Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 69, pp. 1–25. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu A KAM THEOREM FOR HIGHER DIMENSIONAL REVERSIBLE NONLINEAR SCHRÖDINGER EQUATIONS ZHAOWEI LOU, YINGNAN SUN Abstract. In this article we prove an abstract Kolmogorov-Arnold-Moser (KAM) theorem for infinite dimensional reversible systems. Using this theo- rem, we obtain the existence of quasi-periodic solutions for a class of reversible (non-Hamiltonian) coupled nonlinear Schrödinger systems on a d-torus. 1. Introduction and main result Among various techniques for studying the existence of quasi-periodic solutions of nonlinear partial differential equations (PDEs), the Kolmogorov-Arnold-Moser (KAM) theory is one of the most powerful tools. Kuksin [12] and Wayne [18] first developed a Newtonian scheme to investigate quasi-periodic solutions of Hamilton- ian PDEs in one dimensional space. The general idea is that Hamiltonian function is thought of as a normal form plus a real analytic perturbation, then constructing an infinite symplectic transformation sequences to make the perturbation smaller and smaller and construct a converged local normal form. The normal form is help- ful to understand the dynamics around the quasi–periodic solutions, for example, one sees the linear stability and zero Lyapunov exponents. The feasibility of the KAM method in one dimensional Hamiltonian PDEs, how- ever, depends crucially on the second Melnikov condition. Because of the mul- tiple eigenvalues of the linear operator, such condition is not naturally available in higher dimensional case, and the KAM method is in general not easy to ap- ply. In 1998, Bourgain [3] first made a breakthrough. He used multi-scale analysis method to avoid the cumbersome second Melnikov condition and thus obtained small-amplitude quasi-periodic solutions of two dimensional nonlinear Schrödinger equations (NLS). Later, he improved his method and studied quasi-periodic solu- tions of NLS and nonlinear wave equations in any dimensional space. Following the idea and methods in [3], many works [4, 2, 17] have been done. There are strong hopes to develop the KAM theory for higher dimensional PDEs, because multi-scale analysis can not help us to understand the dynamics around quasi-periodic solutions. Geng and You [9] first built a KAM theorem for higher dimensional beam equation and nonlocal smooth NLS. They used momentum con- servation condition which means the nonlinearity is independent of spatial variable x to avoid the difficulty of multiple eigenvalues. In 2010, Eliasson and Kuksin 2020 Mathematics Subject Classification. 37K55, 35B15. Key words and phrases. KAM theorem; reversible vector field; quasi-periodic solution; nonlinear Schrödinger equation. ©2022. This work is licensed under a CC BY 4.0 license. Submitted May 2, 2022. Published October 10, 2022. 1 2 Z. LOU, Y. SUN EJDE-2022/69 [6] studied a class of higher dimensional NLS with convolution type potential and nonlinearity containing spatial variable x. They used the block diagonal normal form structure to deal with multiple eigenvalues of linear operator. Also, they introduced Lipschitz domain property of perturbation to handle infinitely many resonances at each KAM step. By developing Töplitz-Lipschitz property of per- turbation and constructing appropriate tangential sites on Z2, Geng, Xu and You [8] obtained quasi-periodic solutions of two-dimensional completely resonant NLS. Later on, Geng and You [11] simplified the proof of [6] via momentum conservation condition. Procesi and Procesi [15] extended the result in [8] to the d-dimensional case by a very ingenious choice of tangential sites. See [5, 16, 19, 10] for further studies. Recently, KAM theory for Hamiltonian PDEs has been generalized to reversible ones in one dimensional space [20, 1]. In fact, reversible PDEs are a class of phys- ically important PDEs as well as Hamiltonian ones. For example, the following coupled NLS system arising from nonlinear optics (see [13]): iut −∆u+Mξu+ ∂ūG1(|u|2, |v|2) = 0, ivt −∆v +Mξ̃v + ∂v̄G2(|u|2, |v|2) = 0, x ∈ Td := Rd/2πZd, (1.1) where Mξ and Mξ̃ are real Fourier multiplier, Gi = o(|u|3 + |v|3), i = 1, 2 are real analytic functions near (u, v) = (0, 0). When G1 = G2, equation (1.1) is not only reversible (with respect to the involution S0(u(x), v(x)) = (ū(−x), v̄(−x))) but also Hamiltonian. When G1 6= G2, equation (1.1) is no longer Hamiltonian but still reversible. This motives us to develop reversible KAM theory for equation (1.1). As in the Hamiltonian case, the major difficulty in constructing KAM scheme for (1.1) is to deal with infinitely many resonances. In this article, by introducing the class of Töplitz-Lipschitz vector fields (inspired by [6, 8, 1]) and momentum conservation condition, the difficulty can be overcome. Töplitz-Lipschitz vector field plays the most essential role and it reduces infinitely many resonances to only finitely many ones. Momentum conservation condition can simplify the proof. We mention that Töplitz-Lipschitz vector field introduced here is the generalization of Töplitz-Lipschitz functions in [8]. Following [6], we could study more general equation (1.1) with nonlinearities Gi containing the spatial variable x explicitly, but the proof would be more complicated since we have to deal with block diagonal normal form. This article is working on NLS with the external parameters, and the completely resonant case (i.e. no Fourier multiplier in equation (1.1)) is in [7]. As in the Hamiltonian case (see [8, 15]), the construction of Birkhoff normal form will be a new challenge. 1.1. Main result. Let I1 = {i(1), i(2), . . . , i(n)} ⊂ Zd, I2 = {̃i(1), ĩ(2), . . . , ĩ(m)} ⊂ Zd be two sets of distinguished sites of Fourier modes. For technical convenience, we suppose 0 ∈ I1∩I2. We denote by λi, i ∈ Zd (resp. λ̃i) the eigenvalues of −∆+Mξ (resp. −∆ +Mξ̃) under periodic boundary conditions: λi(j) = ωj = |i(j)|2 + ξj , 1 ≤ j ≤ n, λi = |i|2, i /∈ I1, EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 3 λ̃ĩ(j) = ω̃j = |̃i(j)|2 + ξ̃j , 1 ≤ j ≤ m, λ̃i = |i|2, i /∈ I2, and the corresponding eigenfunctions φi(x) = 1√ (2π)d ei〈i,x〉. Assume the parameters (ξ, ξ̃) ∈ O := [0, 1]n × [0, 1]m ⊂ Rn+m. Then we have the following main result. Theorem 1.1. For any 0 < γ � 1, there exists a Cantor subset Oγ ⊂ O with meas(O \ Oγ) = O(γ1/4), such that for any (ξ, ξ̃) ∈ Oγ , equation (1.1) with re- versible perturbation G1 = |u|4|v|2 + o((|u|2 + |v|2)3) 6= G2 = |u|2|v|2 + o((|u|2 + |v|2)2) possesses a small amplitude quasi-periodic solution of the form u(t, x) = ∑ i∈Zd ui(ώ1t, . . . , ώnt)φi(x), v(t, x) = ∑ i∈Zd vi(´̃ω1t, . . . , ´̃ωmt)φi(x), (1.2) where ui : Tn → R (resp. vi : Tm → R ) and ώ1, . . . , ώn (resp. ´̃ω1, . . . , ´̃ωm ) are close to the unperturbed frequencies ω1, . . . , ωn (resp. ω̃1, . . . , ω̃m ). Moreover, the quasi-periodic solutions are real analytic and may not be linear stable. Remark 1.2. In the following proof, we set the nonlinearities G1 = |u|4|v|2 + o((|u|2 + |v|2)3) and G2 = |u|2|v|2 + o((|u|2 + |v|2)2) to simplify notations. Our method also applies to the general cases Gi = (|u|2 + |v|2)ni + o((|u|2 + |v|2)ni), ni ∈ Z, ni ≥ 2, i = 1, 2. We point out that the reversible coupled nonlinearities may lead to the linear in- stability of KAM tori. Note that because G1 6= G2 in equation (1.1), the eigenvalues of Mj,∞ = ( Ωj,∞ Aj,∞ Ãj,∞ Ω̃j,∞ ) , j ∈ Zd1 ∩ Zd2 after KAM iteration (see (5.16)) may not be real numbers. Thus we can not obtain the linear stability of quasi-periodic solution (1.2). But for the case of Hamiltonian coupled nonlinearities G1 = G2, Aj,∞ = Ãj,∞ (i.e., Mj,∞ is symmetric), this means the eigenvalues of Mj,∞ are real numbers and thus their result is linearly stable. For the single (non-coupled) reversible PDEs in [1], Mj,∞ is the scalar normal frequency Ωj,∞, thus their result is also linearly stable. The rest of this article is organized as follows. In Section 2, we give the definitions of weighted norms for functions and vector fields. An abstract KAM theorem (Theorem 3.2) for infinite dimensional reversible systems is presented in Section 3. In Section 4, we use the KAM theorem to prove Theorem 1.1. The proof of Theorem 3.2 is given in Section 5. Some properties of reversible system and technical lemmas are listed in the Appendix. 2. Preliminaries For the sake of completeness, we first introduce some definitions and notation. Let I ⊂ Zd be a finite subset and ρ > 0. We introduce the Banach space `ρI of all 4 Z. LOU, Y. SUN EJDE-2022/69 complex sequences z = (zj)j∈Zd\I with ‖z‖ρ = ∑ j∈Zd\I e|j|ρ|zj | <∞, where |j| = √ |j1|2 + · · ·+ |jd|2. Given two subsets of Zd : I1 = {i(1), i(2), . . . , i(n)} and I2 = {̃i(1), ĩ(2), . . . , ĩ(m)}, we denote Zdl := Zd \ Il and `ρl := `ρIl , (l = 1, 2). We consider the phase space Pρ := Tn × Tm × Rn × Rm × `ρ1 × ` ρ 2 × ` ρ 1 × ` ρ 2 3 y := (θ, ϕ, I, J, z, w, z̄, w̄). We introduce a complex neighborhood Dρ(r, s) = { y : | Im θ| < r, | Imϕ| < r, |I| < s, |J | < s, ‖z‖ρ < s, ‖w‖ρ < s, ‖z̄‖ρ < s, ‖w̄‖ρ < s } of T n+m 0 := Tn×Tm×{I = 0}×{J = 0}×{z = 0}×{w = 0}×{z̄ = 0}×{w̄ = 0}, where | · | is the sup-norm. Suppose O ⊂ Rn+m is a compact parameter subset. A function f : Dρ(r, s) × O → C is real analytic in y and C4 W (i.e., C4-smooth in the sense of Whitney) in ζ ∈ O and has Taylor-Fourier series expansion f(y; ζ) = ∑ k∈Zn,l∈Nn,α,β∈NZd1 , k̃∈Zm,l̃∈Nm,α̃,β̃∈NZd2 fklαβ,k̃l̃α̃β̃(ζ)ei(〈k,θ〉+〈k̃,ϕ〉)I lJ l̃zαz̄βwα̃w̄β̃ , where 〈k, θ〉 = ∑n i=1 kiθi, I l = ∏n i=1 I li i , and zαz̄β = ∏ j∈Zd1 z αj j z̄ βj j . α, β have only finitely many nonzero components, and similarly for the other indexes. We define the weighted norm of f as ‖f‖Dρ(r,s)×O = sup |z|ρ 0, such that |〈k, ω〉+ 〈k̃, ω̃〉| ≥ γ (|k|+ |k̃|)τ , (k, k̃) 6= 0, (3.5) |〈k, ω〉+ 〈k̃, ω̃〉+ Ωi| ≥ γ (|k|+ |k̃|)τ , i ∈ Zd1 \ Zd2, (3.6) |〈k, ω〉+ 〈k̃, ω̃〉+ Ω̃i| ≥ γ (|k|+ |k̃|)τ , i ∈ Zd2 \ Zd1, (3.7) |〈k, ω〉+ 〈k̃, ω̃〉+ Ωi ± Ωj | ≥ γ (|k|+ |k̃|)τ , (k, k̃) 6= 0, i, j ∈ Zd1 \ Zd2, (3.8) |〈k, ω〉+ 〈k̃, ω̃〉+ Ωi ± Ω̃j | ≥ γ (|k|+ |k̃|)τ , i ∈ Zd1 \ Zd2, j ∈ Zd2 \ Zd1, (3.9) |〈k, ω〉+ 〈k̃, ω̃〉+ Ω̃i ± Ω̃j | ≥ γ (|k|+ |k̃|)τ , (k, k̃) 6= 0, i, j ∈ Zd2 \ Zd1, (3.10) |det((〈k, ω〉+ 〈k̃, ω̃〉)I2 ±Mi)| ≥ γ (|k|+ |k̃|)τ , i ∈ Zd1 ∩ Zd2, (3.11) |det((〈k, ω〉+ 〈k̃, ω̃〉+ Ωi)I2 ±Mj)| ≥ γ (|k|+ |k̃|)τ , (i, j) or (j, i) ∈ (Zd1 ∩ Zd2)× (Zd1 \ Zd2), (3.12) |det((〈k, ω〉+ 〈k̃, ω̃〉+ Ω̃i)I2 ±Mj)| ≥ γ (|k|+ |k̃|)τ , (i, j) or (j, i) ∈ (Zd1 ∩ Zd2)× (Zd2 \ Zd1), (3.13) |det((〈k, ω〉+ 〈k̃, ω̃〉)I4 +Mi ⊗ I2 ± I2 ⊗MT j )| ≥ γ (|k|+ |k̃|)τ , (k, k̃) 6= 0, i, j ∈ Zd1 ∩ Zd2. (3.14) where Ib is b × b identity matrix, det(·) is the determinant, ⊗ is the the tensor product, and (·)T is the transpose of matrices. EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 7 (A4) Regularity: A + P is real analytic in y and C4 W -smooth in ζ. Moreover, ‖A‖s;D(r,s)×O < 1, ε0 := ‖P‖s;D(r,s)×O <∞. (A5) Momentum conservation: The perturbation P satisfies [P,Ml] = 0 (l = 1, . . . , d), where Ml = n∑ b=1 i (b) l ∂ ∂θb + m∑ b=1 ĩ (b) l ∂ ∂ϕb + ∑ %=± ∑ j∈Zd1 %ijlz % j ∂ ∂z%j + ∑ %=± ∑ j∈Zd2 %ijlw % j ∂ ∂w%j . (A6) Töplitz-Lipschitz property: Let Λ = ∑ σ=± σi( ∑ j∈Zd1 Ω0 j (ζ)zσj ∂ ∂zσj + ∑ j∈Zd2 Ω̃0 j (ζ)wσj ∂ ∂wσj ). For fixed i, j ∈ Zd, c ∈ Zd \ {0}, the following limits exist and satisfy: ‖ lim t→∞ ∂P (x) ∂uσi+tc ‖s;Dρ(r,s)×O ≤ ε0, x = θb, ϕb, Ib, Jb, u = z, w; (3.15) ‖ lim t→∞ ∂(Λ + P )(uσi+tc) ∂u±σj±tc ‖s;Dρ(r,s)×O ≤ ε0e−|i∓j|ρ, u = z, w; (3.16) ‖ lim t→∞ ∂(A+ P )(uσi+tc) ∂v±σj±tc ‖s;Dρ(r,s)×O ≤ ε0e−|i∓j|ρ, (u, v) = (z, w), (w, z). (3.17) Furthermore, there exists K > 0 such that when |t| > K, the following estimates hold. ‖ ∂P (x) ∂uσi+tc − lim t→∞ ∂P (x) ∂uσi+tc ‖s;Dρ(r,s)×O ≤ ε0, x = θb, ϕb, Ib, Jb;u = z, w; (3.18) ‖∂(Λ + P )(uσi+tc) ∂u±σj±tc − lim t→∞ ∂(Λ + P )(uσi+tc) ∂u±σj±tc ‖s;Dρ(r,s)×O ≤ ε0 |t| e−|i∓j|ρ, u = z, w; (3.19) ‖∂(A+ P )(uσi+tc) ∂v±σj±tc − lim t→∞ ∂(A+ P )(uσi+tc) ∂v±σj±tc ‖s;Dρ(r,s)×O ≤ ε0 |t| e−|i∓j|ρ, (u, v) = (z, w), (w, z). (3.20) Remark 3.1. In (A6), the conditions (3.16)-(3.17) and (3.19)-(3.20) are the most important for measure estimates. The role played by the conditions (3.15) and (3.18) is to preserve Töplitz-Lipschitz property after the KAM iteration (see Lem- mas 5.5 and 5.6 below). Now we state our KAM theorem. Theorem 3.2. Suppose the S-reversible vector field X = N+A+P in (3.3) satisfies (A1)–(A6). γ > 0 is small enough. Then there exists a positive ε depending only on n,m,L,K, τ, r, s and ρ such that if ‖P‖s;D(r,s)×O ≤ ε, the following holds: There exist (1) a Cantor subset Oγ ⊂ O with Lebesgue measure meas(O \Oγ) = O(γ1/4); (2) a C4 W -smooth family of real analytic torus embeddings Ψ : Tn+m ×Oγ → Dρ(r, s) 8 Z. LOU, Y. SUN EJDE-2022/69 which is ε γ20 -close to the trivial embedding Ψ0 : Tn+m ×O → T n+m 0 ; (3) a C4 W -smooth map φ : Oγ → Rn+m which is ε−close to the unperturbed frequency (ω, ω̃) such that for every ζ ∈ Oγ and (θ, ϕ) ∈ Tn+m the curve t 7→ Ψ((θ, ϕ) + φ(ζ)t; ζ) is a quasi-periodic solution of the equation governed by the vector field X = N +A+ P . 4. Application to the coupled NLS 4.1. Lattice form of equation (1.1). Let I1 = {i(1), i(2), . . . , i(n)} ⊂ Zd and I2 = {̃i(1), ĩ(2), . . . , ĩ(m)} ⊂ Zd and 0 ∈ I1∩I2. Under periodic boundary conditions, we denote the eigenvalues of −∆ +Mξ and −∆ +Mξ̃ by λi, i ∈ Zd and λ̃i, i ∈ Zd, respectively, satisfying ωj = λi(j) = |i(j)|2 + ξj , 1 ≤ j ≤ n, Ωi = λi = |i|2, i /∈ I1, ω̃j = λ̃ĩ(j) = |̃i(j)|2 + ξ̃j , 1 ≤ j ≤ m, Ω̃i = λ̃i = |i|2, i /∈ I2, and the corresponding eigenfunctions φi(x) = (2π)−d/2ei〈i,x〉. Without loss of generality, we consider (1.1) when G1 = |u|4|v|2, G2 = |u|2|v|2 since the higher order terms of nonlinearities will not make any difference. Let u(t, x) = ∑ h∈Zd qh(t)φh(x), v(t, x) = ∑ h∈Zd ph(t)φh(x), then we obtain the equivalent lattice reversible equations q̇h = iλhqh +Q(qh)(q, p, q̄, p̄), ṗh = iλ̃hph + Q̃(ph)(q, p, q̄, p̄) ˙̄qh = −iλhq̄h +Q(q̄h)(q, p, q̄, p̄), ˙̄ph = −iλ̃hp̄h + Q̃(p̄h)(q, p, q̄, p̄), h ∈ Zd, (4.1) which is reversible with respect to S(q, p, q̄, p̄) = (q̄, p̄, q, p), where Q(qh) = ∑ i,j,k,l,m∈Zd Q (qh) ijklmqiqjpkq̄lp̄m, Q(q̄h) = Q(qh), (4.2) Q̃(ph) = ∑ i,j,k∈Zd Q̃ (ph) ijk qipj q̄k, Q̃(p̄h) = Q̃(ph) (4.3) with Q (qh) ijklm = 2i ∫ Td φiφjφkφ̄lφ̄mφ̄hdx = { 2i (2π)2d , i+ j + k − l −m− h = 0, 0, i+ j + k − l −m− h 6= 0, (4.4) and Q̃ (ph) ijk = i ∫ Td φiφj φ̄kφ̄hdx = { i (2π)d , i+ j − k − h = 0, 0 i+ j − k − h 6= 0, (4.5) By direct computation, one can verify that the perturbations Q(q) = (Q(qh))h∈Zd1 and Q̃(p) = (Q̃(ph))h∈Zd2 have the following regularity properties. EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 9 Lemma 4.1. For each fixed ρ > 0, Q(q) (resp. Q̃(p)) is real analytic as a map in a neighborhood of the origin with ‖Q(q)‖ρ ≤ c‖(q, p)‖5ρ, (resp. ‖Q̃‖(p)ρ ≤ c‖(q, p)‖3ρ). Let P 0 = ∑ %=± (Q(q%) ∂ ∂q% + Q̃(p%) ∂ ∂p% ). Lemma 4.2. (1) [P 0,M0 l ] = 0, for l = 1, . . . , d, where M0 l = ∑ %=± ∑ j∈Zd %ijlq % j ∂ ∂q%j + ∑ %=± ∑ j∈Zd %ijlp % j ∂ ∂p%j ; (4.6) (2) P 0 satisfies Töplitz-Lipschitz property. Proof. (1) If we write Q(q%h) = ∑ α,β,α̃,β̃ Q (q%h) αβα̃β̃ qαq̄βpα̃p̄β̃ , Q̃(p%h) = ∑ α,β,α̃,β̃ Q̃ (p%h) αβα̃β̃ qαq̄βpα̃p̄β̃ , (4.7) then by (4.4) and (4.5), we have Q (q%h) αβα̃β̃ = 0 and Q̃ (p%h) αβα̃β̃ = 0 when πl(αβ, α̃β̃; v) 6= 0, v = q%h, p % h. where πl(αβ, α̃β̃; v) = ∑ j∈Zd (αj − βj)jl + ∑ j∈Zd (α̃j − β̃j)jl − %hl. Note that by elementary computations, we have [qαq̄βpα̃p̄β̃ ,M0 l ] = iπl(αβ, α̃β̃; v)qαq̄βpα̃p̄β̃ , which implies [P 0,M0 l ] = 0. (2) We only consider limt→∞ ∂Q(qi+tc) ∂qj+tc . It follows from (4.2) and (4.4) that ∂Q(qi) ∂qj = ∑ n+k+l−m=i−j 4i (2π)2d qnpkq̄lp̄m, then ∂Q(qi+tc) ∂qj+tc = ∂Q(qi) ∂qj = lim t→∞ ∂Q(qi+tc) ∂qj+tc . � 4.2. Verification of assumptions (A1)–(A6). We introduce the action-angle variables (θ, ϕ, I, J) and normal coordinates (z, w, z̄, w̄) by the following transfor- mation Ψ on some D(r, s), (r, s > 0): qi(j) = √ Ij + I0 j eiθj , q̄i(j) = √ Ij + I0 j e−iθj , j = 1, . . . , n, pĩ(j) = √ Jj + J0 j eiϕj , p̄ĩ(j) = √ Jj + J0 j e−iϕj , j = 1, . . . ,m, qh = zh, q̄h = z̄h, h /∈ I1, ph = wh, p̄h = w̄h, h /∈ I2, (4.8) where the I0 j and J0 j are fixed numbers satisfying 0 < s < I0 j and J0 j < 2s. We obtain a new vector field X(θ, ϕ, I, J, z, z̄, w, w̄) = N(θ, ϕ, I, J, z, z̄, w, w̄) + P (θ, ϕ, I, J, z, z̄, w, w̄), (4.9) 10 Z. LOU, Y. SUN EJDE-2022/69 where N = ω ∂ ∂θ + ω̃ ∂ ∂ϕ + iΩz ∂ ∂z + iΩ̃w ∂ ∂w − iΩz̄ ∂ ∂z̄ − iΩ̃w̄ ∂ ∂w̄ and P = ∑ w∈{θ,ϕ,I,J,z,z̄,w,w̄} P (w)(θ, ϕ, I, J, z, z̄, w, w̄; ξ, ξ̃) ∂ ∂w (4.10) with ωb = |i(b)|2 + ξb, 1 ≤ b ≤ n, ω̃b = |̃i(b)|2 + ξ̃b, 1 ≤ b ≤ m, Ωh = |h|2, h /∈ I1, Ω̃h = |h|2, h /∈ I2; and P (θb) = 1 2iqi(b) Q(q i(b) ) ◦Ψ− 1 2iq̄i(b) Q(q̄ i(b) ) ◦Ψ. (4.11) P (Ib) = q̄i(b)Q (q i(b) ) ◦Ψ + qi(b)Q (q̄ i(b) ) ◦Ψ, (4.12) P (zσh) = Q(qσh) ◦Ψ, σ = ±; (4.13) P (ϕb) = 1 2ipĩ(b) Q̃(p ĩ(b) ) ◦Ψ− 1 2ip̄ĩ(b) Q̃(p̄ ĩ(b) ) ◦Ψ; (4.14) P (Jb) = p̄ĩ(b)Q̃ (p ĩ(b) ) ◦Ψ + pĩ(b)Q̃ (p̄ ĩ(b) ) ◦Ψ; (4.15) P (wσh) = Q̃(pσh) ◦Ψ, σ = ±. (4.16) Here X is reversible with respect to the involution S(θ, ϕ, I, J, z, w, z̄, w̄) = (−θ,−ϕ, I, J, z̄, w̄, z, w). Now we verify assumptions (A1)–(A6) for (4.9): (A1) Set ζ = (ξ, ξ̃), it is obvious as the Jacobian matrix ∂(ω, ω̃) ∂ζ = ∂(ω, ω̃) ∂(ξ, ξ̃) = In+m. (A2) It is also obvious. (A3) One can refer to [8, Section 3.2], their proof is similar. (A4) Suppose vector field (4.9) is defined on the domain D(r, s) with 0 < r < 1, s = ε2. It follows from (4.11)-(4.16) that P (θb) = O(s2), P (Ib) = O(s3), P (zσh) = O(s 5 2 )e−ρ|h|, P (ϕb) = O(s), P (Jb) = O(s2), P (wσh) = O(s 3 2 )e−ρ|h|. Then ‖P‖s;Dρ(r,s)×O ≤ cs1/2 ≤ cε. (A5) Through the transformation Ψ in (4.8), the vector fields M0 l in (4.6) are transformed into Ml = Ψ∗M0 l , then [P,Ml] = [Ψ∗P 0,Ψ∗M0 l ] = Ψ∗[P 0,M0 l ] = 0. EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 11 (A6) We only consider ∂P (θb) ∂zj+tc and ∂P (zi+tc) ∂zj+tc and the others can be verified similarly. ∂P (θb) ∂zj+tc = O(s2)e−ρ|j+tc| → 0, t→∞. then ‖∂P (θb) ∂zj+tc − lim t→∞ ∂P (θb) ∂zj+tc ‖s;Dρ(s,r)×O ≤ ε. It follows from (4.13) and (2) in Lemma 4.2 that ∂P (zi) ∂zj = O(s5/2)e−ρ|i−j| and ∂P (zi+tc) ∂zj+tc = ∂P (zi) ∂zj = lim t→∞ ∂P (zi+tc) ∂zj+tc . Then ‖∂P (zi+tc) ∂zj+tc − lim t→∞ ∂P (zi+tc) ∂zj+tc ‖s;Dρ(s,r)×O ≤ ε |t| e−ρ|i−j|. 5. Proof of Theorem 3.2 At the νth step of the KAM iteration, we consider an S-reversible vector field on Dρν (rν , sν)×Oν : Xν = Nν +Aν + Pν satisfying (A1)–(A6), where Nν and Aν have the same form as N and A in (3.1) and (3.2). We shall construct an S-invariant transformation Φν : Dρν+1(rν+1, sν+1)×Oν → Dρν (rν , sν)×Oν such that Φ∗νXν := (DΦν)−1 ·Xν ◦ Φν = Nν+1 +Aν+1 + Pν+1 with a new normal form Nν+1, Aν+1 and a much smaller perturbation term Pν+1 and still satisfying (A1)–(A6). In the sequel, for simplicity, we drop the subscript ν and write the symbol ‘+’ for ‘ν + 1’. Then we have the vector field X = N +A+ P (5.1) with N = n∑ b=1 ωb(ζ) ∂ ∂θb + m∑ b=1 ω̃b(ζ) ∂ ∂ϕb + ∑ %=± ( ∑ j∈Zd1 %iΩj(ζ)z%j ∂ ∂z%j + ∑ j∈Zd2 %iΩ̃j(ζ)w%j ∂ ∂w%j ), A = ∑ %=± ∑ j∈Zd1∩Zd2 (%iAj(ζ)w%j ∂ ∂z%j + %iÃj(ζ)z%j ∂ ∂w%j ), Let 0 < r+ < r and s+ = 1 4 sε1/3, ε+ = cγ−5(2r − 2r+)−1K5τ+19ε5/3 + ε7/6, where c is some suitable (possible different) constant independent of the iterative steps. Then our goal is to find a set O+ ⊂ O and an S-invariant transformation 12 Z. LOU, Y. SUN EJDE-2022/69 Φ : Dρ(r+, s+)×O+ → Dρ(r, s)×O such that it transforms the above X in (5.1) into X+ = N+ +A+ + P+ with N+ = n∑ b=1 ω+,b(ζ) ∂ ∂θb + m∑ b=1 ω̃+,b(ζ) ∂ ∂ϕb + ∑ %=± ( ∑ j∈Zd1 %iΩ+,j(ζ)z%j ∂ ∂z%j + ∑ j∈Zd2 %iΩ̃+,j(ζ)w%j ∂ ∂w%j ), A+ = ∑ %=± ∑ j∈Zd1∩Zd2 (%iA+,j(ζ)w%j ∂ ∂z%j + %iÃ+,j(ζ)z%j ∂ ∂w%j ). 5.1. Solving the homological equations. For K > 0, we define the truncation operator TK as follows: for f on D(r) = {(θ, ϕ) ∈ Cn×Cm : | Im θ| < r, | Imϕ| < r}, TKf(θ, ϕ) := ∑ (k,k̃)Zn×Zm, |k|+|k̃|≤K fk,k̃ei(〈k,θ〉+〈k̃,ϕ〉). The average of f with respect to (θ, ϕ) is defined as [f ] = f0,0 = 1 (2π)n+m ∫ Tn+m f(θ, ϕ) dθ dϕ We write the reversible vector field P as Taylor-Fourier series P (y; ζ) = ∑ v∈V ∑ k,k̃,l,l̃,α,β,α̃,β̃ P (v) klαβ,k̃l̃α̃β̃ (ζ)ei(〈k,θ〉+〈k̃,ϕ〉)I lJ l̃zαz̄βwα̃w̄β̃ ∂ ∂v , Let R = ∑ v∈V R(v)(y; ζ) ∂∂v be the truncation of P with R(v) = ∑ |l|+|l̃|+|α|+|β|+|α̃|+|β̃|≤1 TKP (v) lαβ,l̃α̃β̃ (θ, ϕ)I lJ l̃zαz̄βwα̃w̄β̃ for v ∈ {θb, ϕb, Ib, Jb, zj , z̄j , wj , w̄j}. We rewrite R(v) as follows: for v ∈ {θb, ϕb, Ib, Jb, zj , z̄j , wj , w̄j}, R(v) = Rv(θ, ϕ) + ∑ u∈{Ia,Ja,zj ,z̄j ,wj ,w̄j} Rvu(θ, ϕ)u. Remark 5.1. For the usual KAM procedure, we only need to eliminate terms Rθ, Rϕ in R(θ), R(ϕ). In this paper, we need to control the first derivatives of perturbation vector field P − R in lemma 5.6. So we must eliminate all the linear terms Rθuu, Rϕuu in R(θ), R(ϕ). We define the normal form of R as [R] = n∑ b=1 [Rθb ] ∂ ∂θb + m∑ b=1 [Rϕb ] ∂ ∂ϕb + ∑ j∈Zd1 [Rzjzj ]zj ∂ ∂zj + ∑ j∈Zd1∩Zd2 [Rzjwj ]wj ∂ ∂zj + ∑ j∈Zd1 [Rz̄j z̄j ]z̄j ∂ ∂z̄j + ∑ j∈Zd1∩Zd2 [Rz̄jw̄j ]w̄j ∂ ∂z̄j EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 13 + ∑ j∈Zd2 [Rwjwj ]wj ∂ ∂wj + ∑ j∈Zd1∩Zd2 [Rwjzj ]zj ∂ ∂wj + ∑ j∈Zd2 [Rw̄jw̄j ]w̄j ∂ ∂w̄j + ∑ j∈Zd1∩Zd2 [Rw̄j z̄j ]z̄j ∂ ∂w̄j In the sequel, we denote by φtX the flow generated by vector field X and φ1 X = φtX |t=1. Suppose vector field F has the same form as R, Φ∗X = (φ1 F )∗X, Φ∗X = (φ1 F )∗(N +A) + (φ1 F )∗R+ (φ1 F )∗(P −R) = N +A+ [N +A, F ] + ∫ 1 0 (1− t)(φtF )∗[[N +A, F ], F ]dt +R+ ∫ 1 0 (φtF )∗[R,F ]dt+ (φ1 F )∗(P −R). We solve the homological equation [N +A, F ] +R = [R]. (5.2) We denote ∂(ω,ω̃)f(θ, ϕ) := ∂ωf(θ, ϕ) + ∂ω̃f(θ, ϕ). By the definition of Lie bracket, the homological equation (5.2) is equivalent to the following scalar forms (5.3)–(5.6): ∂(ω,ω̃)f = g (5.3) where (f, g) = (Fu, Ru − [Ru]), (F v, Rv), u ∈ {θa, ϕa}; v ∈ {Ia, Ja, θaIb, θaJb, ϕaIb, ϕaJb, IaIb, IaJb, JaIb, JaJb}. ∂(ω,ω̃)f − iλf = g, (5.4) where (f, g;λ) ∈ {(Ful , Rul ;λl) : l = 1, . . . , 6}, with u1 ∈ {θaz−%i , ϕaz −% i , Iaz −% i , Jaz −% i , z%i , z % i Ia, z % i Ja : i ∈ Zd1 \ Zd2}, λ1 = %Ωi, u2 ∈ {θaw−%j , ϕaw −% j , Iaw −% j , Jaw −% j , w%j , w % j Ia, w % jJa : j ∈ Zd2 \ Zd1}, λ2 = %Ω̃j , u3 = z%i z σ j , λ3 = %Ωi − σΩj , i, j ∈ Zd1 \ Zd2, i 6= j, % 6= σ, u4 = z%i w σ j , λ4 = %Ωi − σΩ̃j , i ∈ Zd1 \ Zd2, j ∈ Zd2 \ Zd1, u5 = w%i z σ j , λ5 = %Ω̃i − σΩj , i ∈ Zd2 \ Zd1, j ∈ Zd1 \ Zd2, u6 = w%iw σ j , λ6 = %Ω̃i − σΩ̃j , i, j ∈ Zd2 \ Zd1, i 6= j, % 6= σ. ∂(ω,ω̃)F + iMF = G, (5.5) where (F ,G;M) ∈ {((Ful F vl ) , ( Rul Rvl ) ;Ml ) : l = 1, . . . , 6 } , with (u1, v1) ∈ {(uz%j , uw % j ) : u ∈ {θa, ϕa, Ia, Ja}, j ∈ Zd1 ∩ Zd2}, M1 = %MT j , (u2, v2) ∈ {(z%i , w % i ), (z%i Ia, w % i Ia), (z%i Ja, w % i Ja) : i ∈ Zd1 ∩ Zd2}, M2 = −%Mi, (u3, v3) = (z%i z σ j , z % i w σ j ), M3 = −%ΩiI2 + σMT j , i ∈ Zd1 \ Zd2, j ∈ Zd1 ∩ Zd2, (u4, v4) = (w%i z σ j , w % iw σ j ), M4 = −%Ω̃iI2 + σMT j , i ∈ Zd2 \ Zd1, j ∈ Zd1 ∩ Zd2, 14 Z. LOU, Y. SUN EJDE-2022/69 (u5, v5) = (z%i z σ j , w % i z σ j ), M5 = σΩjI2 − %Mi, i ∈ Zd1 ∩ Zd2, j ∈ Zd1 \ Zd2, (u6, v6) = (z%i w σ j , w % iw σ j ), M6 = σΩ̃jI2 − %Mi, i ∈ Zd1 ∩ Zd2, j ∈ Zd2 \ Zd1. ∂(ω,ω̃)F z%i z σ j − %iΩiF z % i z σ j + σiF z % i z σ j Ωj − %iAiFw % i z σ j + σiF z % i w σ j Ãj = Rz % i z σ j − δ%σδij [Rz % i z σ j ], ∂(ω,ω̃)F z%i w σ j − %iΩiF z % i w σ j + σiF z % i w σ j Ω̃j − %iAiFw % iw σ j + σiF z % i z σ j Aj = Rz % i w σ j − δ%σδij [Rz % i w σ j ], ∂(ω,ω̃)F w%i z σ j − %iΩ̃iFw % i z σ j + σiFw % i z σ j Ωj − %iÃiF z % i z σ j + σiFw % iw σ j Ãj = Rw % i z σ j − δ%σδij [Rw % i z σ j ], ∂(ω,ω̃)F w%iw σ j − %iΩ̃iFw % iw σ j + σiFw % iw σ j Ω̃j − %iÃiF z % i w σ j + σiFw % i z σ j Aj = Rw % iw σ j − δ%σδij [Rw % iw σ j ], i, j ∈ Zd1 ∩ Zd2. (5.6) here δµν = 1 if µ = ν, and 0 otherwise. Suppose that for ζ ∈ O, |k|+ |k̃| ≤ K, |〈k, ω〉+ 〈k̃, ω̃〉| ≥ γ Kτ , (k, k̃) 6= 0, |〈k, ω〉+ 〈k̃, ω̃〉+ Ωi| ≥ γ Kτ , i ∈ Zd1 \ Zd2, |〈k, ω〉+ 〈k̃, ω̃〉+ Ω̃i| ≥ γ Kτ , i ∈ Zd2 \ Zd1, |〈k, ω〉+ 〈k̃, ω̃〉+ Ωi ± Ωj | ≥ γ Kτ , (k, k̃) 6= 0, i, j ∈ Zd1 \ Zd2, |〈k, ω〉+ 〈k̃, ω̃〉+ Ωi ± Ω̃j | ≥ γ Kτ , i ∈ Zd1 \ Zd2, j ∈ Zd2 \ Zd1, |〈k, ω〉+ 〈k̃, ω̃〉+ Ω̃i ± Ω̃j | ≥ γ Kτ , (k, k̃) 6= 0, i, j ∈ Zd2 \ Zd1, |det((〈k, ω〉+ 〈k̃, ω̃〉)I2 ±Mi)| ≥ γ Kτ , i ∈ Zd1 ∩ Zd2, |det((〈k, ω〉+ 〈k̃, ω̃〉+ Ωi)I2 ±Mj)| ≥ γ Kτ , (i, j) or (j, i) ∈ (Zd1 ∩ Zd2)× (Zd1 \ Zd2), |det((〈k, ω〉+ 〈k̃, ω̃〉+ Ω̃i)I2 ±Mj)| ≥ γ Kτ , (i, j) or (j, i) ∈ (Zd1 ∩ Zd2)× (Zd2 \ Zd1), |det((〈k, ω〉+ 〈k̃, ω̃〉)I4 +Mi ⊗ I2 ± I2 ⊗MT j )| ≥ γ Kτ , (k, k̃) 6= 0, i, j ∈ Zd1 ∩ Zd2. (5.7) Lemma 5.2. Suppose that on O, the non-resonance conditions in (5.7) hold uni- formly. Then there exists a positive c = c(n,m, τ) such that (5.2) has a unique solution F with [F ] = 0, which is regular on D(r, s)×O. Moreover, (1) ‖F‖s;Dρ(r,s)×O ≤ cγ−5K5τ+19ε; (2) F ◦ S = DS · F ; (3) [F,Ml] = 0 for l = 1, . . . , d; (4) F satisfies (A6) with ε2/3 in place of ε on D(r−δ, s/2), where 0 < δ < r/2. Proof. (1) As we have mentioned above, (5.2) is equivalent to (5.3)–(5.6). Below we only consider the most difficult equation (5.6) with % = σ, i 6= j since the other ones can be solved similarly. EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 15 By Fourier expansion, we have ((〈k, ω〉+ 〈k̃, ω̃〉)I4 − %Mi ⊗ I2 + %I2 ⊗MT j )  iF z%i z % j k,k̃ iF z%i w % j k,k̃ iF w%i z % j k,k̃ iF w%iw % j k,k̃  =  R z%i z % j k,k̃ R z%i w % j k,k̃ R w%i z % j k,k̃ R w%iw % j k,k̃  , where (〈k, ω〉+ 〈k̃, ω̃〉)I4 − %Mi ⊗ I2 + %I2 ⊗MT j = (〈k, ω〉+ 〈k̃, ω̃〉)I4 +  −%Ωi + %Ωj %Ãj −%Ai 0 %Aj −%Ωi + %Ω̃j 0 −%Ai −%Ãi 0 −%Ω̃i + %Ωj %Ãj 0 −%Ãi %Aj −%Ω̃i + %Ω̃j  . Using the non-resonance conditions in (5.7), we obtain that for |k|+ |k̃| ≤ K, |F z % i z % j k,k̃ |O, |F z % i w % j k,k̃ |O, |Fw % i z % j k,k̃ |O, |Fw % iw % j k,k̃ |O ≤ cγ−5K5τ+19 ( |Rz % i z % j k,k̃ |O + |Rz % i w % j k,k̃ |O + |Rw % i z % j k,k̃ |O + |Rw % iw % j k,k̃ |O ) . Then according to the definition of vector fields, we have ‖F‖s;Dρ(r,s)×O ≤ cγ−5K5τ+19‖R‖s;Dρ(r,s)×O ≤ cγ−5K5τ+19ε. (2) F ◦S = DS ·F can be implied by the uniqueness of solutions of homological equation. (3) We verify that [F,Ml] = 0. For l = 1, . . . , d, consider πl(kαβ; k̃α̃β̃; v) = { πl(kαβ; k̃α̃β̃), v = θb, ϕb, Ib, Jb, πl(kαβ; k̃α̃β̃)− %jl, v = z%j , w % j , (5.8) where πl(kαβ; k̃α̃β̃) = n∑ b=1 i (b) l kl + m∑ b=1 ĩ (b) l k̃l + ∑ j∈Zd1 (αj − βj)jl + ∑ j∈Zd2 (α̃j − β̃j)jl. As in the proof of Lemma (4.2), one can verify that a vector field X satisfying [X,Ml] = 0 is equivalent to X (v) klαβ,k̃l̃α̃β̃ = 0, if πl(kαβ; k̃α̃β̃; v) 6= 0. Thus to prove [F,Ml] = 0, it suffices to verify that F (v) klαβ,k̃l̃α̃β̃ = 0, if πl(kαβ; k̃α̃β̃; v) 6= 0. This can be implied by [P,M] = 0 since F is determined by R. (4) Fr the proof we can follow that of [8, Lemma 4.3] since there is no essential difference. � 5.2. Estimates on the coordinate transformation. Lemma 5.3. If ε� δγ5K−5τ−19, then for every −1 ≤ t ≤ 1, we have φtF : Dρ(r − 2δ, s/4)→ Dρ(r − δ, s/2), ‖φtF − id‖s;Dρ(r−2δ,s/4)×O ≤ cγ−5K5τ+19ε, ‖DφtF − Id‖s;Dρ(r−2δ,s/4)×O ≤ cγ−5δ−1K5τ+19ε. 16 Z. LOU, Y. SUN EJDE-2022/69 Proof. Using Cauchy’s inequality, we obtain ‖DF‖s;Dρ(r−δ,s/2)×O ≤ c δ ‖F‖s;Dρ(r,s)×O ≤ c δ γ−5K5τ+19ε, then if ε� δγ5K−5τ−19, for every −1 ≤ t ≤ 1, φtF : Dρ(r − 2δ, s/4)→ Dρ(r − δ, s/2) is well-defined. Thus by Gronwall’s inequality and the estimate for DF , we have ‖φtF − id‖s;D(r−2δ,s/4)×O ≤ c‖F‖s;D(r,s)×O ≤ cγ−5K5τ+19ε, ‖DφtF − Id‖s;D(r−2δ,s/4)×O ≤ c‖DF‖s;D(r−δ,s/2)×O ≤ cδ−1γ−5K5τ+19ε. � 5.3. New normal form. Through the time-1 map Φ = φ1 F defined above, the vector field X is transformed into X+ = Φ∗X = N+ +A+ + P+ with new normal form N+,A+ and new perturbation P+. In this subsection, we consider the new normal form N+ = N + N̂ , A+ = A+ Â, where N̂ = n∑ b=1 [Rθb ] ∂ ∂θb + m∑ b=1 [Rϕb ] ∂ ∂ϕb + ∑ j∈Zd1 ([Rzjzj ]zj ∂ ∂zj + [Rz̄j z̄j ]z̄j ∂ ∂z̄j ) + ∑ j∈Zd2 ([Rwjwj ]wj ∂ ∂wj + [Rw̄jw̄j ]w̄j ∂ ∂w̄j ), (5.9)  = ∑ j∈Zd1∩Zd2 ([Rzjwj ]wj ∂ ∂zj + [Rz̄jw̄j ]w̄j ∂ ∂z̄j + [Rwjzj ]zj ∂ ∂wj + [Rw̄j z̄j ]z̄j ∂ ∂w̄j ) and ω+,b = ωb + [Rθb ], (b = 1, . . . , n), ω̃+,b = ω̃b + [Rϕb ], (b = 1, . . . ,m), Ω+,i = Ωi − i[Rzizi ], (i ∈ Zd1), Ω̃+,i = Ω̃i − i[Rwiwi ], (i ∈ Zd2), A+,i = Ai − i[Rziwi ], Ã+,i = Ãi − i[Rwizi ], (i ∈ Zd1 ∩ Zd2). It follows form (5.9) that ‖N̂‖s;Dρ(r,s)×O ≤ ‖R‖s;Dρ(r,s)×O. Then |[Rzσi zσi ]|O ≤ ‖R‖s;Dρ(r,s)×O ≤ ε; thus |Ω+,i − Ωi|O = |[Rzizi ]|O ≤ ε, for i ∈ Zd1. Similarly, |Ω̃+,i − Ω̃i|O, |A+,i −Ai|O, |Ã+,i − Ãi|O, |ω+,b − ωb|O, |ω̃+,b − ω̃b|O ≤ ε. EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 17 5.4. New perturbation. The new perturbation is P+ = ∫ 1 0 (φtF )∗[R(t), F ]dt+ (φ1 F )∗(P −R), with R(t) = (1− t)[R] + tR. Let η = ε1/3. We now give the estimate of ‖P+‖ηs;D(r−2δ,ηs/4)×O. ‖(φtF )∗[R(t), F ]‖ηs;D(r−2δ,s/4)×O ≤ cδ−1η−1‖R‖s;D(r,s)×O‖F‖s;D(r,s)×O ≤ cδ−1η−1γ−5K5τ+19ε2. Consider the estimate for ‖(φ1 F )∗(P − R)‖ηs;D(r−2δ,ηs/4)×O. Rewrite P − R as P −R = P(1) + P(2), where P(1) = ∑ v∈{θb,ϕb,Ib,Jb,z%j ,w % j } ∑ |l|+|l̃|+|α| +|β|+|α̃|+|β̃|≤1 (1− TK)P (v) lαβ,l̃α̃β̃ (θ, ϕ)I lJ l̃zαz̄βwα̃w̄β̃ ∂ ∂v , P(2) = ∑ v∈{Ib,Jb,z%j ,w % j } ∑ |l|+|l̃|+|α| +|β|+|α̃|+|β̃|≥2 P (v) lαβ,l̃α̃β̃ (θ, ϕ)I lJ l̃zαz̄βwα̃w̄β̃ ∂ ∂v . Then ‖P(1)‖ηs;D(r−δ,ηs/2)×O ≤ η−1e−Kδ‖P‖s;D(r,s)×O ≤ η−1e−Kδε, ‖P(2)‖ηs;D(r−δ,ηs/2)×O ≤ cη‖P‖s;D(r,s)×O ≤ cηε. This implies that ‖P −R‖ηs;D(r−δ,ηs/2)×O ≤ e−Kδ‖P‖ηs;D(r,ηs/2)×O + cη‖P‖s;D(r,s)×O ≤ η−1e−Kδε+ cηε. Therefore, ‖P+‖ηs;D(r−2δ,ηs/4)×O ≤ ‖(φtF )∗[R(t), F ]‖ηs;D(r−2δ,ηs/4)×O + ‖(φ1 F )∗(P −R)‖ηs;D(r−2δ,ηs/4)×O ≤ cδ−1η−1γ−5K5τ+19ε2 + η−1e−Kδε+ cηε = cδ−1γ−5K5τ+19ε5/3 + e−Kδε2/3 + cε 4 3 ≤ ε+. The following lemma ensures that the new perturbation P+ satisfies reversibility and momentum conservation condition. Lemma 5.4. (1) P+ is S-reversible; (2) [P+,Ml] = 0, l = 1, . . . , d; Proof. (1) We conclude from (2) in Lemma 5.2 that Φ ◦ S = S ◦ Φ, which implies X+◦S = −DS ·X+. It is obvious that N+◦S = −DS ·N+, thus P+◦S = −DS ·P+. (2) We know that [N,Ml] = 0 and [P,Ml] = 0. From Lemma 5.2, we obtain [F,Ml] = 0. This together with P+ = P −R+ [P, F ] + 1 2! [[N,F ], F ] + 1 2! [[P, F ], F ] + · · ·+ 1 i! [. . . [N,F ] . . . , F︸ ︷︷ ︸ i ] + 1 i! [. . . [P, F ] . . . , F︸ ︷︷ ︸ i ] + . . . (5.10) implies that [P+,Ml] = 0. � 18 Z. LOU, Y. SUN EJDE-2022/69 Lemma 5.5. Suppose P satisfies (A6), and F satisfies (A6) with ε2/3 in place of ε. Also assume that for σ = ±, |j| > K, ∂F (Ib) ∂zσj = 0, ∂F (Ib) ∂wσj = 0, ∂F (Jb) ∂zσj = 0, ∂F (Jb) ∂wσj = 0, ∂F (θb) ∂zσj = 0, ∂F (θb) ∂wσj = 0, ∂F (ϕb) ∂zσj = 0, ∂F (ϕb) ∂wσj = 0, and for |i∓ j| > K, ∂F (zσi ) ∂z±σj = 0, ∂F (wσi ) ∂w±σj = 0, ∂F (zσi ) ∂w±σj = 0, ∂F (wσi ) ∂z±σj = 0. Then [P, F ] also satisfies (A6) with ε+ in place of ε. Proof. By the definition of Lie bracket, the zi-component of [P, F ] is [P, F ](zi) = ∑ u∈V ( ∂P (zi) ∂u F (u) − ∂F (zi) ∂u P (u)), where V = {θa, ϕb, zi, wj , z̄i, w̄j : a = 1, . . . , n; b = 1, . . . ,m; i ∈ Zd1; j ∈ Zd2}. To verify that [P, F ] satisfies (A6), we only consider ∂ ∂zj [P, F ](zi) and the deriva- tives with respect to the other components are similarly analyzed. It suffices to consider ∑ h ∂2P (zi) ∂zh∂zj F (zh) and ∑ h ∂P (zi) ∂zh ∂F (zh) ∂zj in ∂ ∂zj [P, F ](zi) since the other terms can be similarly studied. Let pzzij = limt→∞ ∂P (zi+tc) ∂zj+tc and fzzij = limt→∞ ∂F (zi+tc) ∂zj+tc . Then ‖ ∑ h ( ∂2P (zi+tc) ∂zh∂zj+tc F (zh) − lim t→∞ ∂2P (zi+tc) ∂zh∂zj+tc F (zh))‖s;Dρ(r−δ,s/2) ≤ ‖F‖s;Dρ(r,s)‖ ∂P (zi+tc) ∂zj+tc − pzzij ‖s;Dρ(r,s) ≤ cε 5/3 |t| e−ρ|i−j| ≤ ε+ 30|t| e−ρ+|i−j| (5.11) and ‖ ∑ h ( ∂P (zi+tc) ∂zh+tc ∂F (zh+tc) ∂zj+tc − pzzihfzzhj )‖s;Dρ(r−δ,s/2) ≤ ∑ h ‖fzzhj ‖s;Dρ(r−δ,s/2)‖ ∂P (zi+tc) ∂zh+tc − pzzih‖s;Dρ(r−δ,s/2) + ∑ h ‖pzzih‖s;Dρ(r−δ,s/2)‖ ∂F (zh+tc) ∂zj+tc − fzzhj ‖s;Dρ(r−δ,s/2) + ∑ h ‖∂P (zi+tc) ∂zh+tc − pzzih‖s;Dρ(r−δ,s/2)‖ ∂F (zh+tc) ∂zj+tc − fzzhj ‖s;Dρ(r−δ,s/2) ≤ cKd ε 5/3 |t| e−ρ|i−j| + cKd ε 5/3 t2 e−ρ|i−j| ≤ ε+ 30|t| e−ρ+|i−j|. (5.12) EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 19 Note that h is bounded by cKd in the above inequality since |i − h| ≤ K and |j − h| ≤ K. � The following lemma follows from (5.10) and Lemma 5.5. Lemma 5.6. P+ satisfies (A6) with K+, ε+, ρ+ in place of K, ε, ρ. 5.5. Iteration and convergence. For given r > 0, s > 0, L > 0 and 0 < γ, ε < 1, consider c a positive constant depending only on n,m, τ . For ν ≥ 0, we define the iterative sequences: δν = r 2ν+3 , rν+1 = rν − 2δν , r0 = r, εν+1 = cγ−5δ−1 ν K5τ+19 ν ε5/3 ν + ε7/6 ν , ε0 = ε, e−Kνδν = ε1/2 ν , ην = ε1/3 ν , sν+1 = 1 4 ηνsν , s0 = s, Lν+1 = Lν + εν , L0 = L, ρν = ρ(1− ν+1∑ i=2 2−i). 5.5.1. Iteration lemma. According to the preceding analysis, we obtain the follow- ing lemma. Lemma 5.7. Let ε be small enough and ν ≥ 0. Suppose that (1) The normal form Nν +Aν = ων(ζ) ∂ ∂θ + ω̃ν(ζ) ∂ ∂ϕ + ∑ σ=± σi(Ων(ζ)zσ ∂ ∂zσ + Ω̃ν(ζ)wσ ∂ ∂wσ ) +Aν , with ζ ∈ Oν satisfies (5.7) with ων , ω̃ν ,Ων , Ω̃ν , Aν , Ãν and Kν ; (2) ων , ω̃ν ,Ων,j , Ω̃ν,j are C4 W smooth in ζ and satisfy |ων − ων−1|O, |ω̃ν − ω̃ν−1|O, |Ων,j − Ων−1,j |O, |Ω̃ν,j − Ω̃ν−1,j |O ≤ εν−1; (3) Nν +Aν + Pν satisfies (A5) and (A6) with Kν , εν , ρν and ‖Pν‖sν ;Dρν (rν ,sν)×Oν ≤ εν . Then there exists a real analytic, S-invariant transformation Φν : Dρν (rν+1, sν+1)×Oν → Dρν (rν , sν) satisfying ‖Φν − id‖sν+1;Dρν (rν+1,sν+1)×Oν ≤ cε 1/2 ν , (5.13) ‖DΦν − Id‖sν+1;Dρν (rν+1,sν+1)×Oν ≤ cε 1/2 ν , (5.14) and a closed subset Oν+1 = Oν \ ∪Kν<|k|+|k̃|≤Kν+1 Rν+1 kk̃ (γ), (5.15) where Rν+1 kk̃ (γ) is defined in (5.18), such that Xν+1 = (Φν)∗Xν = Nν+1 +Aν+1 + Pν+1 satisfies the same assumptions as Xν with ‘ν + 1’ in place of ‘ν’. 20 Z. LOU, Y. SUN EJDE-2022/69 5.5.2. Convergence. We now complete the proof of Theorem 3.2. Let X0 = N0 +A0 + P0 = N +A+ P be an initial S-reversible vector field and satisfies the assumptions of Theorem 3.2. Recall that ε0 = ε, r0 = r, s0 = s, ρ0 = ρ, L0 = L. Suppose O is a compact set of positive Lebesgue measure and all the conditions in the iterative lemma with ν = 0 hold. Then we inductively obtain the following sequences Oν+1 ⊂ Oν , Ψν = Φ0 ◦ Φ1 ◦ · · · ◦ Φν : Dρν (rν+1, sν+1)×Oν → Dρ0(r0, s0), Xν+1 = (Ψν)∗X = Nν+1 +Aν+1 + Pν+1. Let Õ = ∩∞ν=0Oν . Using (5.13), (5.14) and following from [14], we obtain that Nν +Aν ,Ψ ν , DΨν converge uniformly on D ρ 2 ( r2 , 0)× Õ with N∞ +A∞ = ω∞ ∂ ∂θ + ω̃∞ ∂ ∂ϕ ∑ σ=± σi(Ω∞z σ ∂ ∂zσ + Ω̃∞w σ ∂ ∂wσ ) +A∞. which corresponds to a motion equation, θ̇ = ω∞, ϕ̇ = ω̃∞, İ = 0, J̇ = 0, żσj = σiΩ∞,jz σ j , σ = ±, j ∈ Zd1 \ Zd2, ẇσj = σiΩ̃∞,jw σ j , j ∈ Zd2 \ Zd1,( żσj ẇσj ) = σi ( Ω∞,j A∞,j Ã∞,j Ω̃∞,j )( zσj wσj ) , j ∈ Zd1 ∩ Zd2. (5.16) By the choice of εν and Kν , we have εν+1 = O(ε 7/6 ν ), thus εν → 0, ν → ∞. And we also have ∑∞ ν=0 εν ≤ 2ε. Consider the flow φtX of X. It follows from Xν+1 = (Ψν)∗X that φtX ◦Ψν = Ψν ◦ φtXν+1 . (5.17) Thanks to the uniform converge of Xν ,Ψ ν and DΨν , we can take limits on both sides of (5.17). Therefore, on D ρ 2 ( r2 , 0)× Õ, we have φtX ◦Ψ∞ = Ψ∞ ◦ φtX∞ , Ψ∞ : D ρ 2 ( r 2 , 0)× Õ → Dρ(r, s)×O. It follows that for each ζ ∈ Õ, the set Ψ∞(Tn+m × {ζ}) is and embedded torus which is invariant for the original perturbed reversible system at ζ ∈ Õ. 5.6. Measure estimate. Let O−1 = O and K−1 = 0. At the νth step of the KAM iteration, the resonant set Rν ⊂ Oν−1 need to be excluded. Rν = ∪Kν−1<|k|+|k̃|≤KνR ν kk̃ , (5.18) with Rν kk̃ = R0ν kk̃ ∪ (∪iR1ν kk̃,i ) ∪ (∪iR2ν kk̃,i ) ∪ (∪ijR11,±ν kk̃,ij ) ∪ (∪ijR12,±ν kk̃,ij ) (∪ijR22,±ν kk̃,ij ) ∪ (∪iR3ν kk̃,i ) ∪ (∪ijR13,±ν kk̃,ij ) ∪ (∪ijR23,±ν kk̃,ij ) ∪ (∪ijR34,±ν kk̃,ij ), EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 21 where R0ν kk̃ = {ζ ∈ Oν−1 : |〈k, ων(ζ)〉+ 〈k̃, ω̃ν(ζ)〉| < γ Kτ ν }; (5.19) R1ν kk̃,i = {ζ : |〈k, ων〉+ 〈k̃, ω̃ν〉+ Ων,i| < γ Kτ ν }, i ∈ Zd1; (5.20) R2ν kk̃,i = {ζ : |〈k, ων〉+ 〈k̃, ω̃ν〉+ Ω̃ν,i| < γ Kτ ν }, i ∈ Zd2; (5.21) R11,±ν kk̃,ij = {ζ : |〈k, ων〉+ 〈k̃, ω̃ν〉+ Ων,i ± Ων,j | < γ Kτ ν }, i, j ∈ Zd1 \ Zd2; (5.22) R12,±ν kk̃,ij = {ζ : |〈k, ων〉+ 〈k̃, ω̃ν〉+ Ων,i ± Ω̃ν,j | < γ Kτ ν }, i ∈ Zd1 \ Zd2, j ∈ Zd2 \ Zd1; (5.23) R22,±ν kk̃,ij = {ζ : |〈k, ων〉+ 〈k̃, ω̃ν〉+ Ω̃ν,i ± Ω̃ν,j | < γ Kτ ν }, i, j ∈ Zd2 \ Zd1; (5.24) R3ν kk̃,i = {ζ : |det((〈k, ων〉+ 〈k̃, ω̃ν〉)I2 +Mν,i)| < γ Kτ ν }, i ∈ Zd1 ∩ Zd2; (5.25) R13,±ν kk̃,ij = {ζ : |det((〈k, ων〉+ 〈k̃, ω̃ν〉+ Ων,i)I2 ±Mν,j)| < γ Kτ ν }, (i, j) or (i, j) ∈ (Zd1 ∩ Zd2)× (Zd1 \ Zd2); (5.26) R23,±ν kk̃,ij = {ζ : |det((〈k, ων〉+ 〈k̃, ω̃ν〉+ Ω̃ν,i)I2 ±Mν,j)| < γ Kτ ν }, (i, j) or (i, j) ∈ (Zd1 ∩ Zd2)× (Zd2 \ Zd1); (5.27) R34,±ν kk̃,ij = {ζ : |det((〈k, ων〉+ 〈k̃, ω̃ν〉)I4 +Mν,i ⊗ I2 ± I2 ⊗MT ν,j)| < γ Kτ ν }, i, j ∈ Zd1 ∩ Zd2. (5.28) Note that R11,−ν kk̃,ij , R22,−ν kk̃,ij , and R34,−ν kk̃,ij are the most complicated three case, and the former two have been studied in [11]; thus it suffices to consider the last case. We denote Dν = (〈k, ων〉+ 〈k̃, ω̃ν〉)I4 +Mν,i ⊗ I2 − I2 ⊗MT ν,j . Lemma 5.8. For any given i, j ∈ Zd1 ∩ Zd2 with |i− j| ≤ Kν , either |det(Dν)| ≥ 1 or there are i0, j0, c1, . . . , cd−1 ∈ Zd with |i0|, |j0|, |c1|, . . . , |cd−1| ≤ 3K2 ν and t1, . . . , td−1 ∈ Z such that i = i0 +t1c1 +· · ·+td−1cd−1, j = j0 +t1c1 +· · ·+td−1cd−1. For a proof of the above lemma see [11]. In the following, for convenience of notation, let t := (t1, t2, . . . , td−1), c := (c1, c2, . . . , cd−1) and t ·c := t1c1 +t2c2 + · · ·+td−1cd−1. By Lemma 5.8, we have the following result. Lemma 5.9. ∪i,j∈Zd1∩Zd2R 34,−ν kk̃,ij ⊂ ∪i0,j0,c1,c2,...,cd−1∈Zd; t1,t2,...,td−1∈Z R34,−ν kk̃,i0+t·c,j0+t·c , where |i0|, |j0|, |c1|, |c2|, . . . , |cd−1| ≤ 3K2 ν . 22 Z. LOU, Y. SUN EJDE-2022/69 Lemma 5.10. Let τ > 4(d−1)(d+1)! (d−1)!(d+1)−1 . Then for fixed k, k̃, i0, j0, c1, . . . , cd−1, meas ( ∪t1,t2,...,td−1∈Z R 34,−ν kk̃,i0+t·c,j0+t·c ) ≤ c γ1/4 K τ (d+1)! ν . Proof. Without loss of generality, we assume |t1| ≤ |t2| ≤ · · · ≤ |td−1|. Let Ων,j = |j|2 + Ω0 ν,j , Ω̃ν,j = |j|2 + Ω̃0 ν,j , and Dν(t) = (〈k, ων〉+ 〈k̃, ω̃ν〉)I4 +Mν,i0+t·c ⊗ I2 − I2 ⊗MT ν,j0+t·c. Using Töplitz-Lipschitz property of Aν + Pν , for l = i0, j0, 1 ≤ j ≤ d − 1, we have |Ω0 ν,l+t·c − lim tj→∞ Ω0 ν,l+t·c| < ε |tj | , |Ω̃0 ν,l+t·c − lim tj→∞ Ω̃0 ν,l+t·c| < ε |tj | , |Aν,l+t·c − lim tj→∞ Aν,l+t·c| < ε |tj | , |Ãν,l+t·c − lim tj→∞ Ãν,l+t·c| < ε |tj | . Then we have |det(Dν(t))− lim tj→∞ det(Dν(t))| < εK4 ν |tj | . We consider the resonant set R34,−ν kk̃,i0j0c∞d−1 = { ζ ∈ Oν−1 : | lim t1→∞ ( lim t2,...,td−1→∞ det(Dν(t)))| < γ K τ d! ν } . For fixed k, k̃, i0, j0, c, its Lebesgue measure satisfies meas(R34,−ν kk̃,i0j0c∞d−1 ) ≤ γ1/4 K τ d! ν , and for ζ ∈ Oν−1 \ R34,−ν kk̃,i0j0c∞d−1 , | lim t1→∞ ( lim t2,...,td−1→∞ det(Dν(t)))| ≥ γ K τ d! ν . Below we consider the following cases: Case 1: |t1| > K τ d! +4 ν . For ζ ∈ Oν−1 \ R34,−ν kk̃,i0j0c∞d−1 , we have |det(Dν(t))| ≥ | lim t1→∞ ( lim t2,...,td−1→∞ det(Dν(t)))| − d−1∑ j=1 εK4 ν |tj | ≥ γ K τ d! ν − (d− 1) ε K τ d! ν ≥ γ 2K τ d! ν ≥ γ Kτ ν . Case l: This is the general case 2 ≤ l ≤ d − 1. We consider |t1| ≤ K τ d! +4 ν , |t2| ≤ K 2τ d! +4 ν , . . . , |tl−1| ≤ K (l−1)!τ d! +4 ν , |tl| > K l!τ d! +4 ν . We define the resonant set R34,−ν kk̃,i0j0ct1t2...tl−1∞d−l = { ζ ∈ Oν−1 : | lim tl,...,td−1→∞ det(Dν(t))| < γ K l!τ d! ν } . EJDE-2022/69 KAM FOR HIGHER DIMENSIONAL REVERSIBLE NLS 23 Then for fixed k, k̃, i0, j0, c, t1, t2, . . . , tl−1, its Lebesgue measure satisfies meas(R34,−ν kk̃,i0j0ct1...tl−1∞d−l) ≤ γ1/4 K l!τ d! ν , meas ( ∪ |t1|,...,|tl−1|≤K (l−1)!τ d! +4 ν R34,−ν kk̃,i0j0ct1...tl−1∞d−l ) ≤ 2l−1K (l−1)!(l−1)τ d! +4(l−1) ν γ1/4 K l!τ d! ν ≤ 2l−1γ1/4 K (l−1)!τ d! −4(l−1) ν . Thus for |t1| ≤ K τ d! +4 ν , |t2| ≤ K 2τ d! +4 ν , . . . , |tl−1| ≤ K (l−1)!τ d! +4 ν , |tl| > K l!τ d! +4 ν , ζ ∈ Oν−1 \ R34,−ν kk̃,i0j0ct1...tl−1∞d−l , we have |det(Dν(t))| ≥ | lim tl,...,td−1→∞ det(Dν(t))| − d−1∑ j=l εK4 ν |tj | ≥ γ K l!τ d! ν − (d− l) ε K l!τ d! ν ≥ γ 2K l!τ d! ν ≥ γ Kτ ν . Case d: |t1| ≤ K τ d! +4 ν , |t2| ≤ K 2τ d! +4 ν , . . . , |td−1| ≤ K (d−1)!τ d! +4 ν . We define the resonant set R34,−ν kk̃,i0j0ct1t2...td−1 = { ζ ∈ Oν−1 : |det(Dν(t))| < γ Kτ ν } . For fixed k, k̃, i0, j0, c, t1, t2, . . . , td−1, its Lebesgue measure satisfies meas(R34,−ν kk̃,i0j0ct1t2...td−1 ) ≤ γ1/4 Kτ ν , meas ( ∪ |t1|,...,|td−1|≤K (d−1)!τ d! +4 ν R34,−ν kk̃,i0j0ct1t2...td−1 ) ≤ 2d−1K (d−1)!(d−1)τ d! +4(d−1) ν γ1/4 Kτ ν ≤ 2d−1γ1/4 K τ d−4(d−1) ν . Therefore, if τ > 4(d−1)(d+1)! (d−1)!(d+1)−1 , we obtain meas(∪t1,t2,...,td−1∈ZR 34,−ν kk̃,i0+t·c,j0+t·c) ≤ c γ1/4 K τ (d+1)! ν . � According to the above analysis, we obtain the following lemma. 24 Z. LOU, Y. SUN EJDE-2022/69 Lemma 5.11. Let τ > d!(2d(d + 1) + n + m + 1) + 4(d−1)(d+1)! (d−1)!(d+1)−1 . Then the total measure of resonant set should be excluded during the KAM iteration is meas(∪ν≥0Rν) = O(γ1/4). 6. Appendix Suppose the vector field X(θ, I, z, z̄) is defined on Dρ(r, s) = {y = (θ, I, z, z̄) : | Im θ| < r, |I| < s, ‖z‖ρ < s, ‖z̄‖ρ < s}. Definition 6.1. Suppose S is an involution map: S2 = id. Vector field X is called reversible with respect to S (or S-reversible), if DS ·X = −X ◦ S, i.e., (DS(y))X(y) = −X(S(y)), y ∈ Dρ(r, s), where DS is the tangent map of S. Definition 6.2. Suppose S is an involution map: S2 = id. Vector field X is called invariant with respect to S (or S-invariant), if DS ·X = X ◦ S. Definition 6.3. A transformation Φ is called invariant with respect to above in- volution S (or S-invariant), if Φ ◦ S = S ◦ Φ. Lemma 6.4. (1) If X and Y are both S-reversible (or S-invariant), then [X,Y ] is S-invariant. (2) 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 (Cauchy’s inequality, [11]). Let 0 < δ < r. For an analytic function f(θ, I, z, z̄) on Dρ(r, s), it holds ‖ ∂f ∂θb ‖s;Dρ(r−δ,s) ≤ c δ ‖f‖s;Dρ(r,s), ‖ ∂f ∂Ib ‖s;Dρ(r,s/2) ≤ c s ‖f‖s;Dρ(r,s), ‖ ∂f ∂zσi ‖s;Dρ(r,s/2) ≤ c s ‖f‖s;Dρ(r,s)e ρ|i|, σ = ±. Acknowledgments. Z. Lou was supported by the National Natural Science Foun- dation of China (NSFC) (Grant No. 11901291), and by the Natural Science Foun- dation of Jiangsu Province, China (Grant No. BK20190395). Y. Sun was supported by the NSFC (Grant No. 11971012). References [1] M. Berti, L. Biasco, M. Procesi; KAM for reversible derivative wave equations. Arch. Ration. Mech. Anal., 212(3) (2014), 905–955. [2] M. Berti, P. Bolle; Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential. J. 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Pöschel; On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z., 202(4) (1989), 559–608. [15] C. Procesi, M. Procesi. A KAM algorithm for the resonant non-linear Schrödinger equation. Adv. Math., 272 (2015), 399–470. [16] M. Procesi, X. Xu; Quasi-Töplitz functions in KAM theorem. SIAM J. Math. Anal., 45(4) (2013), :2148–2181. [17] W. Wang; Energy supercritical nonlinear Schrödinger equations: quasiperiodic solutions. Duke Math. J., 165(6) (2016), :1129–1192. [18] C. Wayne; Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Comm. Math. Phys., 127(3) (1990), :479–528. [19] X. Xu, J. Geng; KAM tori for higher dimensional beam equation with a fixed constant potential. Sci. China Ser. A, 52(9) (2009), 2007–2018. [20] J. Zhang, M. Gao, X. Yuan; KAM tori for reversible partial differential equations. Nonlin- earity, 24(4) (2011), 1189–1228. Zhaowei Lou School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China Email address: zwlou@nuaa.edu.cn Yingnan Sun (corresponding author) School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China Email address: sunyingnan@nuaa.edu.cn 1. Introduction and main result 1.1. Main result 2. Preliminaries 3. A KAM theorem for infinite dimensional reversible systems 4. Application to the coupled NLS 4.1. Lattice form of equation (??) 4.2. Verification of assumptions (A1)–(A6) 5. Proof of Theorem ?? 5.1. Solving the homological equations 5.2. Estimates on the coordinate transformation 5.3. New normal form 5.4. New perturbation 5.5. Iteration and convergence 5.6. Measure estimate 6. Appendix Acknowledgments References