Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 02, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.02 A KAM THEOREM FOR DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS ZHAOWEI LOU, YOUCHAO WU Abstract. In this article, we establish a Kolmogorov-Arnold-Moser (KAM) theorem for degenerate infinite-dimensional reversible systems under a non- degenerate condition of Rüssmann type. This theorem broadens the scope of applicability of degenerate KAM theory, previously confined to Hamilton- ian systems, by incorporating infinite-dimensional reversible systems. Using this theorem, we obtain the existence and linear stability of quasi-periodic solutions for a class of non-Hamiltonian but reversible beam equations with non-linearities in derivatives. 1. Introduction In Kolmogorov-Arnold-Moser (KAM) theory, the non-degeneracy condition typi- cally refers to the regularity assumption of tangent frequencies with respect to exter- nal parameters. This condition plays a vital role in ensuring the persistence of KAM tori under small perturbations. Degenerate KAM theory for finite-dimensional Hamiltonian systems has been widely developed since the works of Arnold [1] and Pjartly [15]. A complete geometric definition of non-degeneracy condition was given by Rüssmann [18], which is not only applicable to maximal dimensional KAM tori but also to lower dimensional elliptic tori. See also other important works by Brjuno [5], Cheng and Sun [6] and Xu, Qiu, and You [22]. KAM theory for degenerate infinite-dimensional Hamiltonian systems was ini- tially established by Xu, Qiu, and You [21]. They proved a KAM theorem under an analytic Rüssmann degeneracy assumption. Subsequently, Bambusi, Berti, and Magistrelli [3] introduced a different Rüssmann degeneracy condition from that in [21] and established a new KAM theorem for a class of nonlinear wave equations. Building upon the non-degeneracy condition in [3], Baldi, Berti, Haus and Montalto derived a new degenerate KAM theorem particularly applicable to gravitational wave equations. The proof is based on the Nash-Moser iteration, KAM reduction and pseudo-differential calculus. Furthermore, Gao and Liu[8] recently developed a KAM theorem for degenerate infinite-dimensional Hamiltonian systems with dou- ble normal frequencies, successfully applying it to nonlinear wave equations and nonlinear Schrödinger equations with periodic boundary conditions. 2020 Mathematics Subject Classification. 37K55, 35B15. Key words and phrases. KAM theorem; infinite-dimensional reversible system; Rüssmann non-degeneracy condition. ©2024. This work is licensed under a CC BY 4.0 license. Submitted August 1, 2023. Published January 3, 2024. 1 2 Z. LOU, Y. WU EJDE-2024/02 On the other hand, KAM theory for infinite-dimensional Hamiltonian systems has also been extended to infinite-dimensional reversible systems [2, 4, 7, 9, 12, 11, 13, 14, 24]. However, to the best of our knowledge, these results are all based on classical non-degeneracy conditions, and there is still no works on KAM theory for degenerate infinite-dimensional reversible systems. In contrast, some progress has been made in the KAM theory for finite-dimensional reversible systems under Rüssmann degeneracy conditions [19, 20, 23]. The primary motivation of this article is to bridge the gap in the literature concerning the degenerate KAM theory for infinite-dimensional reversible systems. The study aims to generalize the existing theorems and to provide new insights into the applicability of degenerate KAM theory in a broader range of systems, particularly in infinite-dimensional reversible settings. By doing so, we hope to advance the understanding of the existence and stability properties of quasi-periodic motions in more complex systems that were previously not well-covered by the classical non-degeneracy conditions. In this article, we establish a KAM theorem for degenerate infinite-dimensional reversible systems, which is, to the best of our knowledge, a new result in the field. Our method combines and extends the techniques from previous works on degenerate KAM theory for infinite-dimensional Hamiltonian systems and finite- dimensional reversible systems, allowing us to study new systems by the combi- nation of degeneracy and reversibility in an infinite-dimensional setting. As an application, we use our KAM theorem to prove the existence and linear stabil- ity of quasi-periodic solutions for a class of non-Hamiltonian but reversible beam equations with derivative non-linearities. We consider a family of vector fields with normal form N = ω(ξ) ∂ ∂θ + iΩ(ξ)z ∂ ∂z − iΩ(ξ)z̄ ∂ ∂z̄ = n∑ j=1 ωj(ξ) ∂ ∂θj + ∑ j≥1 ( iΩj(ξ)zj ∂ ∂zj − iΩj(ξ)z̄j ∂ ∂z̄j ) , on the phase space Pa,p = Tn × Rn × la,p × la,p 3 (θ, I, z, z̄), where Tn(1 ≤ n < +∞) is the standard n−torus, la,p is the Hilbert space of all complex sequence z = (z1, z2, . . . , ) with the norm defined by ‖z‖2a,p = ∑ j≥1 |zj |j2pe2aj < +∞, a ≥ 0, p ≥ 0. The parameter ξ ∈ Π ⊂ Rn, where Π is a bounded, connected closed domain in Rn. Note that N is reversible with respect to the involution map S : (θ, I, z, z̄)→ (−θ, I, z̄, z): N ◦ S = −DS ·N, where DS is the tangent map of S. The motion equations of S-reversible vector fields N are θ̇ = ω(ξ), İ = 0, ż = iΩ(ξ)z, ˙̄z = −iΩ(ξ)z̄. (1.1) EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 3 Obviously, for each ξ ∈ Π, T n = Tn × {0, 0, 0} ⊂Pa,p is an invariant torus of the reversible systems (1.1) with the frequencies ω(ξ) = (ω1(ξ), . . . , ωn(ξ)). Now we consider the small perturbations of N : X = N + P , where the pertur- bations P are S-reversible. To vector fields X, the associated reversible equations are θ̇ = ω(ξ) + P (θ), İ = P (I), ż = iΩ(ξ)z + P (z), ˙̄z = −iΩ(ξ)z̄ + P (z̄). (1.2) The definition of P (θ), P (I), P (z), P (z̄) will be given below. We are concerned with the persistence of invariant tori for reversible systems (1.2). For this purpose, we first give some notations and assumptions. Let CN ,1(Π) be the N -order Lipschitz continuously differentiable function space. If Π is a closed set, the derivatives of function on Π are understood in the sense of Whitney. Consequently, the space CN ,1(Π) is also understood in the sense of Whitney, where the integer N will be decided below. We use the following assumptions. (A1) (Non-degeneracy condition) Suppose for all ξ ∈ Π, rank{∂ω ∂ξ } = r, rank{∂ βω ∂ξβ : ∀β, 1 ≤ |β| ≤ n− r + 1} = n, where ∂ω ∂ξ is a function vector group of all 1-order partial derivatives of ω, and ∂βω ∂ξβ = ( ∂βω1 ∂ξβ , . . . , ∂ βωn ∂ξβ ) . Moreover, for some N ≥ n − r + 1, ω ∈ {CN ,1(Π)}n with ‖ω‖CN ,1(Π) , max1≤j≤n ‖ωj‖CN ,1(Π) ≤M1. (A2) (Spectral asymptotic) There exist d ≥ 1 and δ < d− 1 such that Ωj(ξ) = bjd + b′jd ′ + · · ·+O(jδ), b > 0, d′ < d, where the dots stand for finite lower order terms of j and bjd + b′jd ′ + . . . are independent of the parameter ξ. Moreover, Ωj satisfies ‖Ωj − bjd − b′jd ′ − . . . ‖CN ,1(Π) ≤M2j δ, ∀j ≥ 1. (A3) (Regularity of perturbations) For all ξ ∈ Π, the S-reversible perturbation vector field P : Pa,p →Pa,p̄ is real analytic with p ≤ p̄ and p− p̄ < d− 1. Without loss of generality, suppose p− p̄ < δ. We denote by D(s, r) a complex neighborhood of Tn in Pa,p: D(s, r) = {(θ, I, z, z̄) ∈Pa,p : | Im θ| < s, |I| < r, ‖z‖a,p < r, ‖z̄‖a,p < r}, where | Im θ| is the imaginary part of θ and | · | is the sup-norm for n-dimensional vector, r > 0 is a radius of neighborhood (Note that it is different from that in assumption (A1.). Let ‖ · ‖∗ = ‖ · ‖CN ,1(Π). Now we consider the vector field P (v) = (P (θ)(v), P (I)(v), P (z)(v), P (z̄)(v)), 4 Z. LOU, Y. WU EJDE-2024/02 where v = (θ, I, z, z̄) ∈ D(s, r). It is convenient to rewrite it as a partial differential operator P (v) = P (θ)(v) ∂ ∂θ + P (I)(v) ∂ ∂I + P (z)(v) ∂ ∂z + P (z̄)(v) ∂ ∂z̄ , where P (θ)(v) ∂ ∂θ = n∑ j=1 P (θj)(v) ∂ ∂θj , and similarly for P (I)(v) ∂∂I , P (z)(v) ∂∂z , P (z̄)(v) ∂∂z̄ . Suppose the vector field W (ξ) = (W (θ)(ξ),W (I)(ξ),W (z)(ξ),W (z̄)(ξ)) ∈ Pa,p̄, with the weighted norm ‖W‖∗r = ‖W‖p̄,r = |W (θ)|∗ + 1 r |W (I)|∗ + 1 r ‖W (z)‖∗a,p̄ + 1 r ‖W (z̄)‖∗a,p̄, where |W (θ)|∗ = |(|W (θ) 1 |∗, . . . , |W (θ) n |∗)|, ‖W (z)‖∗a,p̄ = ‖(‖W (z) 1 ‖∗, ‖W (z) 2 ‖∗, . . . )‖∗a,p̄. For a function f(v, ξ) on D ×Π, we define its norm by ‖f‖∗D = sup v∈D ‖f(v, ·)‖∗. Similarly we define the norm of vector field W as ‖W‖∗r,D = ‖W‖p̄,r,D = |W (θ)|∗D + 1 r |W (I)|∗D + 1 r ‖W (z)‖∗a,p̄,D + 1 r ‖W (z̄)‖∗a,p̄,D. For k ∈ Zn and l ∈ Z+∞, we denote |k| = ∑n j=1 |kj |, |l| = ∑∞ j=1 |lj |, |l|δ =∑∞ j=1 |lj |jδ, and [l]d = {1, | ∑∞ j=1 ljj d|}. We denote Ak = (1 + |k|)τ , where τ ≤ τ0 with τ0 = { (n+ 2 d−1 )(n− r + 1), if d > 1, (n+ 1 + κ+n−r+1 κ )(n− r + 1), if d = 1, where κ = min{d, d− 1− δ, d− d′}. Theorem 1.1. Suppose the S-reversible vector field X = N + P satisfies (A1)– (A3). Then there exist K and L such that for k, l with 0 6= |k| ≤ K, |l| ≤ 2 and |l|k ≤ L, the inequality |〈k, ω(ξ)〉+ 〈l,Ω(ξ)〉| ≥ α0 > 0 holds for all ∀ξ ∈ Π. Then for sufficiently small α > 0(α > α0), there exists sufficiently small ε0 = ε0 (α) > 0 such that if ε = ‖P‖∗r,D(s,r) < ε0, then there exists a nonempty Cantorian subset Π∗ of Π and embeddings Φ∗(·, ·) : Tn × Π∗ → Pa,p̄ satisfying ‖Φ∗ − Φ0‖∗r ≤ cε, where Φ0 is the trivial embedding Tn ×Π∗ → T n = Tn × {0, 0, 0} ⊂Pa,p̄, and for all ξ ∈ Π∗,Φ∗(Tn, ξ) is a real analytic embedding of a rotational torus for reversible systems (1.2) at ξ with its frequencies ω∗(ξ) satisfying |ω∗ − ω|∗ ≤ cε. Moreover, the Lebesgue measure meas(Π−Π∗) ≤ c(diamPi)n−1αµ, where µ = { 1 n−r+1 , if d > 1, κ (κ+n−r+1)(n−r+1) , if d = 1. EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 5 The Cantorian subset Π∗ in Theorem1.1 arises from the small divisor conditions encountered during the KAM step, |〈k, ω(ξ)〉+ 〈l,Ω(ξ)〉| ≥ α [l]d Ak ,∀k ∈ Zn, l ∈ Z+∞, |l| ≤ 2, |k|+ |l| 6= 0. (1.3) In other words, the KAM iteration can only be performed for those parameters that satisfy the inequality (1.3). Hence, we need to exclude certain points where this inequality does not hold. The subsequent Theorem 1.2 ensures that the set Π is not empty if α is sufficiently small. Theorem 1.2. Let Rk,l(α) = { ξ : |〈k, ω(ξ)〉+ 〈l,Ω(ξ)〉| < α [l]d Ak } , where k ∈ Zn, l ∈ Z+∞, |l| ≤ 2, |k|+ |l| 6= 0. If (A1) and (A2) hold, then there exist K and L such that for sufficiently small α > 0, meas ( ∪|k|≥Kor|l|k≥L Rk,l(α) ) ≤ c(diam(Pi)n−1αµ, where µ is defined in Theorem 1.1. The proof of the above theorem follows a similar approach as in [21] and is omitted here. In the subsequent sections we focus on the proof and application of Theorem 1.1. In Section 2, we derive and solve the homological equation. Section 3 is dedicated to constructing one step of the iterative scheme. The complete iteration sequences, forming the basis of the proof of Theorem 1.1, are presented in Section 4. Finally, in Section 5, we apply Theorem 1.1 to a class of non-Hamiltonian but reversible beam equations. 2. Homological equation At each step of the KAM iteration, we obtain the S-invariant change of variables Φ through the time 1-map ΦtF |t=1 of the flow generated by the S-invariant vector field F . The vector field F and the correction N̂ to the normal form N are solutions of the homological equation [F,N ] + N̂ = R, (2.1) where the symbol [·, ·] represents the Lie bracket of vector fields, and the vector field R is the truncation of the Taylor polynomial of the reversible vector field P , R = Rθ ∂ ∂θ + ( RI(θ) +RII(θ)I +RIz(θ)z +RIz̄(θ)z̄ ) ∂ ∂I + ( Rz(θ) +RzI(θ)I +Rzz(θ)z +Rzz̄(θ)z̄ ) ∂ ∂z + ( Rz̄(θ) +Rz̄I(θ)I +Rz̄z(θ)z +Rz̄z̄(θ)z̄ ) ∂ ∂z̄ . (2.2) The reversiblity of P implies that Rθ(θ) = Rθ(−θ), RI(θ) = −RI(−θ), Rz(θ) = −Rz̄(−θ), RII(θ) = −RII(−θ), RIz(θ) = −RIz̄(−θ), RzI(θ) = −Rz̄I(−θ), Rzz(θ) = −Rz̄z̄(−θ), Rzz̄(θ) = −Rz̄z(−θ). (2.3) 6 Z. LOU, Y. WU EJDE-2024/02 The normal form of R is defined as [R] = [Rθ] ∂ ∂θ + [diagRzz]z ∂ ∂z + [diagRz̄z̄]z̄ ∂ ∂z̄ = n∑ b=1 [Rθb ] ∂ ∂θb + ∑ j≥1 [Rzjzj ]zj ∂ ∂zj + ∑ j≥1 [Rz̄j z̄j ]z̄j ∂ ∂z̄j . We choose the normal correction N̂ = [R]. Thus (2.1) is a homological equation of F . Below we solve this homological equation and estimate the generating vector field F . Lemma 2.1. Suppose that uniformly on Π+ ⊂ Π, |〈k, ω(ξ)〉+ 〈l,Ω(ξ)〉| ≥ α [l]d Ak , (2.4) for all k ∈ Zn, l ∈ Z+∞ with |l| ≤ 2, |k| + |l| 6= 0. Then the homological equation (2.1) has solution F and N̂ that are normalized by [F ] = 0, [N̂ ] = N̂ . Moreover, they satisfy F ◦ S = DS · F and ‖N̂‖∗r,D(s,r) 6 ‖R‖ ∗ r,D(s,r), ‖F‖∗r,D(s−σ,r) 6 cM αN+2σb ‖R‖∗r,D(s,r), where b = (N + 2)τ +N + 1 + n 2 , M = (M1 +M2)N+2, ‖ · ‖∗ = ‖ · ‖CN ,1(Π+). Proof. The homological equation (2.1) splits into: ∂ωF θ i = Rθi − [Rθi ], i = 1, . . . , n; (2.5) ∂ωF I i = RIi , i = 1, . . . , n; (2.6) ∂ωF II ij = RIIij , i, j = 1, . . . , n; (2.7) ∂ωF Iz ij + iΩjF Iz ij = RIzij , i = 1, . . . , n, j ≥ 1; (2.8) ∂ωF Iz̄ ij − iΩjF Iz̄ ij = RIz̄ij , i = 1, . . . , n, j ≥ 1; (2.9) ∂ωF z i − iΩiF z i = Rzi , i ≥ 1; (2.10) ∂ωF zI ij − iΩiF zI ij = RzIij , i ≥ 1, j = 1, . . . , n; (2.11) ∂ωF zz ij + iΩjF zz ij − iΩiF zz ij = Rzzij − [Rzzij ]δij , i, j ≥ 1; (2.12) ∂ωF zz̄ ij − iΩiF zz̄ ij − iΩjF zz̄ ij = Rzz̄ij , i, j ≥ 1; (2.13) ∂ωF z̄ i + iΩiF z̄ i = Rz̄i , i ≥ 1; (2.14) ∂ωF z̄I ij + iΩiF z̄I ij = Rz̄Iij , i ≥ 1, j = 1, . . . , n; (2.15) ∂ωF z̄z ij + iΩjF z̄z ij + iΩiF z̄z ij = Rz̄zij , i, j ≥ 1; (2.16) ∂ωF z̄z̄ ij − iΩiF z̄z̄ ij + iΩjF z̄z̄ ij = Rz̄z̄ij − [Rz̄z̄ij ]δij , i, j ≥ 1. (2.17) Below we only consider the homological equations (2.10) and (2.12), the other equations can be analyzed similarly. Firstly, we consider the equation (2.10). Note that ‖Rz‖∗a,p,D(s,r) ≤ r‖R‖ ∗ r,D(s,r). Let F zi = ∑ k∈Zn F zi (k)ei〈k,θ〉, Rzi = ∑ k∈Zn Rzi (k)ei〈k,θ〉, EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 7 where {F zi (k) | k ∈ Zn} are the Fourier coefficients of F zi . The vector Rz is an analytic mapping from D(s, r) into la,p̄, its Fourier coefficients Rz(k) satisfy the l2-estimates (ref. [16]):∑ k ( ‖Rz(k)‖∗a,p̄ )2 e2|k|s 6 2n(‖Rz‖∗a,p̄,D(s)) 2. By Lemma 6.8 and the non-resonant condition (2.4), for ξ ∈ Π+, we have ω̄(ξ) = (ω(ξ),Ωi(ξ)) ∈ O+, and 1 〈k,ω(ξ)〉+Ωj(ξ) = Gk̄[ω̄(ξ)], where k̄ = (k, 1) ∈ Zn+1. By deriving function Gk̄[ω̄(ξ)], we have ‖Gk̄[ω̄(ξ)]‖∗ 6 cAN+2 k |k|N+1M αN+2 . By Lemma 6.9, we have ‖F zi (k)‖∗ 6 cAN+2 k |k|N+1M αN+2 ‖Rzi (k)‖∗, ‖F z(k)‖∗ 6 cAN+2 k |k|N+1M αN+2 ‖Rz(k)‖∗. Thus ‖F z‖∗a,p̄,D(s−σ) ≤ cMBσ αN+2 ‖Rz‖∗a,p̄,D(s,r), where Bσ = (∑ k A 2(N+2) k |k|2(N+1)e−2|k|σ )1/2 6 c σb . Then 1 r ‖F z‖∗a,p̄,D(s−σ) 6 cM αN+2σb ‖R‖∗r,D(s,r). We now consider the homological equation (2.12) given by ∂ωF zz ij + iΩjF zz ij − iΩiF zz ij = Rzzij − [Rzzij ]δij , i, j ≥ 1. Since Rzz = ∂ ∂zR (z)|(z,z̄,I)=0, by the Cauchy estimate, we have |||Rzz|||D(s) ≤ 1 r ‖R(z)‖∗a,p̄,D(s,r) ≤ ‖R‖ ∗ r,D(s,r), where ||| · ||| denotes the operator norm from la,p → la,p̄. Note that the operator Rzz is equivalent to the operator R̃ = ( viR zz ij wj ) from l2 → l2 and |||R̃|||2,D(s) = |||Rzz|||D(s), where vi and wj are some weights, ||| · |||2 is the operator norm from l2 → l2. Thus we need only consider the norm of l2-operator R̃. Let R̃ = ∑ k R̃(k)ei〈k,θ〉, F̃ = ∑ k F̃ (k)ei〈k,θ〉, we have iF̃ij(k) = R̃ij(k) 〈k, ω〉+ Ωj − Ωi , |k|+ |i− j| 6= 0, where {F̃ij(k) : k ∈ Zn} are the Fourier coefficients of the (i, j)−components of the F̃ . By Lemma 6.8 and taking ω̄ = (ω,Ωi − Ωj) ∈ O+, by the same way as the above and combining 2|id − jd| > |i− j| |id−1 + jd−1|, we know ‖ 1 〈k, ω〉+ Ωi − Ωj ‖∗ 6 cAN+2 k |k|N+1M αN+2|i− j| , i 6= j. 8 Z. LOU, Y. WU EJDE-2024/02 Again by Lemma 6.9, ‖F̃ij(k)‖∗ ≤ cAN+2 k |k|N+1M αN+2|i− j| ‖R̃ij(k)‖∗, i 6= j. Hence, by Lemma 6.6, |||F̃ (k)|||2 6 cAN+2 k |k|N+1M αN+2 |||R̃(k)|||2. Summing up for k as the above, we have |||F̃ |||2,D(s−α) 6 cBσM αN+2 |||R̃|||2,D(s). Going back to the original operator norm we have 1 r ‖F zzz‖∗a,p̄,D(s−σ,r) 6 cM αN+2σb ‖R‖∗r,D(s,r). The proof is complete. � 3. KAM step At the ν-th step of the iteration scheme, we are given an S-reversible vector field Xν = Nν +Pν , where Nν = ων ∂ ∂θ +iΩνz ∂ ∂z − iΩν z̄ ∂ ∂z̄ is a normal S-reversible vector field, and Pν is an S-reversible the perturbation vector field. To simplify notation, we drop the index “ν′′ and write “+′′ for “ν + 1′′. Thus, P = Pν , P+ = Pν+1, and so on. To proceed with the next step of the iteration, we assume that the perturbation vector field is sufficiently small, and we can choose 0 ≤ η ≤ 1 8 such that |P |∗r,D(s,r) ≤ ασb+1η c0 , (3.1) where c0 is some suitably large constant depending only on n and τ . 3.1. Approximation estimates. We approximate P by its Taylor polynomial R of the form (2.2). This approximation extends to the corresponding components P (θ), P (I), P (z), and P (z̄) of the vector field P . Since P is analytic, the components R(θ), R(I), R(z), R(z̄), and their remainders are given by certain Cauchy integrals. We then obtain the estimates ‖R‖∗r,D(s,r) ≤ c‖P‖ ∗ r,D(s,r) , (3.2) ‖P −R‖∗ηr,D(s,4ηr) ≤ cη‖P‖ ∗ r,D(s,r) . (3.3) 3.2. Solution of the homological equation. By Lemma 2.1 and the above es- timates, we obtain ‖N‖∗r,D(s,r) ≤ c‖P‖ ∗ r,D(s,r) , (3.4) ‖F‖∗r,D(s−σ,r) 6 cM αN+2σb ‖P‖∗r,D(s,r), (3.5) ‖DF‖∗r,r,D(s−2σ, r2 ) 6 cM αN+2σb+1 ‖P‖∗r,D(s,r). (3.6) where DF is the differential of F with respect to (θ, I, z, z̄), and the operator norm ‖ · ‖r̄,r is defined by ‖L‖r̄,r = supw 6=0 ‖Lw‖p̄,r̄ ‖w‖p,r . EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 9 3.3. Coordinate transformation. From Lemma 2.1, F is defined on D(s−σ, r)× Π+. By Lemma 6.11 and Cauchy’s inequality, we have th4e following lemma. Lemma 3.1. If ‖F‖∗r,D(s−σ,r) ≤ σ, then for ξ ∈ Π+, the flow φtF (·, ξ) exists on D(s − 2σ, r2 ) for |t| ≤ 1 and it maps D(s − σ, r2 ) into D(s − σ, r). Moreover, for |t| ≤ 1, ‖φtF − id‖∗r,D(s−2σ,r/2), σ‖DφtF − Id‖∗r,r,D(s−3σ,r/4) 6 c‖F‖ ∗ r,D(s−σ,r), (3.7) where id and Id are identity mapping and unit operator, respectively. We choose E = ‖P‖∗r,D(s,r) αN+2σb+1 , by Lemmas 2.1 and 3.1, it follows that for |t| ≤ 1, ‖ΦtF − id‖∗r,D(s−2σ,r/2), σ‖DΦtF − Id‖∗r,r,D(s−3σ,r/4) 6 cME. (3.8) Lemma 3.2. If the S-reversible vector field W (·, ξ) on V = D(s−4σ, 2ηr) depends on the parameter ξ ∈ Π+ with ‖W‖∗r,V < +∞, F is an S-invariant vector field, and Φ = ΦtF : U = D(s − 5σ, ηr) → V , then Φ∗W = (DΦ)−1W ◦ Φ is also an S-reversible vector field and if E is small, we have ‖Φ∗W‖∗ηr,U ≤ c‖W‖∗ηr,V . Proof. See [11] for the verification of the reversibility of Φ∗W . Since Φ∗W = (DΦ)−1W ◦ Φ, we have ‖Φ∗W‖∗ηr,U ≤ ‖(DΦ)−1‖∗ηr,ηr,U‖W ◦ Φ‖∗ηr,U , By (3.8) and η2 = E, 1 σ ‖Φ− id‖∗ηr,U , ‖DΦ− Id‖∗ηr,ηr,U ≤ 1, then by Lemma 6.10 it follows that ‖W ◦ Φ‖∗ηr,U ≤ ‖W‖∗ηr,v. Again we have ‖(DΦ)−1‖∗ηr,ηr,U ≤ 1 + ‖(DΦ)−1 − id‖∗ηr,ηr,U ≤ 2, Hence ‖Φ∗W‖∗ηr,U ≤ c‖W‖∗ηr,v. � 3.4. New vector field. The map Φ = ΦtF |t=1 defined above transforms X into X+ = Φ∗X = N+ + P+ on D(s− 5σ, ηr), where N+ = N + N̂ , P+ = (Φ1 F )∗(P −R) + ∫ 1 0 (ΦtF )∗[R(t), F ]dt, with R(t) = (1− t)N̂ + tR. By Lemma 3.2, we have ‖(Φ1 F )∗(P −R)‖∗ηr,D(s−5σ,ηr) ≤ ‖P −R‖ ∗ ηr,D(s−4σ,2ηr), ‖(Φ1 F )∗[R(t), F ]‖∗ηr,D(s−5σ,ηr) ≤ ‖[R(t), F ]‖∗ηr,D(s−4σ,2ηr). In (3.3), we had already estimated ‖P −R‖∗ηr,D(s,4ηr), so it remains to consider the Lie bracket [R(t), F ]. Using the Cauchy estimate we obtain ‖[R(t), F ]‖∗ηr,D(s−4s,2ηr) 6 c η ‖DR(t) · F + DF ·R(t)‖∗r,D(s−4σ,2ηr) 6 c ησ ‖R(t)‖∗r,D(s,r) · ‖F‖ ∗ r,D(s−σ,r). Then we arrive at the estimate ‖P+‖∗ηr,D(s−5σ,ηr) ≤ cη‖P‖ ∗ r,D(s,r) + cMη‖P‖∗r,D(s,r) ≤ cMη‖P‖∗r,D(s,r). (3.9) 10 Z. LOU, Y. WU EJDE-2024/02 3.5. New normal form. We consider N+ = N + N̂ = w+ ∂ ∂θ + iΩ+z ∂ ∂z − iΩ+z̄ ∂ ∂z̄ with ‖N̂‖∗r,D(s,r) ≤ ‖P‖ ∗ r,D(s,r), where ω+ = ω + ω̂, Ω+ = Ω + Ω̂. This implies |ω̂|∗D ≤ ‖P‖∗r,D(s,r), ‖Ω̂‖ ∗ −δ,D(s,r) ≤ ‖P‖ ∗ r,D(s,r), where ‖Ω̂‖∗−δ = supj≥1 j −δ‖Ω̂j‖∗. With −δ ≤ p̄− p, we have |ω̂|∗ + ‖Ω̂‖∗−δ,D(s,r) ≤ ‖P‖ ∗ r,D(s,r). (3.10) For |k| ≤ K, we observe that |l|δ ≤ |l|d−1 ≤ [l]d, hence |〈k, ω̂〉+ 〈l, Ω̂〉 ≤ |k| · |ω̂|+ |l|δ · |Ω̂|−δ ≤ c|k|Ak [l]d Ak ‖P‖∗r ≤ cα̂ [l]d Ak . with α̂ ≥ c|k|Ak‖P‖∗r . Using the bound for the old divisors, the new ones then satisfy |〈k, ω+(ξ)〉+ 〈l,Ω+(ξ)〉| ≥ α+ [l]d Ak , |k| ≤ K, (3.11) on Π with α+ = α− α̂. 4. Iteration and convergence In Section 3, we present a detailed process of one cycle of the KAM iteration. Now, we choose a sequence of relevant parameters that allow us to perform infinitely many iterations of the KAM step. For ν ≥ 0, define αν = α0 ν 2 ν , σν+1 = σν 2 , εν+1 = c1Mνε 3 2 ν (αN+2 ν σb+1 ν )1/2 , Mν = (M1 +M2 + c1(ε1 + · · ·+ εν−1))N+1, λν = αν Mν , η2 ν = εν αN+2 ν σb+1 ν , Eν = εν αN+2 ν σb+1 ν . Furthermore, sν+1 = sν − 5σν , rν+1 = ηvrν , and Dν = D(sν , rν). As initial value fix σ0 = s0 40 ≤ 1 4 so that s0 > s1 > · · · ≥ s0 2 , and assume that ε0 ≤ γ0α N+2 0 σa+1 0 , γ0 ≤ (c1 + c2)−2, where c1 is twice as large as the constants that appear in Section 3, c2 , c · M̂ ≥ c1Mν , M̂ = (M1 + M2 + c1)N+1. Let Ak = (1 + |k|)τ , (1 + k0)τ+1 = 1 c1γ0 and 1 +Kν = (1 +K0) · 2ν . Lemma 4.1 (Iterative lemma). Assume the reversible vector field Xν = Nν + Pν is regular on Dν × Πν , where Nν is a normal form with coefficients satisfying |ων |∗ + ‖Ων‖∗−δ ≤Mν , |〈k, ων(ξ)〉+ 〈l,Ων(ξ)〉| ≥ αν [l]d Ak , 0 6= |k| ≤ K, |l| ≤ 2, |l|k ≤ L, on Πν . Pν satisfies ‖Pν‖∗rν ,Dν ≤ εν . Then there exists a family of real analytic, S-invariant transformations Φν+1 : Dν+1 ×Πν+1 → Dν , and a closed subset Πν+1 = Πν� ∪|k|≥K Rν+1 k,l (αν+1), EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 11 where Rν+1 k,l (αν+1) = { ξ ∈ Πν ||〈k, ων+1〉+ 〈l,Ων〉| < αν+1 [l]d Akν } such that for Xν+1 = (Φν+1)∗Xν = Nν+1 + Pν+1, the estimates |ων+1 − ων |∗ ≤ cεν , ‖Ων+1 − Ων‖∗−δ ≤ cεν . hold and the same assumptions as above are satisfied with ‘ν + 1’ in place of ‘ν’. Proof. By induction one verifies that εν ≤ γ0α N+2 ν σb+1 ν /22b+2+2v+ 4(N+2) ν for all ν ≥ 0. With the definition of ην this implies εν ≤ αN+2 ν σb+1 ν ην/c1. So the small- ness condition (3.1) of the KAM step is satisfied, and there exists an S-invariant transformation Φν+1 : Dν+1 × Πν+1 → Dν taking Xν into Xν+1 = Nν+1 + Pν+1. The new error satisfies the estimate ‖Pν+1‖∗rν+1,Dν+1 ≤ cMνηνεν = cMνε 3 2 ν (αN+2 ν σb+1 ν )1/2 = εν+1. One verifies that c1εν ≤ c1γ0α N+2 ν σb+1 ν ≤ αν − αν−1/Kν ·AKν , hence ckAkν‖Pν‖∗rν ≤ αν − αν−1. So by (3.11) the small divisor estimates hold for the new frequencies with parameter αν+1 up to |k| ≤ Kν . Removing from Πν the union of resonance zones Rν+1 k,l (αν+1) for |k| ≥ Kν we obtain the parameter domain Πν+1 ⊂ Πν with the desired proper- ties. � With (3.7), (3.8) and (3.10), we also obtain the following estimates. Lemma 4.2. For ν ≥ 0, 1 σν ‖Φν − id‖∗rν ,Dν+1 , ‖DΦ− Id‖∗rν ,rν ,Dν+1 ≤ cεν αN+2 ν σb+1 ν , (4.1) |ων+1 − ων |∗, ‖Ων+1 − Ων‖∗−δ ≤ c1εν , (4.2) |ων − ω|∗, ‖Ων − Ω‖∗−δ ≤ c1(ε1 + · · ·+ εν). (4.3) Proof of Theorem 1.1. Suppose the assumption of Theorem 1.1 are satisfied. To apply the Lemma 4.1 with ν = 0, set N0 = N , P0 = P , s0 = s, r0 = r, E0 = E, α0 = α, and γ = γ0α N+1 0 . The smallness condition is satisfied because ‖P0‖∗r0,D(s0,r0) = ‖P‖∗r,D(s,r) ≤ γα = γ0σ a+1 0 αN+1 0 = ε0. The small divisor conditions are satisfied by setting Π0 = Π \ ∪k,lR0 k,l(α0). Then the Iterative Lemma can be applied, and we can obtain a decreasing sequence of domains Dν ×Πν and S-invariant transformations {Φν = Φ1 ◦ · · · ◦ Φν} for ν ≥ 1, such that (Φν)∗X = Nν + Pν . Moreover, the estimates in Lemma 4.2 hold. To prove the convergence of {Φν}, we note that the operator norm ‖·‖r,s satisfies ‖A ·B‖r,s ≤ ‖A‖r,r · ‖B‖s,s for r ≥ s. We thus obtain ‖Φν+1 − Φν‖∗r1,Dν+1 ≤ ‖DΦν‖∗r1,rν ,Dν · ‖Φν+1 − id‖∗rν ,Dν+1 , (4.4) and ‖DΦν‖∗r1,rν ,Dν+1 ≤ ‖DΦ1‖∗r1,r1,D2 · · · ‖DΦν‖∗rν ,rν ,Dν+1 ≤ ν∏ j=1 ‖DΦj‖∗rj ,rj ,Dj+1 ≤ ν∏ j=1 (1 + 2−j) < +∞, 12 Z. LOU, Y. WU EJDE-2024/02 for all ν ≥ 0. Then we have ‖Φν+1 − Φν‖∗r1,Dν+1 ≤ c‖Φν+1 − id‖∗rν ,Dν+1 , So {Φν} is convergent on D∗ ×Π∗ = ⋂ ν≥1Dν ×Πν . Let limν→∞Φν = Φ∗ and (Φ∗) ∗X = N∗ + P∗, where N∗ = w∗ ∂ ∂θ + iΩ∗z ∂ ∂z − iΩ∗z̄ ∂ ∂z̄ . Since ω∗ = limν→∞ ων by (4.3) it follows that |ω∗ − ω|∗ ≤ cε. Since ‖Pν‖∗rν ,Dν ≤ εν and limν→∞ ‖Pν−P∗‖∗rν ,Dν = 0, it follows that P∗ = 0 on D∗×Π∗. By (4.4) it follows that ‖Φ∗ − id‖∗ ≤ cε. Now we estimate the measure of Π∗. By Theorem 1.2, meas(Π−Πν) ≤ c(diam(Pi)n−1α ν ν2 , so meas(Π−Π∗) ≤ ∑∞ ν=1 meas(Π−Πν) ≤ c · (diam(Pi)n−1αν . This implies that if α is sufficiently small, Π∗ is nonempty. � 5. Application to the beam equation 5.1. Reversible seting of beam equation. We consider the beam equation utt + uxxxx + m̄u+ uu2 t = 0, m̄ > 0, x ∈ [0, π], (5.1) with Navier boundary condition u(t, 0) = 0 = u(t, π), uxx(t, 0) = 0 = uxx(t, π). (5.2) We denote the operator D as Du = (∂xxxx + m̄)1/2u and introduce v = ut, then (5.1) takes the form ut = v, vt = −D2u− uu2 t . (5.3) We set w = 1√ 2 (Du− iv), w̄ = 1√ 2 (Du+ iv). (5.4) We then obtain wt = iDw − i 4 D−1(w + w̄) · (w − w̄)2, w̄t = −iDw + i 4 D−1(w + w̄) · (w − w̄)2. (5.5) The eigenvalues of the operator D are λj = √ j4 + m̄, j ≥ 1, and the corresponding eigenfunctions are φj(x) = √ 2 π sin(jx), which form a complete orthonormal basis in L2[0, π]. Let w = ∑ j≥1 qjφj(x) and w̄ = ∑ j≥1 qjφj(x). System (5.5) is then transformed into the lattice form q̇j = iλjqj +Q(qj), ˙̄qj = −iλj q̄j +Q(q̄j), (5.6) where Q(qσj ) = ∑ l,m,n≥1 ∑ α,β,γ∈{+,−} Q (qσj ) αβγ,lmnq α l q β mq γ n, (5.7) EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 13 Q (qσj ) αβγ,lmn = − iσ 4 λ−1 l almnj , σ ∈ {+,−}, (5.8) almnj = ∫ π 0 φl(x)φm(x)φn(x)φj(x)dx = { constant, ±l ±m± n = j, 0, otherwise. (5.9) In particular, alljj = { 3 2π , if l = j, 1 π , if l 6= j. The system (5.6) can be reinterpreted as a reversible context with respect to the involution S0(q, q̄) = (q̄, q). To this system we associate the S0-reversible vector field X̄(q, q̄) = Λ(q, q̄) +Q(q, q̄) (5.10) where Λ = ∑ σ=± Λ(qσ)(q, q̄) ∂ ∂qσ , Q = ∑ σ=± Q(qσ)(q, q̄) ∂ ∂qσ with Λ(qσ)(q, q̄) = (Λ(qσj ))j≥1 = (−σiλjq σ j )j≥1, Q(qσ)(q, q̄) = (Q(qσj )(q, q̄))j≥1. Now we establish the regularity of the nonlinear reversible vector field Q. To this end, let l2b and L2 be the Hilbert spaces of all bi-infinite, square summable sequences with complex coefficients and all square-integrable complex valued functions on [−π, π], respectively. Define the inverse discrete Fourier transform F : l2b → L2 by Fq = 1√ 2π ∑ j∈Z qje ijx, q = (qj)j∈Z ∈ l2b . This defines an isometry between the two spaces. For a ≥ 0 and p ≥ 0, define the subspaces la,pb ⊂ l2b consisting of all bi-infinite sequences q = (qj)j∈Z with norm ‖q‖2a,p = |q0|2 + ∑ j 6=0 |qj |2|j|2pe2|j|a. These subspaces induce subspaces W a,p ⊂ L2 under F , normed by ‖Fq‖a,p = ‖q‖a,p. Lemma 5.1. For a ≥ 0, p ≥ 0, the vector field Q is an analytic map from a neighborhood of the origin of la,p2 × la,p2 into la,p2 × la,p2 , with |||Q|||a,p = O(‖q‖3a,p), where the norm |||Q|||a,p = ‖Q(q)‖a,p + ‖Q(q̄)‖a,p. Proof. Let q ∈ la,p2 . Considered as the functions on [−π, π], w and w̄ are in W a,p with ‖w‖a,p ≤ ‖q‖a,p, ‖w̄‖a,p ≤ ‖q̄‖a,p. D−1w and D−1w̄ are in W a,p+2 with ‖D−1w‖a,p+2 ≤ ‖q‖a,p, ‖D−1w̄‖a,p+2 ≤ ‖q̄‖a,p, for m̄ > 0. We denote f(w, w̄) = − i 4 D−1(w + w̄) · (w − w̄)2. 14 Z. LOU, Y. WU EJDE-2024/02 By the algebraic property and the analyticity of f(w, w̄), the function f(w, w̄) also belongs to W a,p with ‖f(w, w̄)‖a,p = ‖ − i 4 D−1(w + w̄) · (w − w̄)2‖a,p ≤ c(‖D−1w · w2‖a,p + 2‖D−1w · ww̄‖a,p + ‖D−1w · w̄2‖a,p + ‖D−1w̄ · w2‖a,p + 2‖D−1w̄ · ww̄‖a,p + ‖D−1w̄ · w̄2‖a,p) ≤ c(‖D−1w‖a,p+2‖w‖a,p‖w‖a,p + 2‖D−1w‖a,p+2‖w‖a,p‖w̄‖a,p + ‖D−1w‖a,p+2‖w̄‖a,p‖w̄‖a,p + ‖D−1w̄‖a,p+2‖w‖a,p‖w‖a,p + 2‖D−1w̄‖a,p+2‖w‖a,p‖w̄‖a,p + ‖D−1w̄‖a,p+2‖w̄‖a,p‖w̄‖a,p) ≤ c‖q‖3a,p, in a sufficiently small neighborhood of the origin. By Q(qj) = ∫ π 0 f(w, w̄)φj(x)dx the components of the Fourier sine coefficients of f(w, w̄) . Therefore Q(q) belongs to la,p2 with ‖Q(q)‖a,p = ‖f(w, w̄)‖a,p ≤ c‖q‖3a,p, i.e., |||Q|||a,p = O(‖q‖3a,p). � 5.2. Birkhoff normal form. In this section, to apply the KAM theorem, we transform the reversible vector field X̄ into a Birkhoff normal form. Before proving the proposition, we give the following lemma. Lemma 5.2 ([10, Lemma 4.1]). If l,m, n, j ∈ Z0 satisfy αl + βm + γn − σj = 0 and ((l, α), (m,β), (n, γ), (j,−σ)) 6= ((p,+), (p,−), (q,+), (q,−)), then |αλl + βλm + γλn − σλj | ≥ 1√ (h4 + m̄) 3 , with h = min{l,m, n, j}. Proposition 5.3. There exists a real analytic, S0-invariant transformation Γ in a neighborhood of the origin of la,p2 × la,p2 , which transforms the S0-reversible vector field X̄ into Γ ∗X̄ = (DΓ )−1X̄ ◦ Γ = Λ + Q̄+ Q̂+K such that the corresponding fields Q̄, Q̂ and K are analytic maps from a neighbor- hood of the origin of la,p2 × la,p2 into la,p2 × la,p2 where |||K|||a,p = O(‖q‖5a,p). Proof. We denote Aā,j = {(l,m, n) ∈ N3 : min{l,m, n} ≤ ā}, Bā,σj = {(l,m, n) ∈ Aā,j : ((l, α), (m,β), (n, γ)) = ((p,+), (p,−), (q, σ)). We write Q = Q̄+Q̃+Q̂ where Q̄ = ∑ σ=±,j ∑ (l,m,m)∈Bā,σj ,α,β,γ Q (qσj ) αβγ,lmnq α l q β mq γ n ∂ ∂qσj , Q̃ = ∑ σ=±,j ∑ (l,m,m)∈Aā,j\Bā,σj ,α,β,γ Q (qσj ) αβγ,lmnq α l q β mq γ n ∂ ∂qσj , EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 15 Q̂ = ∑ σ=±,j ∑ (l,m,m)∈Acā,j ,α,β,γ Q (qσj ) αβγ,lmnq α l q β mq γ n ∂ ∂qσj . Then the reversible vector field X̄ becomes X̄ = Λ + Q̄+ Q̃+ Q̂. (5.11) Let Γ = φ1 F be the time−1-map of the flow generated by the vector field F = ∑ σ=± ∑ l,m,m,j≥1,α,β,γ F (qσj ) αβγ,lmnq α l q β mq γ n ∂ ∂qσj with coefficients F (qσj ) αβγ,lmn =  iQ (qσj ) αβγ,lmn αλl+βλm+γλn−σλj , if (l,m, n, α, β, γ) ∈ Aā,j \Bā,σj , 0, otherwise. Note that F is well-defined because of Lemma 5.2. By the Taylor series expansion of (φtF )∗X at t = 0, we have Γ∗X = (φ1 F )∗X = Λ + [Λ, F ] +Q+ ∫ 1 0 (1− t) ( φtF )∗ [[Λ, F ], F ] dt+ ∫ 1 0 ( φtF )∗ [Q,F ] dt. The last line constitutes the higher order term K. By a direct calculation, we have [Λ, qαl q β mq γ n ∂ ∂qσj ] = i(αλl + βλm + γλn − σλj)qαl qβmqγn ∂ ∂qσj . It follows that [Λ, F ] +Q = ∑ σ=± ∑ l,m,n,j≥1,α,β,γ [σi(αλl + βλm + γλn − σλj)F (qσj ) αβγ,lmn +Q (qσj ) αβγ,lmn]qαl q β mq γ n ∂ ∂qσj = Q̄+ Q̂. Hence formally we have Γ ∗X̄ = Λ + Q̄+ Q̂+K as claimed. Now we investigate the analyticity of Γ . We first prove that F is an analytic vector field on la,p2 × la,p2 of the order three at the origin. Indeed, from Lemma 5.2 we have |F (qj)| = | ∑ l,m,n≥1 ∑ α,β,γ F (qj) αβγ,lmnq α l q β mq γ n| ≤ ∑ ±l±m±n=j ∑ α,β,γ |F (qj) αβγ,lmn||q α l ||qβm||qγn| ≤ c1 ∑ ±l±m±n=j ∑ α,β,γ |qαl ||qβm||qγn|. Let wσj = (|qσj |), j ≥ 1, σ = ±, we have∑ ±l±m±n=j ∑ α,β,γ |qαl ||qβm||qγn| = ∑ ±l±m±n=j ∑ α,β,γ |wαl||wβm||wγn| 16 Z. LOU, Y. WU EJDE-2024/02 = ∑ ±l±m±n=j,l,m,n 6=0 |wl||wm||wn|. Let w̃j = |wj |+ |w−j |, j 6= 0, and for ∀j, w̃j ≥ |wj |. Then we prove that∑ ±l±m±n=j,l,m,n 6=0 |wl||wm||wn| ≤ c2(w̃ ∗ w̃ ∗ w̃)j . In fact, for each combination of ±l ±m± n = j, for instance, considering the sum∑ l−m−n=j |wl||wm||wn|. Let l′ = l, m′ = −m, n′ = n. Then |wl| = |wl′ | ≤ w̃l′ , |wm| = |w−m′ | ≤ w̃m′ , |wn| = |w−n′ | ≤ w̃n′ ,∑ l−m−n=j |wl||wm||wn| = ∑ l′+m′+n′=j |wl′ ||w−m′ ||w−n′ | ≤ ∑ l′+m′+n′=j w̃l′w̃m′w̃n′ . By the definition of discrete convolution,∑ l′+m′+n′=j w̃l′w̃m′w̃n′ = (w̃ ∗ w̃ ∗ w̃)j . Therefore, we obtain ‖F (q)‖a,p = √∑ j≥1 e2ajj2p|F (qj)|2 ≤ √∑ j 6=0 e2ajj2pc21c 2 2|(w̃ ∗ w̃ ∗ w̃)j |2 ≤ c1c2‖w̃ ∗ w̃ ∗ w̃‖a,p ≤ c1c2‖w̃‖3a,p ≤ c‖q‖3a,p, where c depends on h and m̄. The analyticity of F follows from the analyticity of each component function and its local boundedness. The estimate for K can get from ‖F (q)‖a,p = O(‖q‖3a,p) and ‖Q(q)‖a,p = O(‖q‖3a,p). By straightforward calculations, one can verify that F ◦ S0 = DS0 · F . So Γ is S0-invariant. This completes the proof. � 5.3. Applying the KAM Theorem. Given an index set J = {j1 ≤ j2 ≤ · · · ≤ jn} ⊆ N, and denote N1 = N\J . We introduce the action-angle coordinates (θ, I) and normal coordinates (z, z̄) by the transformation Ψ: q±jb = √ Ib + ξbe ±iθb , b = 1, 2, . . . , n, q±j = z±j , j ∈ N1, depending on parameters ξ = (ξ1, . . . , ξn) ∈ Rn+. We obtain a new vector field X = Ψ∗(Γ ∗X̄) = N + P (5.12) EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 17 where N = ω(ξ) ∂ ∂θ + iΩ(ξ)z ∂ ∂z − iΩ(ξ)z̄ ∂ ∂z̄ , (5.13) P = ∑ w∈{θ,I,z,z̄} Pw(θ, I, z, z̄; ξ) ∂ ∂w (5.14) with ωb = λjb − 1 4 n∑ k=1 λ−1 jk ajkjkjbjbξk, Ωj = λj − 1 4 n∑ k=1 λ−1 jk ajkjkjjξk (5.15) P (θb) = −1 4 n∑ k=1 λ−1 jk ajkjkjbjbIk − 1 4 ∑ l∈N1 λ−1 l alljbjb |zl|2 + (Q̂(qjb ) +K(qjb )) ◦Ψ 2iqjb − (Q̂(q̄jb ) +K(q̄jb )) ◦Ψ 2iq̄jb , (5.16) P (Ib) = (Q̂(qjb ) +K(qjb )) ◦Ψq̄jb + (Q̂(q̄jb ) +K(q̄jb )) ◦Ψqjb , (5.17) P (zσj ) = −σi 4 ( n∑ k=1 λ−1 jk ajkjkjjIkz σ j + ∑ l∈N1 λ−1 l alljj |zl|2zσj ) + (Q̂(qσj ) +K(qσj )) ◦Ψ. (5.18) Lemma 5.4. The vector field X is reversible with respect to the involution S(θ, I, z, z̄) = (−θ, I, z̄, z). The proof of the above lemma follows a similar approach as in [11, Lemma 6.4] and is omitted here. Verification of Assumptions (A1)–(A3). The frequency vector ω can be rewrit- ten as ω(ξ) = α̂+Aξ where α̂ =  λj1 λj2 ... λjn  , and A = (−1 4 )  λ−1 j1 aj1j1j1j1 λ−1 j2 aj2j2j1j1 λ−1 j3 aj3j3j1j1 · · · λ−1 jn ajnjnj1j1 λ−1 j1 aj1j1j2j2 λ−1 j2 aj2j2j2j2 λ−1 j3 aj3j3j2j2 · · · λ−1 jn ajnjnj2j2 λ−1 j1 aj1j1j3j3 λ−1 j2 aj2j2j3j3 λ−1 j3 aj3j3j3j3 · · · λ−1 jn ajnjnj3j3 ... ... ... . . . ... λ−1 j1 aj1j1jnjn λ−1 j2 aj2j2jnjn λ−1 j3 aj3j3jnjn · · · λ−1 jn ajnjnjnjn  . As we can find that |∂ω ∂ξ | = |A| = (−1 4 )nλ−1 j1 . . . λ−1 jn ( 3 2π + n− 1)( 1 2π )n−1 6= 0. It is easy to see rank{∂ω∂ξ } = n and there exist a positive constant M1 such that ‖ω‖CN ,1(Π) ≤M1. So the Assumption (A1) is satisfied. 18 Z. LOU, Y. WU EJDE-2024/02 From Taylor’s formula we know that Ωj = √ j4 + m̄− 1 4π ( ξ1 λj1 + · · ·+ ξn λjn ) = j2 +O(j−2), and there exist a positive constant M2 such that ‖Ωj − j2‖CN ,1(Π) ≤M2j −2, ∀j ≥ 1. So Assumption (A2) is satisfied. From Proposition 5.3 and (5.12) we know that the perturbation P is analytic, thus Assumption (A3) holds. And we have verified all the assumption of Theorem 1.1. 6. Appendix Definition 6.1 (Infinite-dimensional reversible system). Suppose S is an involution map: S(θ, I, z, z̄) = (−θ, I, z̄, z) on Pa,p. We say that an infinite-dimensional system  θ̇ İ ż ˙̄z  = X(θ, I, z, z̄) is called reversible with respect to S (or S-reversible), if X ◦ S = −DS ·X. i.e., X(S(θ, I, z, z̄)) = −DS(θ, I, z, z̄) ·X(θ, I, z, z̄), ∀(θ, I, z, z̄) ∈Pa,p, where DS(θ, I, z, z̄) is the tangent map of S. Definition 6.2. A transformation Φ is called invariant with respect to above in- volution S (or S-invariant) if Φ ◦ S = S ◦ Φ. Definition 6.3. Suppose S is an involution map: S2 = id. A vector field X is called invariant with respect to S (or S-invariant), if DS ·X = X ◦ S. Lemma 6.4. If X is S-reversible, Y is S-invariant and the transformation Φ is S-invariant, then [X,Y ] and Φ∗X are both S-reversible. In particular, the flow φtY of Y are S-invariant, thus (φtY )∗X is S-reversible. Lemma 6.5 ([17]). The convolution w ∗ v of two complex sequences w, v in la,p2 is defined as (w ∗ v)j = ∑ m wj−mvm. If a ≥ 0, p > 1 2 , then la,p2 is a Hilbert algebra with respect to the convolution of sequences, and ‖w ∗ v‖p ≤ c‖w‖p‖v‖p, the constant c depends only on p. Lemma 6.6 ([16, Lemma A.1]). If A = (Aij) is a bounded linear operator on l2, then B = (Bij) with Bij = |Aij | |i− j| , i 6= j, and Bii = 0 is a bounded linear operator on l2, and ‖B‖ ≤ Π√ 3 ‖A‖. Lemma 6.7 (Cauchy’s estimate,[16, Lemma A.3]). Let E and F be two complex Banach spaces with the norms ‖ · ‖E and ‖ · ‖F . The first-order derivatives dvG of G at v is a linear mapping from E to F , whose operator norm is defined by ‖dvG‖ = sup u6=0 ‖dvG(u)‖F ‖u‖E . EJDE-2024/02 DEGENERATE INFINITE-DIMENSIONAL REVERSIBLE SYSTEMS 19 If G is analytic on Br, the open ball of radius r around v in E, then ‖dvG‖F,E 6 1 r sup u∈Br ‖G(u)‖F . Lemma 6.8 ([6, Lemma A.4]). Let k̄ = (k, 1), with k ∈ Zn, ω̄ = (ω,Ω) with ω ∈ Rn and Ω ∈ R1, and O+ = { ω̄|〈k̄, ω̄〉| > α |k|τ ,∀k = (k, 1), k ∈ Zn } . Let Gk̄ = 1 〈k̄,ω̄〉 , then Gk̄ is infinitively differentiable on O+ with respect to ω̄ in the sense of Whitney and the lth derivatives of Gk(ω̄) is ∂l ∂ω̄l Gk(ω̄) = (−1)|l|l!kl[Gk̄(ω̄)]|l|+1. Lemma 6.9 ([21, Lemma A.5]). If f1 and f2 belong to CN ,1(Π), then f1 ·f2 belongs to CN ,1(Π) and ‖f1 · f2‖CN ,1(Π) 6 ‖f1‖CN ,1(Π) · ‖f2‖CN ,1(m). Lemma 6.10 ([21, Lemma A.6]). Let f(Φ, ξ) be analytic in Φ on D ⊂ Pa,p and belong to CN ,1(Π) in ξ, where D is an open set of Pa,p. Let ‖ · ‖ be the norm of Pa,p. Denote by D−ϕ = {Φ ∈ D,dist(Φ, ∂D) > ρ}, where ∂D is the boundary of D. If Φ : ϕ ∈ D0 ⊂ D → Φ(ϕ; ξ)εD−φ, for all ξ ∈ Π, and supϕ∈D0 ‖Φ(ϕ; ·)− id‖∗ 6 ρ, then for ∀ϕ ∈ D0, ‖f [Φ(ϕ; ·), ·]‖∗ 6 c sup Φ∈D ‖f(Φ, ·)‖∗, where ‖ · ‖∗ denotes the norm of CN ,1(Π) and c only depends on n and N . Lemma 6.11 ([21, Lemma A.7]). Let D be an open domain in a complex Banach space E with the norm ‖ · ‖. X : (Φ, ξ) ∈ D × Π → E is a parameter dependent vector field on D. Suppose X to be analytic in Φ on D and belong to CN ,1(Π) in ξ. If supφ∈D ‖X(Φ; ·)‖∗ 6 ρ, then for each ξ ∈ Π, its flow Φt(·; ξ) exists on D−ρ for |t| 6 1 and maps D−2ρ into D−ρ, where D−ρ and ‖ · ‖∗ are defined as those in Lemma6.10. Moreover, on D−2ρ ‖Φ(·; ·)− id‖∗ 6 c sup Φ∈D ‖X(Φ; ·)‖∗. Acknowledgments. We would like to extend our sincere gratitude to the anony- mous reviewers for their valuable comments and suggestions, which significantly improved the quality of this article. This research was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11901291), and by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20190395). References [1] V. Arnold; Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963),no. 6, 91–192. [2] P. Baldi, M. Berti, R. Montalto; KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014),no. 1-2, 471–536. [3] D. Bambusi, M. Berti, E. Magistrelli; Degenerate KAM theory for partial differential equa- tions, J. Differential Equations, 250 (2011),no. 8, 3379–3397. [4] M. Berti, L. Biasco, M. Procesi; KAM for reversible derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014),no. 3, 905–955. 20 Z. LOU, Y. WU EJDE-2024/02 [5] A. Brjuno; Nondegeneracy conditions in the Kolmogorov theorem, Dokl. Akad. Nauk, 33 (1992),no. 6, 1028–1032. [6] C. Cheng, Y. Sun; Existence of KAM tori in degenerate Hamiltonian systems, J. Differential Equations, 114 (1994),no. 1, 288–335. [7] R. Feola, M. Procesi; Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations, 259 (2015),no. 7, 3389–3447. [8] M. Gao, J. Liu; A degenerate KAM theorem for partial differential equations with periodic boundary conditions, Discrete Contin. Dyn. Syst., 40 (2020),no. 10, 5911–5928. [9] C. Ge, J. Geng, Z. Lou; KAM theory for the reversible perturbations of 2D linear beam equations, Math. Z., 297 (2021),no. 3-4, 1693–1731. [10] J. Geng, J. You; KAM tori of Hamiltonian perturbations of 1D linear beam equations, J. Math. Anal. Appl., 227 (2003),no. 1, 104–121. [11] Z. Lou, J. Si; Quasi-periodic solutions for the reversible derivative nonlinear Schrödinger equa- tions with periodic boundary conditions, J. Dynam. Differential Equations, 29 (2017),no. 3, 1031–1069. [12] Z. Lou, J. Si; Periodic and quasi-periodic solutions for reversible unbounded perturbations of linear Schrödinger equations, J. Dynam. Differential Equations, 32 (2020),no. 1, 117–161. [13] Z. Lou, J. Si, S. Wang; Invariant tori for the derivative nonlinear Schrödinger equation with nonlinear term depending on spatial variable, Discrete Contin. Dyn. Syst., 42 (2022),no. 9, 4555–4595. [14] Z. Lou, Y. Sun; A KAM theorem for higher dimensional reversible nonlinear Schrödinger equations, Electron. J. Differential Equations, (2022),no. 69, 25pp. [15] A. Pjartli; Diophantine approximations of submanifolds of a Euclidean space, Funkcional. Anal. i Priložen., 3 (1969),no. 4, 59–62. [16] J. Pöschel; A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996),no. 1, 119–148. [17] J. Pöschel; Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helv., 71 (1996),no. 2, 269–296. [18] H. Rüssmann; Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001),no. 2, 119–204. [19] X. Wang, J. Xu, D. Zhang; Degenerate lower dimensional tori in reversible systems, J. Math. Anal. Appl., 387 (2012),no. 2, 776–790. [20] X. Wang, J. Xu, D. Zhang; On the persistence of degenerate lower-dimensional tori in re- versible systems, Ergodic Theory Dynam. Systems, 35 (2015),no. 7, 2311–2333. [21] J. Xu, Q. Qiu, J. You; A KAM theorem of degenerate infinite-dimensional Hamiltonian systems. I, II, Sci. China Ser. A, 39 (1996),no. 4, 372–383, 384–394. [22] J. Xu,, J. You, Q. Qiu; Invariant tori for nearly integrable Hamiltonian systems with degen- eracy, Math. Z., 226 (1997),no. 3, 375–387. [23] X. Yang, J. Xu; Persistence of multi-dimensional degenerate hyperbolic lower dimensional invariant tori in reversible systems, J. Differential Equations, 346 (2023), 229–253. [24] J. Zhang, M. Gao, X. Yuan; KAM tori for reversible partial differential equations, Nonlin- earity, 24 (2011),no. 4, 1189–1228. Zhaowei Lou School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China Email address: zwlou@nuaa.edu.cn Youchao Wu School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China Email address: 2414483646@qq.com 1. Introduction 2. Homological equation 3. KAM step 3.1. Approximation estimates 3.2. Solution of the homological equation 3.3. Coordinate transformation 3.4. New vector field 3.5. New normal form 4. Iteration and convergence 5. Application to the beam equation 5.1. Reversible seting of beam equation 5.2. Birkhoff normal form 5.3. Applying the KAM Theorem Verification of Assumptions (A1)–(A3) 6. Appendix Acknowledgments References