LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS. Jiansheng Geng, Jiangong You and Zhiyan Zhao

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1 Geom. Funct. Anal. Vol DOI: /s Published online January 28, 2014 c 2014 Springer Basel GAFA Geometric And Functional Analysis LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS Jiansheng Geng, Jiangong You and Zhiyan Zhao Abstract. To investigate localization in one-dimensional quasi-periodic nonlinear systems, we consider the Schrödinger equation i q n + ɛq n+1 + q n 1 +V n α + xq n + q n 2 q n =0, n Z, as a model, with V a nonconstant real-analytic function on R/Z, and α satisfying a certain Diophantine condition. It is shown that, if ɛ is sufficiently small, then for a.e. x R/Z, dynamical localization is maintained for typical solutions in a quasi-periodic time-dependent way. Contents 1 Introduction Preliminaries The quasi-periodic Schrödinger operator Decay property of matrices Abstract KAM Theorem Function space norms Statement of the abstract KAM theorem Application to Equation KAM Step Construction of O ν Homological equation and its approximate solution Verification of assumptions after one sub-step A succession of symplectic transformations Proof of the KAM Theorem Iteration lemma Convergence Measure estimate A: Appendix A.1 Outline of the proof of Proposition Hamiltonian vector field and Poisson bracket References

2 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS Introduction Physical motivations. Localization of particles and waves in disordered media is one of the most intriguing phenomena in solid-state physics. This phenomenon was first analyzed by P.W. Anderson [And58]. He studied the transport of non-interacting electrons in a crystal lattice, described by a single particle with random on-site energy. In his model, he showed that when the amplitude of the disorder becomes higher than a critical value, the diffusion in the lattice of an initially localized wavepacket is suppressed. An Anderson localized state is characterized by an exponential decay of the amplitude of the wave function. In many physics experiments, a relatively weak disorder on the structure of the lattice is introduced by a quasi-periodic potential. This kind of system corresponds to an experimental realization of the so-called Aubry André [AA80] or Harper [H55] model. It is important in the study of Bose-Einstein condensation and nonlinear optics. Anderson localization in such linear systems, especially in the one-dimensional case, has been thoroughly studied [R08], and rigorous mathematical results have been established [J99]. As a well-known model in mathematical physics, the almost Mathieu operator H x,λ, α acting on l 2 Z is defined by H x,λ, α ψ n =ψ n+1 + ψ n 1 +λ cos 2πx + n αψ n, n Z, where n is the primary lattice site index, α is some ratio between the wavenumbers of two lattices, x R/Z is an arbitrary phase, and ψ n is a complex variable whose modulus square gives the probability of finding a particle at the lattice site n. With α a fixed Diophantine number, for a.e. x and λ large enough, H x,λ, α exhibits dynamical localization [G99,G01], i.e., for any ψ l 2 Z with compact support and arbitrary d>0, sup t r d t :=sup t n 2d e ihx,λ, αt ψ n 2 <. n Z In particular, there exists a transparent transition between diffusion and localization for the almost Mathieu operator. From the perspective of spectrum theory, it is shown by Jitomirskaya [J99] that, for a.e. x, H x,λ, α has 1. λ>2: only pure point spectrum with exponentially decaying eigenfunctions; 2. λ = 2: purely singular-continuous spectrum; 3. λ<2: purely absolutely continuous spectrum. There is a perfect agreement with this conclusion in some experiments e.g., [H88]. 5 1 For α =, with an initial δ function wavepacket, the asymptotic spreading 2 of the wavepacket width r 1 t can be parametrized as r 1 t t γ, and one finds three different regimes

3 118 J. GENG ET AL. GAFA 1. λ>2: localized regime, γ =0; 2. λ = 2: sub-diffusive, γ 0.5; 3. λ<2: ballistic regime, γ =1. However, the situation is much less clear in the presence of interactions nonlinearities. It strongly influences the possibility to observe the localization induced by disorder. One can start from the Gross-PitaevskiiGP equation [G61,P61] in Hartree Fock theory, and get a generalized Aubry André model which includes an additional nonlinear term that represents the mean-field interaction. The Hamiltonian is H = [ψ n+1 ψn + ψ n+1 ψ n +λcos 2πn α + x ψ n ] β ψ n 4, n Z and the equation of motion is generated by i ψ n = H ψ n, yielding the nonlinear Schrödinger equation i ψ n +ψ n+1 + ψ n 1 +λ cos 2πn α + xψ n + β ψ n 2 ψ n =0, n Z, 1.1 that can be considered as the GP equation on a discretized lattice. Similar versions of a discretized GP equation have been already used to investigate the dynamics of condensates in different situations see, for instance, [T01]. It is shown experimentally in [L09] that, if the condensate initially occupies a single lattice site, i.e., a δ function ψ n 0 = δ n,0, the dynamics of the interacting gas is dominated by self-trapping in a wide range of parameters, even for weak interaction. Conversely, if the diffusion starts from a Gaussian wavepacket of width σ, ψ n 0 = ce n2 2σ 2, then self-trapping is significantly suppressed and the destruction of localization by interaction is more easily observable. The aim of the present work is to analyze localization in the quasi-periodic nonlinear dynamical systems, which are modeled by the discrete one-dimensional disordered nonlinear Schrödinger equations of the same form as 1.1. When the disorder is sufficiently large, a rigorous mathematical argument for the maintainability of localization is given in this paper, corresponding to the experimental conclusion in [L09]. Related mathematical works. In the theory of mathematical physics, localization in disordered, nonlinear dynamical systems was initiated by Fröhlich Spencer Wayne [F86] Similar work was also done by Pöschel [P90] and Vittot Bellissard [B85], who constructed infinite-dimensional, compact invariant tori for a large class of non-coupling systems i q n + V n q n + m Z ɛ mn q m + q m 2 q n =0, n Z, via the KAM techniques, where V n } n Z are i.i.d. random variables, ɛ mn are sufficiently small and vanish for m n large enough. Solutions which lie on such tori are

4 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 119 localized for all times. Besides the conclusion, they raised the following conjecture in that paper. Conjecture. [F86]. Consider the equation i q n + ɛq n+1 + q n 1 +V n q n + δ q n 2 q n =0, n Z, 1.2 with V n } n Z i.i.d. random variables. If ɛ and δ are small enough, with the equation in a large class, then for most initial conditions Most, e.g., with respect to the uniform measure on finite-dimensional unit spheres., q0 = q n 0 n Z, of finite support, the solutions qt =q n t n Z of 1.2 satisfy lim t t 1 n 2 q n t 2 =0. n Z Recently, there are several breakthroughs on such problem. For a large class of equation in 1.2, Bourgain Wang [BW08] constructed a quasi-periodic solution when ɛ, δ are sufficiently small. They also considered the slightly tempered equations [B07], with the nonlinearity replaced by λ n q n 2 q n, λ n <ɛ1+ n τ for some small τ>0. By constructing symplectic transformations to create energy barriers, they proved that, if ɛ is sufficiently small, then the diffusion bound i.e., the H 1 norm of the solution grows polynomially with t almost surely. Moreover, a Nekhoroshev-type result about Equation 1.2 was given by Wang Zhang [W09], who proved the long time Anderson localization for arbitrary l 2 initial data. By establishing an abstract KAM theorem, Geng Zhao [G13] constructed smallamplitude time quasi-periodic solutions of the lattice Schrödinger equation i q n + ɛq n+1 + q n 1 +tanπx + n αq n + ɛ q n 2 q n =0, n Z, 1.3 for most of x R/Z if ɛ is sufficiently small and α is Diophantine. This is based on the work by Bellissard Lima Scoppola [BLS83], which have studied the linear operator corresponding to Equation 1.3, the well-known Maryland model. The operator, which has dense point spectrum, describes media with no resonance, and this provides convenience for the KAM iteration. Statement of the main result. Inspired by the conjecture in [F86], we try to establish a nonlinear version of dynamical localization in the quasi-periodic potential case. Consider the one-dimensional nonlinear Schrödinger equation i q n + ɛq n+1 + q n 1 +V n α + xq n + q n 2 q n =0, n Z, 1.4 for 0 <ɛ 1, with V a nonconstant real-analytic function on R/Z, and α R is a Diophantine number, i.e., there exist τ >1and γ>0 such that x 1 is the absolute value of x modulo 1 defined so that 0 x n α 1 γ, n n τ

5 120 J. GENG ET AL. GAFA The nonconstant real-analytic potential V, as in[e97], is a smooth function in the Gevrey class sup m V x CL m m! 2, m 0, 1.6 x R/Z for some C, L>0, and satisfying the transversality condition max 0 m s m ϕ V x + ϕ V x ξ >0, x, ϕ, 1.7 max 0 m s m x V x + ϕ V x ξ ϕ 1, x, ϕ, 1.8 for some ξ, s>0. Clearly, the case V x =cos2πx is included. Based on an earlier KAM mechanism which was introduced by Eliasson [E97], we construct an abstract KAM theorem, and apply this theorem to prove welllocalization of Equation 1.4 for typical initial data. From the KAM perspective, the main technical challenges in this work are the following: i Unlike the model in [F86], we need to tackle with the second order perturbation in the Hamiltonian; ii Different from the method in [BW08], our proof is developed from the traditional KAM method; iii Compared with the work in [G13], the main difficulty is that the corresponding linear operator has dense point spectrum with infinitely many resonances. The main result can be stated as follows. Theorem 1. Consider the one-dimensional nonlinear Schrödinger equation i q n + ɛq n+1 + q n 1 +V n α + xq n + q n 2 q n =0, n Z, 1.9 with V a nonconstant real-analytic function on R/Z, and α R a Diophantine number. Given an integer b>1, andanyj = n 1,...,n b } Z, there exists a sufficiently small ɛ = ɛ V, α, J, such that if 0 <ɛ<ɛ, then the following holds for a.e. x R/Z. There exists a Cantor set O ɛ = O ɛ x [0, 1] b with [0, 1] b \O ɛ 0 1 as ɛ 0 such that the solution qt =q n t n Z of Equation 1.9, with initial datum q0 O ɛ supported on J, satisfies, for any fixed d>0, sup t n 2d q n t 2 <. n Z Moreover, for each n Z, q n t is quasi-periodic in time. 1 Hereafter, we use the symbol O to denote the Lebesgue measure of O R b.

6 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 121 Remark 1.1. The quasi-periodic solutions we obtained are not necessarily smallamplitude, since the nonlinearity q n 2 q n is integrable. Moreover, if the nonlinearity is diagonal dominant with some short-range decay, e.g., q n 2 q n + ɛ m n e ϱ m n q m 2 q n, the theorem above can also be obtained. Remark 1.2. Smallness assumption on ɛ is necessary, otherwise the result is not true even for the linear problem. This is different from the random potential case. 2 Preliminaries From now on, in the formulations and proofs of various assertions, we shall encounter absolute constants depending on the Hamiltonian, the dimension and so on. All such constants will be denoted by c, c 1,c 2,..., and sometimes even different constants will be denoted by the same symbol. 2.1 The quasi-periodic Schrödinger operator. Consider the Schrödinger operator T = T x :l 2 Z l 2 Z definedas Tq n := ɛq n+1 + q n 1 +V x + n αq n, n Z, 2.1 with V and α as in Equation 1.9. It is well-known from [E97] that if ɛ is sufficiently small, then for a.e. x R/Z, the spectrum of T x is pure point. We refer to [BG00, C93,F90,J94,K05,S87] for other works on the pure point spectrum and Anderson localization of quasi-periodic Schrödinger operators. Now, we are going to represent the main idea of [E97], which is critical for the KAM iteration in this paper. Let us start with some notations for infinitedimensional matrices. Given an infinite-dimensional matrix D, withd mn R the m, nth entry, for a subset Λ Z, wedefineλ := Z \ Λ, R Λ := n R Z Dmn, m,n Λ : n i =0 if i Λ}, D Λ := δ mn, otherwise. Then D Λ : R Λ + R Λ R Λ + R Λ,actsasR Λ R Z D R Z proj R Λ on the first component and as the identity on the second component. When there is no risk for confusion, we will use D Λ also to denote its first component. Let D 0 =diagv x+n α} n Z and Z 0 = ɛδ with Δ the discrete Laplacian. With ε 0 = ɛ 1 4, σ 0 =1andany 2 s+4 L s+1 s +1! 2 } M 0 max C, 2 τ, 8, N 0 1, ρ 0 = N0 ξ 1,

7 122 J. GENG ET AL. GAFA one can define the following sequences as in [E97], M ν+1 = M sm ν 3 ν, a ν = 1 τ M 3 sm ν 3 ν, ε ν+1 = ε 1 2 ε aν /2 ν N ν+1 = ε aν ν, ρ ν+1 = ε aν ν, σ ν+1 = 1 3 ρ ν. ν, 2.2 These sequences of parameters will be applied in the KAM iteration in this paper. Proposition 1. There exists a constant ɛ 0 = ɛ 0 C, L, ξ, s, γ, τ such that the following holds for the operator 2.1 if 0 <ɛ<ɛ 0. Fix any x R/Z. There exists a sequence of orthogonal matrices U ν, ν =1, 2,..., with U ν I Z mn ε e 3 2 σν m n, such that U ν D 0 + Z 0 U ν = D ν + Z ν, where Z ν is a symmetric matrix satisfying Z ν mn ε ν e ρν m n, and D ν is a symmetric matrix which can be block-diagonalized via an orthogonal matrix Q ν with Q ν mn =0 if m n >N ν. More precisely, there is a disjoint decomposition j Λν j = Z such that D ν = Q νd ν Q ν = j D ν Λ ν j with Λ ν j M ν, diamλ ν j M ν N ν, j. 2 Moreover, there exists a full-measure subset X R/Z such that if we fix x X, then for each k Z, there is a ν 0 k such that Λ ν+1 k =Λ ν k, ν ν 0 k. Proposition 1 shows the pure point spectrum of T. In Appendix A.1, we shall give an outline of the proof. 2.2 Decay property of matrices. Lemma 2.1. Given two matrices G =G mn m,n Z and F =F mn m,n Z.LetK = GF. 1 If G mn c G e σg m n, F mn c F e σf m n for some positive c G, c F, σ G, σ F > 0, then we have K mn c K e σk m n for any 0 <σ K < minσ G,σ F } and c K = c c G c F minσ G,σ F } σ K 1. 2 The disjoint decomposition defines an equivalence relation m n on the integers and, for each n Z, we denote its equivalence class by Λ ν n.

8 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS If G mn c G e σg max m, n }, F mn c F e σf m n, then K mn c K e σk max m, n }. 3 If G mn c G e σg m n, F mn c F e σf max m, n }, then K mn c K e σk max m, n }. 4 If G mn c G e σg max m, n }, F mn c F e σf max m, n }, then K mn c K e σk max m, n }. In particular, if σ G σ F, then the conclusions above hold with σ K =minσ G,σ F } and c K = c c G c F σ G σ F 1. Proof. Since the matrix element of K = GF can be formulated as K mn = l Z G ml F ln, we have that, in Case 1, for any 0 <σ K < minσ G,σ F }, GF mn G ml F ln l Z c G c F e σg m l σf l n e l Z c G c F e σk m n l Z e σg σk m l σf σk l n e c c G c F minσ G,σ F } σ K 1 e σk m n. Here we have applied the basic triangular inequality m l + l n m n. Moreover, if σ G σ F, assume that 0 <σ G <σ F without loss of generality, then GF mn c G c F e σg m l σf l n e l Z c G c F e σg m n l Z σf σg l n e c c G c F σ F σ G 1 e σg m n. As for Case 2 4, the corresponding conclusions can also be obtained by using the trivial facts m l + max l, n } max m, n }, max m, l } + max l, n } max m, n }. Thus Lemma 2.1 has been proved. Remark 2.2. If we replace the matrix F satisfying F mn c F e σf max m, n } with a vector f =f n n Z satisfying f n c f e σf n in Case 3 and 4, then for the vector Gf, we can obtain the conclusion that Gf n c K e σk n, where c K and σ K are the same as in the lemma.

9 124 J. GENG ET AL. GAFA 3 Abstract KAM Theorem 3.1 Function space norms. Given d, ρ > 0, let l 1 d,ρ Z be the space of summable complex valued sequences q =q n n Z, with the norm q d,ρ := n Z q n n d e ρ n <, where n := 1+n 2.Forr, s > 0, let D d,ρ r, s be the complex b-dimensional neighborhood of T b 0} 0} 0} in T b R b l 1 d,ρ Z l1 d,ρ Z, i.e., D d,ρ r, s :=θ, I, q, q : Imθ = Imθ 1,...,θ b <r, I <s 2, q d,ρ = q d,ρ <s}, where is the l 1 -norm of complex vectors. Given a real-analytic function F θ, I, q, q; ξ ond = D d,ρ r, s, C 1 W i.e., C1 in the sense of Whitney parametrized by ξ O 3, a closed region in R b.weexpand F into the Taylor Fourier series with respect to θ, I, q, q: F θ, I, q, q; ξ = F αβ θ, I; ξq α q β, α,β where, for multi-indices α := n Z α ne n, β := n Z β ne n, α n,β n N, with finitely many non-vanishing components, F αβ θ, I; ξ = F klαβ ξi l e i k,θ, q α q β = qn αn q n βn. k Z b,l N b α n,β n 0,0 Here e m denotes the vector with the mth component being 1 and the other components being zero. Definition 3.1. For each non-zero multi-index α, β =,α n,β n, n Z, α n, β n N, with finitely many non-vanishing components, we define n + αβ := maxn Z :α n,β n 0, 0}, n αβ := minn Z :α n,β n 0, 0}, n αβ := max n+ αβ, n αβ }, and α := n α n, β := n β n. In particular, for α = β =0,wedefinen + αβ = n αβ = n αβ =0. let With ξ F klαβ := b i=1 ξ i F klαβ and F klαβ O := sup ξ O F klαβ + ξ F klαβ, F αβ O := k,l F klαβ O I l e k Imθ, F O := k,l,α,β F klαβ O I l e k Imθ q α q β. 3 In this paper, all dependencies on the parameter ξ O are assumed of class C 1 W,thusall derivatives with respective to ξ will be interpreted in this sense.

10 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 125 Define the weighted norm of F as F D, O := sup F O. 4 D For the Hamiltonian vector field X F = I F, θ F, i qn F n Z, i qn F n Z associated F on D O, its norm is defined by X F D, O := I F D, O + 1 s 2 θf D, O +sup D 1 s qn F O + qn F O n d e n ρ. Sometimes, for the sake of notational simplification, we shall not write the subscript O in the norms defined above if it is obvious enough. Given two real-analytic functions F and G on D, let, } denote the Poisson bracket of such functions, i.e., F, G} = I F, θ G θ F, I G +i n Z qn F qn G qn F qn G. n Z Some basic estimates about the Hamiltonian vector field and the Poisson bracket are given in Appendix A Statement of the abstract KAM theorem. Now, we consider the perturbed Hamiltonian H = N + P + P = ex, ξ+ ωx, ξ,i + Ωx, ξq, q + P q, q; x+p θ, I, q, q; x, ξ, 3.1 defined on the domain D = D d,ρ r, s. Our goal is to prove that, for a.e. x R/Z, the Hamiltonian H admits invariant tori for most of the parameter ξ O= Ox, provided that X P +P D,O is sufficiently small. Remark 3.1. From now on, we shall not report x for convenience if it is irrelevant. To this end, we need to impose some conditions on ω, Ω, and the perturbations P + P. A1 Nondegeneracy of tangential frequencies: The map ξ ω is a C 1 W diffeomorphism between O and its image. A2 Regularity of Ω: Ω = T + A + W. T is the symmetric matrix defined in 2.1, independent of ξ. More precisely, with V and α as in Equation 1.9. T =diagv x + n α} n Z + ɛδ, 4 In the case of a vector-valued function F : D O C n n <, the norm is defined as F D, O := n i=1 Fi D, O.

11 126 J. GENG ET AL. GAFA A is Hermitian, independent of ξ, satisfying A mn c, m, n ˆN 0, otherwise 3.2 for some positive ˆN. W is C 1 W parametrized by ξ O,with W mn O pe σ max m, n }, m, n N 0, otherwise 3.3 for some positive p 1, σ ρ and sufficiently large N. Moreover, there exists a subset J Zsuch that Ω mn 0 if m or n J. 3.4 A3 Short range of P : P q, q = α = β 2 P αβ q α q β is real-analytic in q, q, and independent of ξ, with P αβ e ρn+ αβ n αβ, α = β 2, 3.5 qn P = qn P 0, n J. 3.6 A4 Decay property of P : P = α,β P αβθ, I; ξq α q β is real-analytic in θ, I, q, q, CW 1 parametrized by ξ O, and, with ε = ɛ 1 4, εe ρn αβ, P αβ D, O e ρn αβ, α + β 2, α + β qn P = qn P 0, n J. 3.8 A5 Gauge invariance of P :ForP = P klαβ I l e i k,θ q α q β,wehave P klαβ 0 if k Z b,l N b α,β b k j + α β 0. j=1 Our abstract KAM theorem can be stated as follows. Theorem 2. Consider the Hamiltonian H in 3.1, witha1 A5 satisfied. There is a positive constant ε = ε ω, V, α, ˆN, p, σ, N, r, s, d, ρ such that if X P +P D,O ε ε, then for every x X, there exists a Cantor set O ε = O ε x Ox with O \ O ε 0 as ε 0, such that the following holds. a There exists a CW 1 map ω : O ε R b, such that ω ω Oε 0 as ε 0. b There exists a map Ψ:T b O ε D d,0 r/2, 0, real-analytic in θ T b and CW 1 parametrized by ξ O, such that Ψ Ψ 0 Dd,0r/2,0, O ε 0 as ε 0, where Ψ 0 is the trivial embedding: T b O T b 0} 0} 0}.

12 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 127 c For any θ T b and ξ O ε, Ψθ + ωξt, ξ =θ + ωξt, It,qt, qt is a b-frequency quasi-periodic solution of equations of motion associated with the Hamiltonian 3.1. d For each t, qt =q n t n Z l 1 d,0 Z. Remark 3.2. The statement d of Theorem 2 implies that sup t n 2d q n t 2 <c n Z sup t 2 n d q n t <, which is exactly the conclusion of Theorem 1. Remark 3.3. In case that H satisfies A1 A5 at the first step, all assumptions hold for the Hamiltonian at each KAM step with suitable parameters. 3.3 Application to Equation 1.9. The Hamiltonian associated with Equation 1.9 is H = V x + n αq n q n + ɛ q n q n+1 + q n q n n Z n Z n Z Fix J = n 1,...,n b } Z, andz 1 = Z \J.Letε = ɛ 1 4,withɛ sufficiently small such that n i ln ε = 1 ln ɛ, i =1,...,b. 4 We introduce action-angle variables and amplitude parameters to the Hamiltonian 3.9, q n = I n + ξ n e iθn, q n = I n + ξ n e iθn, n J, where I,θ=I n1,...,i nb,θ n1,...,θ nb are the standard action-angle variables in the q n, q n n J -space around ξ, withξ =ξ n1,...,ξ nb O=[ɛ 1 12, 1] [0, 1] b a parameter, and q, q = q n, q n n Z1. Then the Hamiltonian 3.9 becomes H = N θ, I, q, q; x, ξ+ P q, q+p θ, I, q, q; ξ, with N θ, I, q, q; x, ξ := V x + n αξ n ξ2 n+ V x + n α+ξ n I n n J n J + V x + n α q n 2 + ɛ q n q n+1 + q n q n+1, n Z n/ J 1 n+1 / J, P q, q := 1 q n 4, 2 n Z 1 P θ, I, q, q; ξ := 1 In 2 + ɛ Im + ξ m e iθm q n + e iθm q n 2 + ɛ n J m, n J m n =1 m J,n/ J m n =1 n Z Im + ξ m In + ξ n e iθm θn + e iθm θn.

13 128 J. GENG ET AL. GAFA After introducing the action-angle variables, we find that the structure of the linear operator T in 3.9 has been destroyed. To overcome this disadvantage, we need to add b variables q n 1,...,q n b and the corresponding conjugates q n 1,..., q n b into this system. For convenience, omit the prime of the newly-added variables and still use q to denote q n n Z, since there is no confusion. We then rewrite N as N = V x + n αξ n ξ2 n+ V x + n α+ξ n I n n J n J [ + V x + n α q n 2 + V x + n α q n 2 + ɛ ] n q n+1 + q n q n+1 n Z 1 n J n Z q V x + n α q n 2 ɛ q n q n+1 + q n q n+1 n J n or n+1 J = ex, ξ+ ωx, ξ,i + T xq, q + Axq, q, with ex, ξ := V x + n αξ n + 1 ξ 2 n, 2 n J n J ωx, ξ :=Vx + n 1 α+ξ n1,..., Vx + n b α+ξ nb, 3.10 V x + m α, m = n T mn x := ɛ, m n = ±1, , otherwise V x + m α, m = n, m J A mn x := ɛ, m n = ±1, m or n J , otherwise Now, on some domain D d,ρ r, s, the regularity of P + P holds true: Lemma 3.1. For ɛ>0 sufficiently small and s = 1 8 ɛ 1 4,if I <s 2 and q d,ρ <s, then X P +P Dd,ρr,s,O ɛ 1 4 = ε. We need to show that the Hamiltonian H = N + P + P satisfies the assumptions A1 A5 of the KAM theorem, in which A3 and A5 are obviously satisfied. A1: Since V x + n α} n Z is independent of ξ, wehavethat ω ξ I J in view of Thus A1 holds. A2: Here, W 0. Then it is evident that A2 holds with ˆN = 1 4 ln ɛ, by A4: We focus on the formulation of P. Note that terms of P merely correspond to the normal variables q n, q n, n J, n 1orn +1 J, with the coefficients no more than ɛ, andj [ ˆN, ˆN] =[ 1 4 ln ɛ, 1 4 ln ɛ ]. Then, with ρ 1 ˆN 6 1,3.7 is verified since cɛ ɛ 1 4 e ρ ˆN.

14 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 129 Hence, Theorem 1 is a corollary of Theorem 2. 4 KAM Step To start the KAM iteration for the Hamiltonian 3.1, let D 0 = D d,ρ0 r 0,s 0, O 0, N 0 including e 0, ω 0, W 0, p 0, σ 0, N 0, P 0, ε 0 = ɛ 1 4 denote the initial quantities given in the assumptions A1 A5 respectively, and require that ɛ smaller than the ɛ 0 given in Proposition 1. Suppose we have arrived at the ν th step of the KAM iteration, ν =0, 1, 2,..., recalling that several sequences have been given in 2.2. We consider the Hamiltonian on D ν := D d,ρν r ν,s ν ando ν, H ν = N ν + P + P ν = e ν + ω ν,i + Ω ν q, q + P + P ν, 4.1 where Ω ν = T + A + W ν,anda1 A5 are satisfied, including 3.2, 3.5, 3.6 and Ω ν mn 0, m or n J, 4.2 pν e W ν mn Oν σν max m, n }, m, n N ν, 4.3 0, otherwise εν e P ν αβ Dν,O ν ρνn αβ, α + β 2, 4.4 e ρνn αβ, α + β 3 qn P ν = qn P ν 0, n J. 4.5 Moreover, X P +Pν Dν,O ν ε ν. [ ] Choose some r ν+1 such that 0 <r ν+1 <r ν, and let J ν := 5 aν 2 ε 2 ν.forj = 0, 1,...,J ν, we define the quantities at each KAM sub-step as ρ j ν = 1 j ρ ν, r ν j = r ν jr ν r ν+1, s j ν =2 3j ε j 5 ν s ν, 2J ν J ν and D ν j = D d,ρν+1 r ν j,s j ν, ε j ν = ε j +1 5 ν. Our goal is to construct a set O ν+1 O ν and a finite sequence of maps Φ j ν : D j ν D j 1 ν, j =1, 2,...,J ν, so that the Hamiltonian transformed into the ν + 1th KAM cycle H ν+1 = H ν Φ 1 ν Φ Jν ν = N ν+1 + P + P ν+1 = e ν+1 + ω ν+1,i + Ω ν+1 q, q + P + P ν+1

15 130 J. GENG ET AL. GAFA satisfies all the above iterative assumptions A1 A5 on D ν+1 = D ν Jν parametrized by ξ O ν+1, with new suitable parameters. Moreover, and C 1 W X P +Pν+1 Dν+1, O ν+1 ε Jν ν ε 1 2 ε aν /2 ν ν = ε ν+1. In the remaining part of this paper, all constants labeled with c, c 0, c 1,... are positive and independent of the iteration step. 4.1 Construction of O ν+1. As described in Proposition 1, there exists an orthogonal matrix U ν with U ν I Z mn ε e 3 2 σν m n, 4.6 such that U ν TU ν = D ν + Z ν, where Z ν is a symmetric matrix satisfying Z ν mn ε ν e ρν m n, 4.7 and D ν is a symmetric matrix which can be block-diagonalized via an orthogonal matrix Q ν with More precisely, Q ν mn =0 if m n >N ν. 4.8 D ν = Q νd ν Q ν = j D ν Λ ν j with Λ ν j M ν, diamλ ν j M ν N ν, j. To describe U ν Ω ν U ν, we need furthermore to consider U ν AU ν and U ν W ν U ν.in view of 3.2, 4.3 and4.6, there exists a constant c 1 > 0 such that U ν A + W ν U ν mn Oν c 1 max ˆN 2 e 3σν ˆN, p ν σ 2 ν } e σν max m, n }, by a simple application of Lemma 2.1. Define the truncation Âν as Âν mn := U ν A + W ν U ν mn, m, n N ν 0, otherwise. 4.9 It follows that Uν A + W ν U ν Âν mn ε ν e ρν max m, n } 4.10 Oν under the assumption C1 c 1 max ˆN 2 e ˆN, 3σν p ν σν 2 } e σν ρνnν ε ν. Let K ν+1 := N ν+1 M ν +1N ν with the sequences M ν, N ν, ν =0, 1,..., defined in 2.2 and D Λ ν := Dν ν Λ ν j, Ã ν := Q νâνq ν, 4.11 Λ ν j Λ ν

16 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 131 where Λ ν := Λ ν j :Λν j [ K ν+1 + N ν, K ν+1 + N ν ] } [ N ν+1, N ν+1 ]. In view of 4.8 and4.9, we have Ãν mn 0 if m or n > 2N ν. Since both of D Λ ν ν that and Ãν are Hermitian, there is an orthogonal matrix O ν such Oν D Λ ν + ÃνO ν ν =diagμ ν j } j Λ ν, where μ ν j } j Λ ν are eigenvalues of D Λ ν + Ãν. Due to the block-diagonal structure of ν D Λ ν + Ãν, wealsohave ν O ν mn 0 if m n > 2M ν +2N ν Indeed, Dν Λ ν + Ãν can be expressed as D Λ ν + ν Ãν = D Λ ν + Ãν ν Λ ν j [ 2N ν,2n ν]= D ν Λ ν j where Λ ν := Λ ν j :Λν j [ 2N ν, 2N ν ], Λ ν j Λν } with diamλ ν 2M ν +2N ν. The diagonalization of Dν Λ ν + Ãν is just the diagonalization of blocks D Λ ν + Ãν ν and D Λ ν. ν j As for the eigenvalues of D Λ ν +Ãν, it is well-known that μ ν n} ν n Λ ν CW 1 -smoothly depend on ξ and there exist orthonormal eigenvectors ψn ν corresponding to μ ν n, CW 1 - smoothly depending on ξ see e.g. [C93]. In fact, μ ν n = D Λ ν + Ãνψn, ν ψ n ν and ξj μ ν n = ξj D Λ ν + Ãνψ ν n, ψ ν n, j =1,...,b. By the construction of à ν,wehave ξj à ν = Q ν ξj  ν Q ν,withâν the truncation of U ν A + W ν ξu ν. Since D ν, A, U ν and Q ν are all independent of ξ, sup ξ μ ν n c sup ξ W ν mn cp ν ξ O ν ξ Oν m,n Now we defined the new parameter set O ν+1 O ν as k, ω ν > γν k, k 0, τ O ν+1 := ξ O ν : k, ω ν + μ ν n > γν k τ N, k 0, n Λ ν, ν+1 2 k, ω ν + μ ν m ± μ ν n >, k 0, m,n Λ ν 4.14 γν k τ N 4 ν+1 for some 0 <γ ν 1, τ b. These inequalities are famous small-divisor conditions for controlling the solutions of the linearized equations. From now on, to simplify notations, the subscripts or superscripts ν of quantities at the νth step are neglected, and the corresponding quantities at the ν +1th step are labeled with +. In addition, we still use the superscript j to distinguish quantities at various sub-steps.

17 132 J. GENG ET AL. GAFA 4.2 Homological equation and its approximate solution. For P = k,l,α,β P klαβ ξi l e i k,θ q α q β, according to 4.4 and the definition of norm in Subsection 3.1, we have P klαβ O εe ρn αβ e k r, 2 l + α + β 2, k Z b Decompose P = R +P R with R := k 2 l + α + β 2 P klαβ e i k,θ I l q α q β, P R = k 2 l + α + β 3 P klαβ e i k,θ I l q α q β. It follows X R D, O X P D, O ε. Recalling that P q, q isasumofhigh-order terms, there exists a constant c 2 > 0 such that X P Dd,ρr, ηs, O, X P R Dd,ρr, ηs, O c 2 ηs 1 8 ε 6 5, 4.16 with η := ε 1 5, provided C2 c 2 s 1 8 ε. Let e := P 0000 and ω := have P I dθ q= q=0,i=0. With O + defined as in 4.14, we Proposition 2. There exist two real-analytic Hamiltonians F = k l + α + β 2 F klαβ q α q β I l e i k,θ, `P = k 1 α + β 2 `P k0αβ q α q β e i k,θ, and a Hermitian matrix W, all of which are C 1 W parametrized by ξ O +, such that N,F} + R = e + ω,i + W q, q + `P Moreover, both of F and `P have gauge invariance, and for ε sufficiently small, F klαβ O+ ε 4 5 k 2τ+1 e k r e ρn αβ, 4.18 `P k0αβ O+ ε 7 5 k 2τ+1 e k r e ρ1 n αβ, 4.19 W mn εe ρ max m, O+ n }, m, n N +, m,n J, , otherwise qn F = qn F = qn `P = qn `P 0, n J ProofofProposition2: We decompose the proof into the following parts.

18 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 133 Truncation and approximate linearized equations At first, we rewrite R as R = P kl00 e i k,θ I l + k k l 1 + P k02 q, q e i k,θ, P k10,q + P k01, q + P k20 q, q + P k11 q, q where P k10, P k01, P k20, P k11, P k02 respectively denote Pn k10 := P k0en0, Pn k01 := P k00en, Pmn k20 := P k0em+e n0, P mn k11 := P k0eme n, P k02 mn := P k00em+e n. The gauge invariance A5 implies that P 010,P 001,P 020,P We try to construct a Hamiltonian F, of the same form as R, such that N,F} + R = e + ω,i + P 011 q, q By a straightforward calculation and simple comparison of coefficients, Equation 4.22 is equivalent to the following equations for k 0 and l 1, k, ω F kl00 =ip kl00, 4.23 k, ω I Z ΩF k10 =ip k10, 4.24 k, ω I Z +ΩF k01 =ip k01, 4.25 k, ω I Z ΩF k20 F k20 Ω=iP k20, 4.26 k, ω I Z ΩF k11 + F k11 Ω=iP k11, 4.27 k, ω I Z +ΩF k02 + F k02 Ω=iP k In view of the definition of O +, we know that 4.23 issolvedono +,with F kl00 O+ γ 2 k 2τ+1 εe k r. As for , consider the equations k, ω I Z D + Â ˆF k10 =iˆr k10, 4.29 k, ω I Z +D + Â ˆF k01 =iˆr k01, 4.30 k, ω I Z D + Â ˆF k20 ˆF k20 D + Â =iˆr k20, 4.31 k, ω I Z D + Â ˆF k11 + ˆF k11 D + Â =iˆr k11, 4.32 k, ω I Z +D + Â ˆF k02 + ˆF k02 D + Â =iˆr k

19 134 J. GENG ET AL. GAFA instead, where D and  are defined in the previous subsection, and for k 0, ˆR kx n = U P kx n, n K + 0, otherwise, x = 10, 01, 4.34 ˆR kx mn = U P kx U mn, m, n K + 0, otherwise, x = 20, 11, By 4.6 and4.15, combining with Lemma 2.1, there exists c 3 > 0 such that This means U P kx n O c 3 σ ρ 1 εe ρ n e k r, 4.36 U P kx U mn O c 3 σ ρ 2 εe ρ max m, n } e k r U P kx ˆR kx n O 1 4 ε 7 5 e ρ 1 n e k r, 4.38 U P kx U ˆR kx mn O 1 4 ε 7 5 e ρ 1 max m, n } e k r 4.39 under the assumption that C3 c 3 σ ρ 4 e ρ ρ1 K ε 2 5. Equation provide us with approximate solutions to , with the error estimated later. Block-diagonalization and construction of F Consider the equations k, ω I Λ D Λ + à F k10 =i R k10, 4.40 k, ω I Λ + D Λ + à F k01 =i R k01, 4.41 k, ω I Λ D Λ + à F k20 F k20 D Λ + à =i R k20, 4.42 k, ω I Λ D Λ + à F k11 + F k11 D Λ + à =i R k11, 4.43 k, ω I Λ + D Λ + à F k02 + F k02 D Λ + à =i R k02, 4.44 where D Λ, à are defined as in 4.11 via the orthogonal matrix Q, and R kx Q := ˆRkx, x = 10, 01 Q ˆRkx Q, x = 20, 11, 02. Note that Q mn =0if m n >N, then by 4.34 and4.35, we have R n kx 0, if n >K + + N, x = 10, 01, R mn kx 0, if m or n >K + + N, x = 20, 11, 02. Thus, recalling that Λ := Λ j :Λ j [ K + + N, K + + N] }, solutions of these finite-dimensional equations satisfy

20 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 135 F n kx 0, if n Λ, x = 10, 01, F mn kx 0, if m or n Λ, x = 20, 11, 02. Then, in view of the facts k, ω I Z ± D + Ã F kx = k, ω I Z ± D + Ã F kx = k, ω I Λ ± D Λ + Ã F kx, x = 10, 01, k, ω I Λ ± D Λ + Ã F kx, x = 20, 11, 02, F kx D + Ã = F kx D Λ + Ã, x = 20, 11, 02, they are also solutions of k, ω IZ D + Ã F k10 =i R k10, k, ω I Z + D + Ã F k01 =i R k01, k, ω I Z D + Ã F k20 F k20 D + Ã =i R k20, k, ω I Z D + Ã F k11 + F k11 D + Ã =i R k11, k, ω I Z + D + Ã F k02 + F k02 D + Ã =i R k02, which are respectively equivalent to Equations since D can be blockdiagonalized by the orthogonal matrix Q. Now we focus on the following equations k10 k, ω μ n ˇF n =io Rk10 n, k01 k, ω + μ n ˇF n =io Rk01 n, k20 k, ω μ m μ n ˇF mn =io Rk20 O mn, k11 k, ω μ m + μ n ˇF mn =io Rk11 O mn, k02 k, ω + μ m + μ n ˇF mn =io Rk02 O mn, for k 0andm, n Λ, which is transformed from by diagonalizing D Λ + Ã via the orthogonal matrix O. Obviously, these equations can be solved in O +. Hence, are solved with ˆF kx QO ˇF = kx, x = 10, 01 QO ˇF kx O Q, x = 20, 11, 02. Let F kx U ˆF := kx, x = 10, 01 U ˆF kx U, x = 20, 11, 02, then we obtain a Hamiltonian F = F kl00 e i k,θ I l + k 0 k 0 F k10,q + F k01, q + F k20 q, q + F k11 q, q l 1 + F k02 q, q e i k,θ.

21 136 J. GENG ET AL. GAFA It is easy to see that F = F,bynoting F kl00 = F kl00, F k10 = F k01, F k01 = F k10, F k20 = F k02, F k11 = F k11, F k02 = F k20. Estimates for coefficients of F Let us consider Fmn k20 for instance, and the other terms can be treated in an analogous way. By the construction above, one sees that F k20 mn =i F 0 U mn1 Q n1n 2 O n2n 3 O n 3n 4 Q n 4n 5 ˆRk20 n 5n 6 Q n6n 7 O n7n 8 O n 8n 9 Q n 9n 10 U n 10n k, ω μ n3 μ n8, 4.45 where the summation notation F 0 denotes } n1 Z, n 2 n 1 N, n 3 n 2, n 4 n 3 2M +2N, n 5 n 4 N, n 10 Z, n 9 n 10 N, n 8 n 9, n 7 n 8 2M +2N, n 6 n 7 N by virtue of the structure of Q and O, i.e, 4.8 and4.12. Then, by 4.37 and Lemma 2.1, sup Fmn k20 ξ cγ 1 k τ N+σ 4 ρ 4 M 4 N 8 e 4M+10Nρ εe k r e ρ max m, n }. ξ O + Here we have applied the property of the orthogonal matrices Q and O, and used the factor e 4M+10Nρ to recover the exponential decay. To estimate ξj Fmn k20, we need to differentiate both sides of 4.42 with respect to ξ j, j =1, 2,...,b. Then we obtain the equation about ξj F k20 k, ω I Λ D Λ + à ξ j F k20 ξj F k20 D k20 Λ + à = R ξ j, which can also be solved by diagonalizing D Λ + à via O as above, where R ξ k20 j := i Rk20 ξj + F k20 ξj à ξj k, ω I à F k20. We get the formulation ξj F k20 mn = F 1 with F 1 denotes U mn1 Q n1n 2 O n2n 3 On 3n 4 R k20 ξ j n4n 5 O n5n 6 On 6n 7 Q n 7n 8 Un 8n, k, ω μ n3 μ n6 n1 Z, n 2 n 1 N, n 3 n 2, n 4 n 3 2M +2N, n 8 Z, n 7 n 8 N, n 6 n 7, n 5 n 6 2M +2N By the decay property of ˆR k20 and ξj Â, we have that k20 sup R ξ j mn cγ 1 k τ+1 N+σ 4 ρ 4 M 4 ξ O + N 8 e 4M+11Nρ εe k r e ρ max m, n }. }.

22 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 137 Thus there exists c 4 > 0 such that sup Fmn k20 + ξ Fmn k20 ξ O + c 4 γ 2 k 2τ+1 N+σ 8 ρ 6 M 8 N 14 e 8M+20Nρ εe ρ max m, n } e k r ε 4 5 k 2τ+1 e k r e ρ max m, n }, under the assumption C4 c 4 γ 2 σ ρ 6 N+M 8 8 N 14 e 8M+20Nρ ε Suppose that b i=1 k i +2 0, which means P k20 0. Then ˆR k20 0, since it is a truncation of U P k20 U. By the formulation of Fmn k20 in 4.45, F k20 0. Doing the same thing for F k11, F k02, F k10, F k01 as above, we obtain the gauge invariance of F and the inequality Estimates for coefficients of `P and Let W be the truncation of P 011, satisfying W mn P 011 mn, m, n N = +, 0, otherwise `P = `P 011 q, q + k 0 `P k10,q + `P k01, q + `P k20 q, q + `P k11 q, q with + `P k02 q, q e i k,θ `P 011 := P 011 W, `P k10 := P k10 U ˆR k10 ià + `ZF k10, `P k01 := P k01 U ˆR k01 +ià + `ZF k01, `P k20 := P k20 U ˆR k20 U ià + `ZF k20 if k20 À + `Z, `P k11 := P k11 U ˆR k11 U ià + `ZF k11 +if k11 À + `Z, `P k02 := P k02 U ˆR k02 U +ià + `ZF k02 +if k02 À + `Z, where À := A + W UÂU,and `Z := UZU. Then we obtain N,F} + R = e + ω,i + W q, q + `P By 4.4 and4.5, we have 4.20 holds and under the assumption C5 e ρ ρ1 N + ε `P mn O+ εe ρ max m, n } ε 7 5 e ρ 1 max m, n },

23 138 J. GENG ET AL. GAFA As for the case k 0in4.19, we only estimate `P k20, with the others entirely analogous. By 4.39 andc3, combining with Lemma 2.1, P k20 U ˆR k20 U = UU P k20 U ˆR k20 U mn O mn In view of 4.7 and4.10, 1 4 ε 7 5 e ρ 1 max m, n } e k r Àmn O cσ ρ 2 εe ρ max m, n }, `Z mn cσ ρ 2 εe ρ m n. Then, by applying Lemma 2.1 again, there exists c 6 > 0 such that F k20 À + `Z, À + `ZF k20 mn O+ mn O+ c 6 σ ρ 2 ρ ρ 1 1 ε 9 5 k 2τ+1 e k r e ρ1 max m, n } 1 4 ε 7 5 k 2τ+1 e k r e ρ1 max m, n }, 4.48 provided that C6 c 6 σ ρ 2 ρ ρ 1 1 ε Thus, we can obtain the estimate for `P k20 by putting 4.47 and4.48 together. By the construction of `P, the gauge invariance is easily verified. Verification of 4.21 In view of the construction of R and W above, the objects in 4.46 that may depend on the variables q n, q n n J are F and `P.Let F = Fn k10 q n + Fn k01 q n + Fmn k20 q m q n + Fmn k11 q m q n k 0 n J m or n J =: k 0 +F k02 mn q m q n e i k,θ F k10,q + F k01, q + F k20 q, q + F k11 q, q + F k02 q, q e i k,θ. O For m or n J,by4.2, we have k, ω I Z Ω F k20 F k20 Ω mn k20 = k, ω F mn k20 Ω ml F ln l J l J k, ω Fmn k20, m,n J = k, ω Fmn k20 l J Ω mlfln k20, m J, n J k, ω Fmn k20 l J F ml k20 Ω ln, m J, n J = k, ω I Z ΩF k20 F k20 Ω mn. F k20 ml Ω ln

24 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 139 This means, by comparing the coefficients in both side of Equation 4.46, k, ω I Z Ω F k20 F k20 k20 Ω = i `P mn m or n J. Similarly, mn k, ω I Z Ω F k10 k10 = i `P n, n J, n k, ω I Z +Ω F k01 k01 = i `P n, n J, n k, ω I Z Ω F k11 + F k11 k11 Ω = i `P mn, mn k, ω I Z +Ω F k02 + F k02 k02 Ω = i `P mn, mn m or n J, m or n J. Thus, N, F } equals to k10 k01 `P n q n + `P n q n + k20 k11 k02 `P mn q m q n + `P mn q m q n + `P mn q m q n e i k,θ. n J m or n J k 0 Hence, if we substitute F with F F, which is independent of the variables q n, q n n J, then `P in the homological equation 4.46 is replaced correspondingly, independent of q n, q n n J.4.21 is satisfied. 4.3 Verification of assumptions after one sub-step. We proceed to estimate the norm of X F, and to study properties of Φ 1 F on smaller domains D i := D d,ρ+ r 1 + i 4 r r1 i, 4s, i =1, 2, 3, 4. Lemma 4.1. For ε sufficiently small, we have X F D3, O + ε 3 4 and X `P D3, O + ε 5 4. Proof. In view of the decay property of F in Proposition 2, it follows that and 1 sup D 3 s 1 s 2 θf D3,O +, I F D3,O + cr r 1 2τ+b+1 ε 4 5, qn F O+ + qn F O+ n d e ρ+ n n Z sup D 3 c s +sup D 3 n Z k 0 c s n Z k 0 m Z F k10 n O+ + F k01 n O+ e k r 1 4 r r1 n d e ρ+ n F k20 mn O+ + F k11 mn O+ + F k02 mn O+ q m e k r 1 4 r r1 n d e ρ+ n cr r 1 2τ+b+1 ρ ρ + 2 ε 4 5.

25 140 J. GENG ET AL. GAFA Putting together the estimates above, there is a constant c 7 > 0 such that X F D3, O + c 7 r r 1 2τ+b+1 ρ ρ + 2 ε 4 5. In an entirely analogous way, we have X `P D3, O + c 7 r r 1 2τ+b+1 ρ 1 ρ + 2 ε 7 5. Moreover, if C7 c 7 r r 1 2τ+b+1 ρ 1 ρ + 2 ε , then Lemma 4.1 follows. Let D iη = D d,ρ+ r 1 + i 4 r r1, i 4ηs, i =1, 2, 3, 4. Lemma 4.2. For ε sufficiently small, we have Φ t F moreover, : D 2η D 3η, 1 t 1 and DΦ t F I D1η < 2ε 3 4. Let F 1, e 1, ω 1, W 1, `P 1 be the corresponding quantities in 4.17 respectively, which means that we are in the 1st sub-step. Define H 1 as where H 1 := H Φ 1 F 1 =N + P + R Φ 1 F 1 +P R Φ1 F 1 = N + P + N,F 1 } + R + + P 1 := `P tn,f 1 },F 1 } Φ t F 1 dt P + R, F 1 } Φ t F 1 dt +P R Φ1 F 1 = N + P + e 1 + ω 1,I + W 1 q, q + P 1, tn,f 1 } + P + R, F 1 } Φ t F 1 dt +P R Φ1 F 1. Let Rt :=1 te 1 + ω 1,I + W 1 q, q + `P 1 +tr, which satisfies X Rt D3 cε. ThenP 1 can be written as 1 P 1 = `P 1 + Rt+ P, F 1 } Φ t F dt +P R 1 Φ1 F 1. 0

26 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 141 Hence, X P 1 `P 1 = 1 0 Φ t F 1 X Rt+ P,F 1 } dt +Φ1 F 1 X P R. By Lemma A.4, X Rt+ P,F 1 } D 2η cη 2 ε 7 4 = ε Then, combining with 4.16, recalling the conclusion of Lemma 4.1 and 4.2, X P 1 D 1, O ε ε cε ε 6 5 = ε 1. Now we need to show P 1 satisfies assumptions A4 and A5. Note that P 1 = `P 1 + P R + P,F 1 } + P, F 1 } + 1 2! N,F1 },F 1 } + 1 2! P,F 1 },F 1 } + 1 2! P, F 1 },F 1 } n! N,F1 },F 1 } + 1 }} n! P,F 1 },F 1 } }} n n + 1 n! P, F 1 },F 1 } +. }} n Since all of N, P, P, F 1, `P 1 have gauge invariance, independent of variables q n, q n n J,sodoesP 1 due to Lemma A.5 and A.6 in Appendix. For P R = P klαβ e i k,θ I l q α q β,wehave 2 l + α + β 3 P αβ D ε1 e ρn αβ, α + β 2 e ρn αβ, α + β 3. Here we applied the estimate I s ε1 to handle the case that α + β 2 and 2 l + α + β 3. The decay property of remaining terms, which are made up of several Poisson brackets, is covered by the following lemmas. Lemma 4.3. For ε sufficiently small, P, F 1 } satisfies P, F 1 } αβ D3η,O + ε 5 4 e ρ1 n αβ, α + β 2 ε 1 4 e ρ1 n. αβ, α + β 3

27 142 J. GENG ET AL. GAFA Proof. A straightforward calculation yields that P, F 1 } αβ =i n Z ˇα, ˇβ+ ˆα, ˆβ=α,β + ˇα, ˇβ+ˆα, ˆβ=α,β 1 Pˇα+en, ˇβ F ˆα, ˆβ+e n Pˇα, F 1 ˇβ+en ˆα+e n, ˆβ Pˇα 4.49 } 1 ˇβ,F ˆα ˆβ Terms in 4.49 Let us consider terms Pˇα+en, ˇβ F ˆα, ˆβ+e first. n i α + β 2 Since ˆα + ˆβ + e n = 1 or 2 in view of the construction of F 1,wehavethat ˇα + e n + ˇβ = α + β +1 ˆα + ˆβ If ˇα + e n + ˇβ 2, then, noting that n αβ maxn ˇα+e n, ˇβ,n ˆα, ˆβ+e n },wehave 1 Pˇα+en, ˇβ F ˆα, ˆβ+e D3,O + n εe ρn ˇα+en, ˇβ 3 ε 4 e ρn ˆα, ˆβ+en ε 7 4 e ρn αβ If ˇα + e n + ˇβ = 3, then, by 4.51, ˆα, ˆβ =0, 0, ˇα, ˇβ =α, β. By the definition of norm X P D, O and the construction of F 1, P α+en,β D3,O e ρn α+en,β, 1 F 0,e n D3,O + sε 3 4 e ρ n. Thus, noting that n αβ maxn α+e, n }, n,β P α+en,βf 1 0,e n D3,O + sε 3 4 e ρn 1 αβ 4 ε 7 4 e ρn αβ ii α + β 3 By the same argument as above, 1 Pˇα+en, ˇβ F ˆα, ˆβ+e D3,O + n e ρn ˇα+en, ˇβ 3 ε 4 e ρn ˆα, ˆβ+en ε 3 4 e ρn αβ Doing the same for Pˇα, F 1, we finish estimates for terms in ˇβ+en ˆα+e n, ˆβ Terms in 4.50 By Lemma A.3 and the inequality n αβ },wehave maxn ˇα ˇβ,n ˆα ˆβ 1 Pˇα ˇβ,F ˆα ˆβ } D 3η cr r 1 1 η 2 ε 7 4 e ρn αβ, α + β ε 3 4 e ρn αβ, α + β 3 Combining , there exists c 8 > 0 such that P, F 1 } αβ D3η c 8 r r 1 1 η 2 ρ ρ 1 2 ε 7 4 e ρ1 n αβ, α + β 2 ε 3 4 e ρ1 n, αβ, α + β 3 applying the fact that ˆα + ˆβ 2. Moreover, if C8 c 8 r r 1 1 η 2 ρ ρ 1 2 ε , Lemma 4.3 is proved.

28 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 143 By 4.15, 4.19 and4.20, it is evident that the coefficients of N,F 1 } = e 1 + ω 1,I + W 1 q, q + `P 1 R satisfies N,F 1 } αβ D3,O + cεe ρ1 n αβ. Then we have the following lemma, whose proof is analogous to that of Lemma 4.3. Lemma 4.4. For ε sufficiently small, N,F 1 },F 1 } satisfies N,F 1 },F 1 } αβ D3η,O ε 6 5 e ρ 1 n αβ. Lemma 4.5. For ε sufficiently small, P,F 1 } satisfies Proof. It can be calculated that P,F 1 } αβ =i For Pˇα+en, ˇβ F 1 ˆα, ˆβ+e n P,F 1 } αβ D3,O + ε 1 4 e ρ 1 n αβ, α + β 3. n Z ˇα, ˇβ+ ˆα, ˆβ=α,β 1 Pˇα+en, ˇβ F ˆα, ˆβ+e Pˇα, F 1 ˇβ+en n ˆα+e n, ˆβ in 4.56, since ˆα + ˆβ + e n =1or2and ˇα + e n + ˇβ 4 here, it is obvious that α + β = ˇα + ˇβ + ˆα + ˆβ 3. Note that n αβ maxn ˇα+e n, ˇβ,n ˆα, ˆβ+e n },and n ˇα+e n, ˇβ = maxn+ˇα+e n, ˇβ, n ˇα+e n, ˇβ }, n ˆα, ˆβ+e n = maxn +ˆα, ˆβ+e n, n ˆα, ˆβ+e n }. Then n +ˇα+e n, ˇβ n ˇα+e n, ˇβ + n ˆα, ˆβ+e n n αβ, and hence Pˇα+en, ˇβ F 1 ˆα, ˆβ+e n D3,O + e ρn+ˇα+en, ˇβ n ˇα+en, ˇβ ε 3 4 e ρn ˆα, ˆβ+en ε 3 4 e ρn αβ. Doing the estimate for Pˇα, F 1 in 4.56 similarly, we have that ˇβ+en ˆα+e n, ˆβ P,F 1 } αβ D3,O + c 8 ρ ρ 1 2 ε 3 4 e ρ 1 n αβ ε 1 4 e ρ 1 n αβ, α + β 3, if C8 holds. as Summarize the analysis above, then the decay property for P 1 can be expressed Proposition 3. For ε sufficiently small, P 1 = α,β P 1 αβ θ, I; ξqα q β satisfies P 1 αβ ε D 1,O + 1 e ρ1 n αβ, α + β 2 e ρ1 n. αβ, α + β 3

29 144 J. GENG ET AL. GAFA 4.4 A succession of symplectic transformations. With the verification of assumptions A4 and A5 completed, we finish one sub-step of KAM iteration. Suppose that we have arrived at the j th sub-step, j =1,...,J,withJ = [ 5 2 ε ] a 2, then we encounter the Hamiltonian H j 1 = H Φ 1 F 1 Φ1 F j 1 = N + P j 1 + e i + ω i,i + W i q, q + P j 1, i=1 with the superscript 0 labeling quantities before the 1 st sub-step in particular. Let R j 1 := k 2 l + α + β 2 P j 1 klαβ ei k,θ I l q α q β As demonstrated in Proposition 2, ono +, the following homological equation N,F j } + R j 1 = e j + ω j,i + W j q, q + `P j, 4.58 can be solved, with F j, e j, ω j, W j, `P j having properties similar to F 1, e 1, ω 1, W 1, `P 1 respectively. Then we obtain H j = H j 1 Φ 1 F j = N + P + j i=1 e i + ω i,i + W i q, q + P j. The estimates for F j and the verification of assumptions for P j can be done similarly as in Subsection 4.3. The process above can be summarized as Proposition 4. Consider the Hamiltonian H in 4.1. There exist J symplectic transformations Φ j =Φ 1 F j, j =1,...,J, generated by the corresponding realanalytic Hamiltonians F j respectively, such that H j = H Φ 1 Φ j = N + P + G j + P j, is real-analytic on D j = D d,ρ+ r j,s j, with G j = j i=1 e i + ω i,i + W i q, q.fori =1, 2, 3, 4, η = ε 1 5, let D j i = D d,ρ+ r j+1 + i 4 rj r j+1, i 4 sj, D j iη = D d,ρ + r j+1 + i 4 rj r j+1, i 4 ηsj.

30 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 145 a With R j 1 defined in 4.57, F j satisfies the homological equation 4.58 on O +, X F j j 1 D 3, O + ε 1 4 ε j 1, Φ t F : D j 1 j 2η D j 1 3η, 1 t 1, and DΦ t F j I D j 1 1η F j αβ D j 1 3, O + qn F j = qn F j 0, < 2ε 1 4 ε j 1, ε 1 4 ε j 1 e ρj 1 n αβ, α + β 2 0, α + β 3, n J. b G j satisfies that X Gj D j 3, O + cε and for i =1, 2,...,j, ω i O+ ε i 1, W mn i ε O+ i 1 e ρi 1 max m, n }, m, n N +, m,n J 0, otherwise. c P j satisfies X P +P j D j, O + ε j and assumptions A4, A5, whichinclude P j αβ ε D j, O + j e ρj n αβ, α + β 2 e ρj n, αβ, α + β 3 qn P j = qn P j 0, n J. Let s + = s J =2 3J ε J 5 s, Φ=Φ 1 Φ J,and with Ω + = T + A + W +,and e + = e + N + = e + + ω +,I + Ω + q, q, J e j, ω + = ω + j=1 J ω j, W + = W + j=1 J W j. Then Φ : D + O + D O. From the estimates of ω j and W j,wehave ω + ω O+ cε, ε 2 e W + W mn O+ ρ max m, n } 2, m, n N +, m,n J , otherwise Since W = W and W i = W i, W + is still a Hermitian matrix. Then, by 4.59 and 4.60, A1 anda2 holdwithp + = p + ε 1 2 and σ + := 1 3 ρ. Let P + = P J. It has been verified that the assumptions A4 and A5 for P J hold, which is an analogue to the process in Subsection 4.3. This completes one step of KAM iterations. j=1

31 146 J. GENG ET AL. GAFA 5 Proof of the KAM Theorem With ε 0 = ɛ 1 4, σ 0 =1, ˆN = ln ε0,and 2 s+4 L s+1 s +1! 2 } 122τ + b +3 M 0 = max C, 2 τ, 8,, N 0 =6 ln ε 0, ξ τ ρ 0 = N0 1, one can define the following sequences as in [E97], M ν+1 = M sm ν 3 ν, a ν = 1 τ M 3 sm ν 3 ν, ε ν+1 = ε 1 2 ε aν /2 ν N ν+1 = ε aν ν, ρ ν+1 = ε aν ν, σ ν+1 = 1 3 ρ ν. ν, Given p 0 = ε 1 2 0, r 0 = r, s 0 = s, the other sequences are defined as [ p ν+1 = p ν + ε ν, K ν+1 = N ν+1 M ν +1N ν, J ν = ν+1 r ν = r i, s ν+1 =2 3Jν ε Jν 5 ν s ν, γ ν = ε 1 80 ν. i=2 2 ε ν Let D ν and O ν be as defined in Section Iteration lemma. The preceding analysis can be summarized as follows. Lemma 5.1. There exists ε 0 sufficiently small such that the following holds for all ν =0, 1,... a H ν = N ν + P + P ν is real-analytic on D ν,andcw 1 parametrized by ξ O ν, where N ν = e ν + ω ν,i + Ω ν q, q, P ν = α,β P ν αβ θ, Iq α q β, aν 2 ], with Ω ν = T + A + W ν satisfying Ω ν mn 0 if m or n J, W ν mn Oν ω ν+1 ω ν Oν+1 ε ν, W ν+1 W ν mn Oν+1 pν e σν max m, n }, m, n N ν 0, otherwise ε 1 2 ν e ρν max m, n } 2, m, n N ν+1, m,n J 0, otherwise,. Moreover, P ν has gauge invariance and X P +Pν Dν, O ν ε ν, εν e P ν αβ Dν, O ν ρνn αβ, α + β 2, e ρνn αβ, α + β 3 qn P ν = qn P ν 0, n J.

32 GAFA LOCALIZATION IN ONE-DIMENSIONAL QUASI-PERIODIC NONLINEAR SYSTEMS 147 b For each ν, there is a symplectic transformation Φ ν : D ν+1 D ν with such that H ν+1 = H ν Φ ν. DΦ ν Id Dν+1, O ν+1 ε 1 2 ν, Proof. Let c 0 := 8e 20 maxc 1,...,c 8 }. We need to verify the assumptions C1 C8 forν =0, 1,... By noting that ρ j ν N ν+1 = ε aν ν ρ j+1 ν = ρ ν ρ ν+1 2J ν, = ρ 1 ν+1, σ ν+1 = 1 3 ρ ν, r ν j r ν j+1 = r ν r ν+1, 2J ν it is sufficient for us to check: D1 c 0 s ν ε ν, 2τ+b+1 2 D2 c rν r ν+1 ρν ρ ν+1 0 2J ν 2J ν ε 1 20 ν, D3 c 0 Nν+1 8 M ν 8 Nν 20 e 8MνNνρν ε 7 40 ν, D4 e ρν K ν+1 2Jν ε 2 5 ν, for all ν =0, 1,... By the choice of s 0, the condition D1 clearly holds for ν = 0. Suppose that it holds for some ν, then it is easy to see that c 0 s ν+1 =2 3Jν ε Jν 5 ν c 0 s ν < 2 3Jν ε Jν 5 ν ε ν <ε ν+1. Hence D1 holds for all ν. Let us first take ε 0 sufficiently small such that ε 1 1 a02τ+b r0 2τ+b+1 1 ε a 0 0 c Here we have applied M 0 12 τ 2τ +b+3 and a 0 = M 3 sm [ such that ] a 02τ + b +3> 0. Then, recalling that r ν r ν+1 = r0 2 and J 2+ν ν = 5 aν 2 ε 2 ν, 2. r0 r 2τ+b+1 1 ρ0 ρ 2 1 c 0 ε , 2J 0 2J 0 i.e., D2 holds for ν = 0. Since for ν 1 and for ε 0 sufficiently small, ε aν2τ+b+3 ν we have ε 6 5 ν 0 c 0 rν r ν+1 2J ν 1 2 ν2τ+b+1 c 0 r0 20 2τ+b+1 ρν ρ ν+1 2τ+b+1, ε 1 40 ν 2J ν 2 ε 1 20 ν. a ε ν 1 ν 1 2 εaν ν, 5

33 148 J. GENG ET AL. GAFA Thus, D2 holds true. In Section 6 of [E97], the basic smallness assumption of ε ν, i.e., the inequality A.1 in Lemma A.1, has been verified, then all other assumptions are immediate, including the inequality Γ ν Nν 2 e 6MνNνρν ε 1 8 ν, where Γ ν increases superexponentially in M ν. Since all of M ν, N ν, ρ ν and ε ν here are defined in the same way as [E97], we can apply this inequality. So D3 has been verified. By the definition of ρ ν, a ν and ε ν,wehave ρ ν ε 1 2 aν ν > ln 1 ε ν. Then we see that D4 holds for ν =0, 1, Convergence. Now we fix x X,with X defined as in Proposition 1. This means that the blocks mentioned in Proposition 1 are eventually stationary after some step, i.e., for each n Z, there is a ν 0 n such that Λ ν+1 n =Λ ν n, ν ν 0 n. In this case, the local decay rate for n may not shrink with ν necessarilyρ ν is the global upper bound of the rates for all n Z. Define Ψ ν =Φ 0 Φ 1 Φ ν 1, ν =1, 2,... An induction argument shows that Ψ ν : D ν+1 D 0,and H 0 Ψ ν = H ν = N ν + P + P ν. Let O ε0 = ν=0 O ν. As in standard arguments e.g. [K93,P96], thanks to Lemma 4.2, it concludes that H ν, N ν, P ν,ψ ν, e ν, ω ν and W ν converge uniformly on D d,0 1 2 r 0, 0 O ε0 to, say, H, N, P,Ψ, e, ω and W respectively, in which case it is clear that N = e + ω,i + T + A + W q, q, with Ω = T +A+W satisfying Ω mn 0ifmor n J. Since X Pν Dν,O ν ε ν with ε ν 0, it follows that X P Dd,0 1 r0,0, =0. 2 Oε 0 Since H 0 Ψ ν = H ν,wehaveφ t H 0 Ψ ν =Ψ ν Φ t H ν,withφ t H 0 denoting the flow of the Hamiltonian vector field X H0. The uniform convergence of Ψ ν and X Hν implies that one can pass the limit in the above and conclude that Φ t H 0 Ψ =Ψ Φ t H, Ψ : D d,0 1 2 r 0, 0 D 0. Hence, Φ t H 0 Ψ T b ξ} = Ψ Φ t N T b ξ} =Ψ T b ξ}, ξ O ε0. This means that Ψ T b ξ} is an embedded invariant torus of the original perturbed Hamiltonian system at ξ O ε0. Moreover, the frequencies ω ξ associated with Ψ T b ξ} are slightly deformed from the unperturbed ones, ωξ.