 0 ⋃ Ω 1 ⋃ Ω 2. It is evident that Ω

Rijdl,k ,

|k|≥1 (i,j,d,l)∈∆0

and Ω 2 =

⋃

⋃

|k|≥1 (i,j,d,l)∈∆2

Rijdl,k .

Y. WANG / Nonlinear Analysis 189 (2019) 111585

20

1 √ √ √ √ 3 Consider two hyperplanes ± λi ± λj ± λd ± λl = ± |k|ϱεm+2 . We have

measRijdl,k ≤ m|k|

−1

√ 1 √ m−1 2ϱε 31 2( 2)m−1 mε 3 m ( 2ϱ) ≤ ϱ . m+2 m+3 |k| |k|

(6.1)

1 m 3

ϱ ε So, we can attain measRijdl,k ≤ C |k| m+3 , where C depends on m. Case 1. If (i, j, d, l) ∈ ∆0 , then, from (4.21)

measΩ 0 ≤ C

∑ ϱm ε 13 |k| |k|≥1

m+3 (nb

− n1 + 1)4 1

1

∑

≤ C(nb − n1 + 1)4 ϱm ε 3

1≤|k|=l

lm+3

2m lm−1 .

1

Since l≥1 l14 is convergent, measΩ 0 ≤ Cϱm ε 3 , where the constant C depends on n1 , nb and m. Case 2. If (i, j, d, l) ∈ ∆1 , we assume that l ̸∈ I without loss of generality. So, √ √ √ |⟨k, ω⟩ ± λi ± λj ± λd | ≤ 2ϱ|k| + 3n2b . √ When l > 2ϱ|k| + 3n2b + 1 and ε is small enough, √ √ √ √ |fijdl,k | ≥ | ± λl | − |⟨k, ω⟩ ± λi ± λj ± λd | ∑

1

> 2ϱ|k| + 3n2b + 1 − (2ϱ|k| + 3n2b ) = 1 > which implies that one only need to consider the case 1 ≤ l ≤

√

|k|

|k|≥1

m+2

|k|

(nb − n1 + 1)3

1

|k|≥1 1

≤ Cϱm ε 3 (nb − n1 + 1)3

∑

√ 2ϱ|k| + 3n2b + 1

1

∑

≤ Cϱm ε 3 (nb − n1 + 1)3

m+2 ,

|k|

2ϱ|k| + 3n2b + 1. Therefore,

1

ϱm ε 3

∑

measΩ 1 ≤ C

ϱm ε 3

|k|

m+2

2m lm−1

l≥1

1 1 ≤ Cϱm ε 3 , lm+2

where the constant C depends on n1 , nb , ϱ and m. Case 3. If (i, j, d, l) ∈ ∆2 , we assume that l and d ̸∈ I without loss of generality. We divide this case into two cases (a) and (b) below. √ √ √ √ Case a. If ± λd ± λl = 0, then |fijdl,k | = |⟨k, ω⟩ ± λi ± λj |. Set ⋃ ⋃ Ω 2,1 = Rijdl,k √ √ |k|≥1 l,d̸∈I, i,j∈I, ±

λd ±

λl =0

then we have 1

measΩ

2,1

≤C

ϱm ε 3

∑

m+2

|k| |k|≥1

|k|

(nb − n1 + 1)2

1

≤ Cϱm ε 3 (nb − n1 + 1)2

m+3

|k|≥1 1

≤ Cϱm ε 3 (nb − n1 + 1)2

∑ l≥1

where the constant C depends on n1 , nb and m.

1

∑ |k|

2m lm−1

1 1 ≤ Cϱm ε 3 , lm+3

Y. WANG / Nonlinear Analysis 189 (2019) 111585

21

√ √ Case b. If ± λd ± λl = ̸ 0, then d−l ̸= 0. Without loss of generality, we suppose d > l. So d−l := p ≥ 1. Clearly, √ √ | ± λd ± λl | ≥ d2 − l2 = (d − l)(d + l) = p(2l + p). (6.2) √ √ ˜ + 1. Since ˜ := ϱ|k| + n2 , it is derived that | ± λd ± λl | ≥ 2l + p > 2N When l > N b √ √ |⟨k, ω⟩ ± λi ± λj | ≤ 2ϱ|k| + 2n2b (6.3) and |fijdl,k | ≥ | ±

√

λd ±

√

λl | − |⟨k, ω⟩ ±

√

λi ±

˜ + 1 − 2ϱ|k| − 2n2 = 1 > C |fijdl,k | > 2N b

√ λj |,

ϱm ε |k|

(6.4)

1 3

m+2

˜ . When p > N ˜ + 1, (6.2) holds as ε small enough. Therefore, we only need to consider the case 1 ≤ l ≤ N induces that √ √ ˜ + 2. | ± λd ± λl | ≥ 2pl + p2 > 2p = 2N It follows from (6.3) and (6.4) that ˜ + 2 − 2ϱ|k| − 2n2b |fijdl,k | ≥ 2N 1

= 2ϱ|k| + 2n2b + 2 − 2ϱ|k| − 2n2b = 2 >

ϱm ε 3 |k|

m+2

˜ +1. In this case, 1 ≤ d = l+p ≤ 2N ˜ +2. as ε is small enough. So, we only need to consider the case 1 ≤ p ≤ N Set ⋃ ⋃ Ω 2,2 = Rijdl,k √ √ |k|≥1 l,d̸∈I, i,j∈I, ±

λd ±

λl ̸=0

then we have 1

measΩ 2,2 ≤ C

∑ |k|≥1

ϱm ε 3 |k|

m+2

1

≤ Cϱm ε 3 (nb − n1 + 1)2

|k| ∑

|k|≥1 1

≤ Cϱm ε 3 (nb − n1 + 1)2

∑ l≥1

( )( ) (nb − n1 + 1)2 ϱ|k| + n2b 2ϱ|k| + 2n2b + 2 1

m+3 (nb

|k|

2m lm−1

( )( ) − n1 + 1)2 ϱ|k| + n2b 2ϱ|k| + 2n2b + 2

1 1 ≤ Cϱm ε 3 , lm+1

where the constant C depends on ϱ, n1 , nb and m. Thus, for (i, j, d, l) ∈ ∆2 , there is a constant C satisfying 1 that measΩ 2 ≤ Cϱm ε 3 . Overall, 1 measΩ ≥ (1 − C2 ε 3 )ϱm , ˜ . The constant C2 depends on ϱ, n1 , nb and m. The proof is where we suppose that Ω = [ϱ, 2ϱ]m \ Ω completed. □ References [1] P. Baldi, M. Berti, R. Montalto, KAM for quasi-linear and fully nonlinear forced KdV, Mathematics 24 (3) (2012) 437–450. [2] M. Berti, L. Biasco, M. Procesi, Existence and stability of quasi-periodic solutions for derivative wave equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 24 (2013) 199–214.

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