KAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions

Transcription

1 KAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions Luigi Chierchia Dipartimento di Matematica Università Roma Tre Jiangong You Department of Mathematics Nanjing University, Nanjing , China nju.edu.cn June, 1999 Abstract In this paper, one dimensional (1D) nonlinear wave equations u tt u xx + V (x)u = f(u), with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u = 0. It is proved that for most potentials V (x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem which allows for multiple normal frequencies. 1 Introduction and Results In the 90 s the celebrated KAM (Kolmogorov-Arnold-Moser) theory has been successfully extended to infinite dimensional settings so as to deal with certain classes of partial differential equations carrying a Hamiltonian structure, including, as a typical example, wave equations of the form u tt u xx + V (x)u = f(u), f(u) = O(u 2 ); (1.1) see Wayne [17], Kuksin [10] and Pöschel [15]. In such papers, KAM theory for lower dimensional tori [14], [13], [8] (i.e., invariant tori of dimension lower than the number of This research was partly supported by NNSF of China, by the italian CNR grant # and by MURST (Project Metodi Variazionali ed Equazioni Differenziali Non Lineari ). 1

2 degrees of freedom), has been generalized in order to prove the existence of small-amplitude quasi-periodic solutions for (1.1) subject to Dirichlet or Neumann boundary conditions (on a finite interval for odd and analytic nonlinearities f). The technically more difficult periodic boundary condition case has been later considered by Craig and Wayne [7] who established the existence of periodic solutions. The techniques used in [7] are based not on KAM theory, but rather on a generalization of the Lyapunov-Schmidt procedure and on techniques by Fröhlich and Spencer [9]. Recently, Craig and Wayne s approach has been significantly improved by Bourgain [3], [4], [5] who obtained the existence of quasi-periodic solutions for certain kind of 1D and, most notably, 2D partial differential equations with periodic boundary conditions. The technical reason why KAM theory has not been used to treat the periodic boundary condition case is related to the multiplicity of the spectrum of the associated Sturm- Liouville operator A = d2 + V (x). Such multiplicity leads to some extra small denominator problems (related to the so called normal frequencies), which make the KAM dx 2 analysis particularly delicate. The purpose of this paper is to show that, improving the KAM machinery, one can indeed use KAM techniques to deal also with the multiple normal frequency case arising in PDE s with periodic boundary conditions (e.g., 1D wave equations). The advantage of the KAM approach is, from one side, to possibly simplify proofs and, on the other side, to allow the construction of local normal forms close to the considered torus, which could be useful for a better understanding of the dynamics. For example, in general, one can easily check linear stability and the vanishing of Liapounov exponents. A rough description of our results is as follows. Consider the periodic boundary problem for (1.1) with an analytic nonlinearity f and a real analytic (or smooth enough) potential V. Such potential will be taken in a d-dimensional family of functions parameterized by a real d vector ξ, V (x) = V (x, ξ), satisfying general non-degenerate ( non resonance - of eigenvalue ) conditions. Then for most potentials in the family (i.e. for most ξ in Lebesgue measure sense), there exist small-amplitude quasi-periodic solutions for (1.1) corresponding to d-dimensional KAM tori for the associated infinite dimensional Hamiltonian system. Moreover (as usual in the KAM approach) one obtains, for the constructed solutions, a local normal form which provides linear stability in the case the operator A is positive definite. Finally we hope that the technique used in this paper can be generalized so as to deal with more general situations such as, for example, 2D wave equations. The paper is organized as follows: In section 2 we formulate a general infinite dimensional KAM Theorem designed to deal with multiple normal frequency cases; in section 3 we show how to apply the preceding KAM Theorem to the nonlinear wave equation (1.1) with periodic boundary conditions. The proof of the KAM Theorem is provided in sections 4 6. Some technical lemmata are proved in the Appendix. 2 An infinite dimensional KAM Theorem In this section we will formulate a KAM Theorem in an infinite dimensional setting which can be applied to some 1D partial differential equations with periodic boundary conditions. 2

3 We start by introducing some notations. 2.1 Spaces For n N, let d n Z + be positive even integers 1. Let Z n N Cdn : the coordinates in Z are given by z = (z 0, z 1, z 2, ) with z n (zn, 1, zn dn ) C dn. Given two real numbers a, ρ, we consider the (Banach) subspace of Z given by where the norm a,ρ is defined as Z a,ρ = {z Z : z a,ρ < } z a,ρ = z 0 + n Z+ z n n a e nρ, (and the norm in C dn is taken to be the 1 norm z n = d n j=1 zn ). j In what follows, we shall consider either a = 0 and ρ > 0 or a > 0 and ρ = 0 (corresponding respectively to the analytic case or the finitely smooth case). The role of complex neighborhoods in phase space of KAM theory will be played here by the set P a,ρ ˆT d C d Z a,ρ, where ˆT d is the complexification of the real torus T d = R d /2πZ d. For positive numbers r, s we denote by D a,ρ (r, s) = {(θ, I, z) P a,ρ : Im θ < r, I < s 2, z a,ρ < s} (2.1) a complex neighborhood of T d {I = 0} {z = 0}. Finally, we denote by O a given compact set in R d with positive Lebesgue measure: ξ O will parameterize a selected family of potential V = V (x, ξ) in (1.1). 2.2 Functions We consider functions F on D a,ρ (r, s) O having the following properties: (i) F is real for real arguments; (ii) F admits an expansion of the form F = α F α z α, (2.2) where the multi-index α runs over the set α (α 0, α 1,...) n N Ndn with finitely many non-vanishing components 2 α n ; (iii) for each α, the function F α = F α (θ, I, ξ) is real analytic in the variables (θ, I) { Imθ < r, I < s 2 }; (iv) for each α, the dependence of F α upon the parameter ξ is of class C d 2 W (O) for some d > 0 (to be fixed later): here (O) denotes the class of functions which are m times differentiable on the closed set O C m W 1 We use the notations N = {0, 1, 2, }, Z+ = {1, 2, }. 2 Thus n 0 > 0 such that z α n Y 0 n=0 z αn n n Y 0 d n Y n=0 j=1 (z j n) αj n. 3

4 in the sense of Whitney [18] (and the appearance of the square is due to later notational convenience). The convergence of the expansion (2.2) in D a,ρ (r, s) O will be guaranteed by assuming the finiteness of the following weighted norm: F Da,ρ(r,s),O F α z α (2.3) where, if F α = sup z a,ρ s F klα (ξ)i l e i k,θ, (, being the standard inner product in C n ), k Z d,l N d F α is short, here, for α F α k,l F klα O s 2 l e k r, F klα O max p d 2 p F klα ξ p, (2.4) (the derivatives with respect to ξ are in the sense of Whitney). The set of functions F : D a,ρ (r, s) O C verifying (i) - (iv) above with finite Da,ρ(r,s),O norm will be denoted by F Da,ρ(r,s),O. 2.3 Hamiltonian vector fields and Hamiltonian equations To functions F F Da,ρ(r,s),O, we associate a Hamiltonian vector field defined as X F = (F I, F θ, {ij dn F zn } n N ), ( ) 0 I where J dn denotes the standard symplectic matrix dn/2 and i = 1; the I dn/2 0 derivatives of F are defined as the derivatives term by term of the series (2.2) defining F. The appearence of the imaginary unit is due to notational convenience and will be justified later by the use of complex canonical variables. Correspondingly we consider the Hamiltonian equations 3 θ = F I, I = Fθ, ż n = ij dn F zn, n N. (2.5) A solution of such equation is intended to be just a C 1 map from an interval to the domain of definition of F, D a,ρ (r, s), satisfying (2.5). Given a real number ā, we define also a weighted norm for X F by letting 4 X F ā,ρ (2.6) Da,ρ(r,s),O F I Da,ρ(r,s),O + 1 s 2 F θ Da,ρ(r,s),O + 1 s ( F z 0 Da,ρ(r,s),O + F zn nāe nρ Da,ρ(r,s),O ). n Z+ Notational Remark In what follows, only the indices r, s and the set O will change while a, ā, ρ will be kept fixed, therefore we shall usually denote X F ā,ρ D a,ρ(r,s),o by X F r,s,o, D a,ρ (r, s) by D(r, s) and F Da,ρ(r,s),O by F r,s,o. 3 Dot stands for the time derivatives d/dt. 4 The norm Da,ρ(r,s),O for scalar functions is defined in (2.3). For vector P (or matrix valued) functions G : D a,ρ(r, s) O C m m, (m < ) is similarly defined as G Da,ρ(r,s),O = i=1 Gi D a,ρ(r,s),o (for the matrix valued case the sum will run over all entries). 4

5 2.4 Perturbed Hamiltonians and the KAM result The starting point will be a family of integrable Hamiltonians of the form N = ω(ξ), I + 1 A n (ξ)z n, z n, (2.7) 2 where ξ O is a parameter, A n is a d n d n real symmetric matrix and, is the standard inner product; here the phase space P a,ρ is endowed with the symplectic form di dθ + i n d n/2 j=1 z j n dz j+dn/2 n. For simplicity, we shall take, later, ω(ξ) ξ. n N For each ξ O, the Hamiltonian equations of motion for N, i.e., dθ dt = ω, di dt = 0, dz n dt = ij d n A n z n, n N, (2.8) admit special solutions (θ, 0, 0) (θ +ωt, 0, 0) corresponding to an invariant torus in P a,ρ. Consider now the perturbed Hamiltonians with P F r,s,o. H = N + P = ω(ξ), I + 1 A n (ξ)z n, z n + P (θ, I, z, ξ) (2.9) 2 n N Our goal is to prove that, for most values of parameter ξ O (in Lebesgue measure sense), the Hamiltonian H = N + P still admits an invariant torus provided X P is sufficiently small. In order to obtain this kind of result we shall need the following assumptions on A n and the perturbation P. (A1) Asymptotics of eigenvalues: There exist d N, δ > 0 and b 1 such that d n d for all n, and ( ) 0 Idn/2 A n = λ n + B I dn/2 0 n, B n = O(n δ ) (2.10) where λ n are real and independent of ξ while B n may depend on ξ; furthermore, the behaviour of λ n s is assumed to be as follows λ n = n b + o(n b ), λ m λ n m b n b = 1 + o(n δ ), n < m. (2.11) (A2) Gap condition: There exists δ 1 > 0 such that dist ( σ(j di A i ), σ(j dj A j ) ) > δ 1 > 0, i j ; (σ( ) denotes spectrum of ). Note that, for large i, j, the gap condition follows from the asymptotic property. 5

6 (A3) Smooth dependence on parameters: All entries of B n are d 2 Whitney smooth functions of ξ with C d 2 W -norm bounded by some positive constant L. (A4) Non-resonance condition: meas{ξ O : k, ω(ξ) ( k, ω(ξ) + λ)( k, ω(ξ) + λ + µ) = 0} = 0, (2.12) for each 0 k Z d and for any λ, µ n N σ(j d n A n ); meas Lebesgue measure. (A5) Regularity of the perturbation: The perturbation P F Da,ρ(r,s),O is regular in the sense that X P ā,ρ < with ā > a. In fact, we assume that one of the following Da,ρ(r,s),O holds: (a) ρ > 0, ā > a = 0; (b) ρ = 0, ā > a > 0, (such conditions correspond, respectively, to analytic or smooth solutions). In the case of d = 1 (i.e., the periodic solution case) one can allow ā = a. Now we can state our KAM result. Theorem 1 Assume that N in (2.7) satisfies (A1) - (A4) and P is regular in the sense of (A5) and let γ > 0. There exists a positive constant ɛ = ɛ(d, d, b, δ, δ 1, ā a, L, γ) such that if X P ā,ρ Da,ρ(r,s),O < ɛ, then the following holds true. There exists a Cantor set O γ O with meas(o \ O γ ) 0 as γ 0, and two maps (real analytic in θ and Whitney smooth in ξ O) Ψ : T d O γ D a,ρ (r, s) P a,ρ, ω : O γ R d, such that for any ξ O γ and θ T d the curve t Ψ(θ + ω(ξ)t, ξ) is a quasi-periodic solution of the Hamiltonian equations governed by H = N + P. Furthermore, Ψ(T d, ξ) is a smoothly embedded d-dimensional H-invariant torus in P a,ρ. Remarks (i) For simplicity we shall in fact assume that all eigenvalues λ i of A n are positive for all n s. The case of some non positive eigenvalues can be easily dealt with at the expense of a (even) heavier notation. (ii) In the above case (i.e. positive eigenvalues), Theorem 1 yields linearly stable KAM tori. (iii) The parameter γ plays the role of the Diophantine constant for the frequency ω in the sense that, there is τ > 0 such that k Z d \{0}, k, ω > γ 2 k τ. Notice also that O γ is claimed to be nonempty and big only for γ small enough. (iv) The regularity property ā > a is needed for the measure estimates on O\O γ. As we already mentioned, such regularity requirement is not necessary for periodic solution case, i.e., d = 1. Thus the above theorem applies to the construction of periodic solutions for nonlinear Schrödinger equations. (v) The non-degeneracy condition (2.12) (which is stronger than Bourgain s non-degenerate condition [4] but weaker than Melnikov s one [13]) covers the multiple normal frequency case: this is the technical reason that allows to treat PDE s with periodic boundary conditions. 6

7 3 Application to 1D wave equations In this section we show how Theorem 1 implies the existence of quasi-periodic solutions for 1D wave equations with periodic boundary conditions. Let us rewrite the wave equation (1.1) as follows u tt + Au = f(u), Au u xx + V (x, ξ)u, x, t R, u(t, x) = u(t, x + 2π), u t (t, x) = u t (t, x + 2π), (3.1) where V (, ξ) is a real analytic (or smooth) periodic potential parameterized by some ξ R d (see below) and f(u) is a real analytic function near u = 0 with f(0) = f (0) = 0. As it is well known, the operator A with periodic boundary conditions admits an orthonormal basis of eigenfunctions φ n L 2 (T), n N, with corresponding eigenvalues µ n satisfying the following asymptotics for large n µ 2n 1, µ 2n = n π T V (x)dx + O(n 2 ). For simplicity, we shall consider the case of vanishing mean value of the potential V and assume that all eigenvalues are positive: V (x)dx = 0, µ n λ 2 n > 0, n. (3.2) T Following Kuksin [10] and Bourgain [3], we consider a family of real analytic (or smooth) potentials V (x, ξ), where the d-parameters ξ = (ξ 1,, ξ d ) O R d are simply taken to be a given set of d frequencies λ ni µ ni : where µ ni are (positive) eigenvalues of 5 A. ξ i µ ni λ ni, i = 1,, d (3.3) We may also (and shall) require that there exists a positive δ 1 > 0 such that µ k µ h > δ 1, (3.4) for all k > h except when k is even and h = k 1 (in which case µ k and µ h might even coincide). Notice that, in particular, having d eigenvalues as independent parameters excludes the constant potential case V constant (where, of course, all eigenvalues are double: µ 2j 1 = µ 2j = j 2 + V ). In fact, this case seems difficult to be handled by KAM approach even in the finite dimensional case. Such difficulty does not arise, instead, in the remarkable alternative approach developed by Craig, Wayne [7] and Bourgain [3], [4]. Equation (3.1) may be rewritten as u = v, v + Au = f(u), (3.5) 5 Plenty of such potentials may be constructed with, e.g., the inverse spectral theory. 7

8 which, as well known, may be viewed as the (infinite dimensional) Hamiltonian equations u = H v, v = H u associated to the Hamiltonian H = 1 2 (v, v) (Au, u) + g(u) dx, (3.6) where g is a primitive of ( f) (with respect to the u variable) and (, ) denotes the scalar product in L 2. As in [15], we introduce coordinates q = (q 0, q 1, ), p = (p 0, p 1, ) through the relations u(x) = n N q n λn φ n (x), v = n N λn p n φ n (x), T where 6 λ n µ n. System (3.5) is then formally equivalent to the lattice Hamiltonian equations q n = λ n p n, ṗ n = λ n q n G, G g( q n φ n )dx, (3.7) q n T λn n N corresponding to the Hamiltonian function H = n N λ n(q 2 n+p 2 n)+g(q). Rather than discussing the above formal equivalence, we shall, following [15], use the following elementary observation (proved in the Appendix): Proposition 3.1 Let V be analytic (respectively, smooth), let I be an interval and let ) t I (q(t), p(t)) ({q n (t)} n 0, {p n (t)} n 0 be an analytic (respectively, smooth 7 ) solution of (3.7) such that ( ) q n (t) + p n (t) n a e nρ < (3.8) sup t I n N for some ρ > 0 and a = 0 (respectively, for ρ = 0 and a big enough). Then u(t, x) n N q n (t) λn φ n (x), is an analytic (respectively, smooth) solution of (3.1). Before invoking Theorem 1 we still need some manipulations. We first switch to complex variables: w n = 1 2 (q n + ip n ), w n = 1 2 (q n ip n ). Equations (3.7) read then ẇ n = iλ n w n i G w n, 6 Recall that, for simplicity, we assume that all eigenvalues µ n are positive. 7 Regularity refers to the components q n and p n. w n = iλ n w n + i G w n, (3.9) 8

9 where the perturbation G is given by G(w) = T g( n N w n + w n 2λn φ n )dx. (3.10) Next we introduce standard action-angle variables (θ, I) = ((θ 1,, θ d ), (I 1,, I d )) in the (w n1,, w nd, w n1,, w nd )-space by letting, so that the system (3.9) becomes dθ j dt dw n dt I i = w ni w ni, i = 1,, d, di j = ω j + P Ij, dt = P θ j, j = 1,, d, d w n = iλ n w n ip wn, = iλ n w n + ip wn, n n 1, n 2,, n d, (3.11) dt where P is just G with the (w n1,, w nd, w n1,, w nd )-variables expressed in terms of the (θ, I) variables and the frequencies ω = (ω 1,..., ω d ) coincide with the parameter ξ introduced in (3.3): ω i ξ i = λ ni. (3.12) The Hamiltonian associated to (3.11) (with respect to the symplectic form di dθ + i n dw n d w n ) is given by H = ω, I + λ n w n w n + P (θ, I, w, w, ξ), (3.13) n n 1,,n d Remark Actually, in place of H in (3.13) one should consider the linearization of H around a given point I 0 and let I vary in a small ball B (of radius 0 < s I 0 ) in the positive quadrant {I j > 0}. In such a way the dependence of H upon I is obviously analytic. For notational convenience we shall however do not report explicitly the dependence of H on I 0. Finally, to put the Hamiltonian in the form (2.9) we couple the variables (w n, w n ) corresponding to closer eigenvalues. More precisely, we let z n = (w 2n 1, w 2n, w 2n 1, w 2n ) for large 8 n, say n > n > n d and denote by z 0 = ({w n } 0 n n, { w n } 0 n n ) the remaining n n 1,...,n d n n 1,...,n d conjugated variables. The Hamiltonian (3.13) takes the form where 8 Compare (A1). H = ω, I + 1 A n z n, z n + P (θ, I, z, ξ), (3.14) 2 n N ( ) 0 I2 A n = Diag(λ 2n 1, λ 2n, λ 2n 1, λ 2n ) I 2 0 ( ) 0 0 λ 2n 1 λ 2n 0 0 I2 = λ 2n I 2 0 λ 2n 1 λ 2n 0 0 0,

10 ( ) 0 Id0 for n > n d, while A 0 = Diag({λ n }, {λ n }; 1 n n d, n n 1,, n d ) I d0 0 d 0 = n + 1 d. The perturbation P in (3.14) has the following (nice) regularity property. with Lemma 3.1 Suppose that V is real analytic in x (respectively, belongs to the Sobolev space H k (T) for some k N). Then for small enough ρ > 0 (respectively, a > 0), r > 0 and s > 0 one has X P a+1/2,ρ Da,ρ(r,s),O = O( z 2 a,ρ) ; (3.15) here the parameter a is taken to be 0 (respectively, the parameter ρ is taken to be 0). A proof of this lemma is given in the Appendix. In fact, X P is even more regular (a fact, however, not needed in what follows): (3.15) holds with 1 in place of 1/2. The Hamiltonian (3.14) is seen to satisfy all the assumptions of Theorem 1 with: d n = 4, n 1; d 0 = n + 1 d; d = max{d 0, 4}; b = 1; δ = 2; δ 1 choosen as in (3.4); ā a = 1 2. Thus Theorem 1 yields the following Theorem 2 Consider a family of 1D nonlinear wave equation (3.1) parameterized by ξ ω O as above with V (, ξ) real analytic (respectively, smooth). Then for any 0 < γ 1, there is a subset O γ of O with meas(o\o γ ) 0 as γ 0, such that (3.1) ξ Oγ has a family of small-amplitude (proportional to some power of γ), analytic (respectively, smooth) quasi-periodic solutions of the form u(t, x) = n u n (ω 1t,, ω d t)φ n(x) where u n : T d R and ω 1,, ω d are close to ω 1,, ω d. Remark As mentioned above, our KAM theorem (which applies only to the case that not all the eigenvalues are multiple 9 and under the hypothesis that all µ n s are positive) implies that the quasi-periodic solutions obtained are linearly stable. In the case that all the eigenvalues are double (as in the constant potential case), one should not expect linear stability (see the example given by Craig, Kuksin and Wayne [6]). We also notice that, essentially with only notational changes, the proof of the above theorem goes through in the case that some of the eigenvalues are negative. 4 KAM step Theorem 1 will be proved by a KAM iteration which involves an infinite sequence of change of variables. At each step of the KAM scheme, we consider a Hamiltonian vector field with H ν = N ν + P ν, 9 Recall that we require that the torus frequencies are independent parameters. 10

11 where N ν is an integrable normal form and P ν is defined in some set of the form 10 D(s ν, r ν ) O ν. We then construct a map 11 Φ ν : D(s ν+1, r ν+1 ) O ν+1 D(r ν, s ν ) O ν D(r ν, s ν ) O ν so that the vector field X Hν Φ ν defined on D(r ν+1, s ν+1 ) satisfies X Hν Φ ν X Nν+1 rν+1,s ν+1,o ν+1 ɛ κ ν with some new normal form N ν+1 and for some fixed ν-independent constant κ > 1. To simplify notations, in what follows, the quantities without subscripts refer to quantities at the ν th step, while the quantities with subscripts + denotes the corresponding quantities at the (ν + 1) th step. Let us then consider the Hamiltonian H = N + P e + ω, I + 1 A n z n, z n + P, (4.1) 2 defined in D(r, s) O; the A n s are symmetric matrices. We assume that ξ O satisfies 12 (for a suitable τ > 0 to be specified later) n N We also assume that on O, and k, ω 1 < k τ γ, ( k, ω I d i + A i J di ) 1 < ( k τ γ ) d ( k, ω I di d j + (A i J di ) I dj I di (J dj A j )) 1 < ( k τ γ ) d 2, (4.2) We now let 0 < r + < r, and define where max p A n L, (4.3) p d 2 ξp X P r,s,o ɛ. (4.4) s + = 1 2 sɛ 1 3, ɛ+ = γ c Γ(r r + )ɛ 4 3, (4.5) Γ(t) sup u c e 1 4 ut t c. u 1 10 Recall the notations from Section Recall that the parameters a, ρ and ā are fixed throughout the proof and are therefore omitted in the notations. 12 The tensor product (or direct product) of two m n, k l matrices A = (a ij), B is a (mk) (nl) matrix defined by A B = (a ijb) = a11b a1nb a n1b a mnb for matrix denotes the operator norm, i.e., M = sup y =1 My. Recall that ω and the A i s depend on ξ A..

12 for t > 0. Here and later, the letter c denotes suitable (possibly different) constants that do not depend on the iteration step 13. We now describe how to construct a set O + O and a change of variables Φ : D + O + = D(r +, s + ) O + D(r, s) O, such that the transformed Hamiltonian H + = N + + P + H Φ satisfies all the above iterative assumptions with new parameters s +, ɛ +, r +, γ +, L + and with ξ O Solving the linearized equation Expand P into the Fourier-Taylor series P = P klα e i k,θ I l z α k,l,α where k Z d, l N d and α n N N dn with finite many non-vanishing components. Let R be the truncation of P given by R(θ, I, z) P 0 + P 1 + P 2 P kl0 e i k,θ I l + k, l 1 P k0α e i k,θ z α + k, α =1 k, α =2 with 2 l + α = 2 l j + α j 2. j N j=1,,d It is convenient to rewrite R as follows R(θ, I, z) = P kl0 e i k,θ I l k, l 1 P k0α e i k,θ z α, (4.6) + Ri k, z i e i k,θ + jiz k,i k,i,j R k i, z j e i k,θ, (4.7) where R k i, Rk ji are respectively the d i vector and (d j d i ) matrix defined by R k i = Note that R k ij = (Rk ji )T. P z i e i k,θ dθ z=0,i=0, R k ji = 1 + δj i 2 2 P z j z i e i k,θ dθ z=0,i=0. (4.8) Rewrite H as H = N + R + (P R). By the choice of s + in (4.5) and by the definition of the norms, it follows immediately that X R r,s,o X P r,s,o ɛ. (4.9) Moreover s +, ɛ + are such that, in a smaller domain D(r, s + ), we have 13 Actually, here c = d 4 τ + d 2 τ + d X P R r,s+ < c ɛ +. (4.10) 12

13 Then we look for a special F, defined in domain D + = D(r +, s + ), such that the time one map φ 1 F of the Hamiltonian vector field X F defines a map from D + D and transforms H into H +. More precisely, by second order Taylor formula, we have H φ 1 F = (N + R) φ 1 F + (P R) φ 1 F = N + {N, F } + R = N + + P + ds s 0 {{N + R, F }, F } φ t F dt + {R, F } + (P R) φ 1 F. +{N, F } + R P 000 ω, I n N R 0 nnz n, z n, (4.11) where ω = P I dθ I=0,z=0, Rnn 0 2 P = zn 2 dθ I=0,z=0, N + = N + P ω, I + n N R 0 nnz n, z n, P + = ds s We shall find a function F of the form F (θ, I, z) = F 0 + F 1 + F 2 = satisfying the equation 0 {{N + R, F }, F } X t F dt + {R, F } + (P R) φ 1 F. + k + i j =0 l 1, k =0 F kl0 e i k,θ I l + i N F k i, z i e i k,θ F k jiz i, z j e i k,θ, (4.12) {N, F } + R P 000 ω, I n N R 0 nnz n, z n = 0. (4.13) Lemma 4.1 Equation (4.13) is equivalent to F kl0 = (i k, ω ) 1 P kl0, k 0, l 1, ( k, ω I di + A di J di )Fi k = iri k, ( k, ω I di + A di J di )Fij k Fij(J k dj A j ) = irij, k k + i j 0. (4.14) Proof. Inserting F, defined in (4.12), into (4.13) one sees that (4.13) is equivalent to the following equations 14 {N, F0} + P0 ω, I = 0, 14 Recall the definition of P i in (4.6). {N, F 1 } + P 1 = 0, {N, F 2 } + P 2 n Z R 0 nnz n, z n = 0. (4.15) 13

14 The first equation in (4.15) is obviously equivalent, by comparing the coefficients, to the first equation in (4.14). To solve {N, F 1 } + P 1 = 0, we note that 15 {N, F 1 } = I N, θ F 1 + z N, J z F 1 = I N, θ F 1 + i = i k,i = i k,i zi N, ij di zi F 1 ( k, ω F k i, z i + A i z i, J di F k i )e i k,θ ( k, ω I di + A i J di )F k i, z i e i k,θ. (4.16) It follows that F k i are determined by the linear algebraic system i( k, ω I di + A i J di )F k i + R k i = 0, i N, k Z d. Similarly, from {N, F 2 } = I N, θ F 2 + zi N, ij di zi F 2 i = i ( k, ω Fjiz k i, z j + A i z i, J di (Fji) k T z j + A j z j, J dj Fjiz k i )e i k,θ = i = i k + i j =0 k + i j =0 k + i j =0 ( k, ω F k jiz i, z j + (A j J dj F k ji F k jij di A i )z i, z j )e i k,θ ( k, ω F k ji + A j J dj F k ji F k jij di A i )z i, z j e i k,θ (4.17) it follows that, F k ji is determined by the following matrix equation ( k, ω I dj + A j J dj )F k ji F k ji(j di A i ) = ir k ji, k + i j 0 (4.18) where F k ji, Rk ji are d j d i matrices defined in (4.12) and (4.7). Exchanging i, j we get the third equation in (4.14). The first two equations in (4.14) are immediately solved in view of (4.2). In order to solve the third equation in (4.14), we need the following elementary algebraic result from matrix theory. Lemma 4.2 Let A, B, C be respectively n n, m m, n m matrices, and let X be an n m unknown matrix. The matrix equation AX XB = C, (4.19) is solvable if and only if I m A B I n is nonsingular. Moreover, 15 Recall the definition of N in (4.1). X (I m A B I n ) 1 C. 14

15 In fact, the matrix equation (4.19) is equivalent to the (bigger) vector equation given by (I A B I)X = C where X, C are vectors whose elements are just the list (row by row) of the entries of X and C. For a detailed proof we refer the reader to the Appendix in [20] or [12], p Remark. Taking the transpose of the third equation in (4.14), one sees that (Fij k)t satisfies the same equation of Fji k. Then (by the uniqueness of the solution) it follows that Fji k = (F ij k)t. 4.2 Estimates on the coordinate transformation We proceed to estimate X F and Φ 1 F. We start with the following Lemma 4.3 Let D i = D( i 4 s, r + + i 4 (r r +)), 0 < i 4. Then X F D3,O < c γ c Γ(r r + )ɛ. (4.20) Proof. By (4.2), Lemma 4.1 and Lemmata 7.4, 7.5 in the Appendix, we have F kl0 O k, ω 1 P kl < c γ c k c e k (r r +) ɛs 2 2 l, k 0, Fi k O = ( k, ω I di + A i J di ) 1 Ri k ( k, ω I di + A i J di ) 1 Ri k < c γ c k c Ri k, Fij k O ( k, ω I di d j + (A i J di ) I dj I di (J dj A j )) 1 Rij k < c γ c k c R k ij, k + i j 0. (4.21) Where O for matrix is similar to (2.4). It follows that 1 s 2 F θ D2,O + 1 s 2 ( f kl0 I l k e i k,θ + Fi k z i k e i k,θ F k ij z i z j k e i k,θ ) < c γ c Γ(r r + ) X R < c γ c Γ(r r + )ɛ, (4.22) where Γ(r r + ) = sup k k c e k 1 4 (r r +). Similarly, F I D2,O = F kl0 e i k,θ < c γ c Γ(r r + )ɛ. l 1 Now we estimate X F 1 D2,O. Note that F 1 z i D2,O = k F k i e i<k,θ> D2,O < c γ c Γ k,i R k i e k r < c γ c Γ P 1 z i. (4.23) 15

16 It follows that X F 1 D2,O < c Fz 1 i D2,Oi a e iρ i N < c γ c Γ i N P 1 z i i a e iρ < c γ c Γɛ, by the definition of the weighted norm. Note that 16 Similarly, we have F 2 z i D2,O = k,j (F k ij + (F k ij) T )z j e i k,θ D2,O < c γ c Γ P 2 z i. (4.24) X F 2 D2,O < c γ c Γɛ. (4.25) The conclusion of the lemma follows from the above estimates. In the next lemma, we give some estimates for φ t F. The following formula (4.26) will be used to prove that our coordinate transformations is well defined. Inequality (4.27) will be used to check the convergence of the iteration. Lemma 4.4 Let η = ɛ 1 3, D i 2 η = D(r + + i 1 2 (r r +), i 2ηs), i = 1, 2. We then have if ɛ ( 1 2 γ c Γ 1 ) 3 2. Moreover, φ t F : D 1 2 η D η, 0 t 1, (4.26) Dφ 1 F Id D 12 η < c γ c Γɛ. (4.27) Proof. Let D m F D,O = max{ i + l +p θ i I l z α F D,O, i + l + α = m 2}. Note that F is polynomial in I of order 1, in z of order 2. From 17 (4.25) and the Cauchy inequality, it follows that D m F D1,O < c γ c Γɛ, (4.28) for any m 2. To get the estimates for φ t F, we start from the integral equation, φ t F = id + t 0 16 Recall (2.3), the definition of the norm. 17 Recall the definition of the weighted norm in (2.6). X F φ s F ds 16

17 so that φ t F : D 1 2 η D η, 0 t 1, as it follows directly from (4.28). Since Dφ 1 F = Id (DX F )Dφ s F ds = Id J(D 2 F )Dφ s F ds, it follows that Dφ 1 F Id 2 D 2 F < c γ c Γɛ. (4.29) The estimates of second order derivative D 2 φ 1 F follows from (4.28). 4.3 Estimates for the new normal form The map φ 1 F defined above transforms H into H + = N + + P + (see (4.11) and (4.13)) with N + = e + + ω +, y + 1 A + i 2 z i, z i (4.30) i Z+ where e + = e + P 000, ω + = ω + P 0l0 ( l = 1), A + i = A i + 2R 0 ii. (4.31) Now we prove that N + shares the same properties with N. By the regularity of X P and by Cauchy estimates, we have with δ = ā a > 0. It follows that (A + i ) 1 A 1 i 1 2 A 1 i ω + ω < ɛ, R 0 ii < ɛi δ (4.32) Rii 0 2 A 1 i, ( k, ω + P 0l00 I di J di A + i ) 1 provided k dτ ɛ < c (γ d γ d + ). Similarly, we have ( k, ω I d i + A i J di ) 1 1 ( k, ω I di + A i J di ) 1 ɛ ( k τ γ + ) d,(4.33) ( k, ω + P 0l00 I di d j + (A + i J d i ) I dj I di (J dj A + j )) 1 ( k τ γ + ) d 2, (4.34) provided k d 2τ ɛ < c (γ d 2 γ d 2 + ). This means that in the next KAM step, small denominator conditions are automatically satisfied for k < K where K d 2τ ɛ < c (γ d 2 γ 4 d 2 + ). The following bounds wil be used later for the measure estimates: for l d 2 (by definition of the norms). l (ω + ω) ξ l O ɛ, l (A + i A i ) ξ l O < c ɛi δ, (4.35) 17

18 4.4 Estimates for the new perturbation To complete the KAM step we have to estimate the new error term. By the definition of φ 1 F and Lemma 4.4, H φ 1 F = N + + P + is well defined in D 1 η. Moreover, we have the following estimates 2 X P+ D 12 η by (4.9) and Lemma 7.3. = X R 1 0 dt R s 0 {{N+R,F },F } φs F +{R,F }+(P D R) φ1 12 F η X R ( 1 0 dt R t 0 {{N+R,F },F D + X } φs 12 F η (P R) φ 1 D F 12 η X {{N+R,F },F } Dη + X P R Dη < c γ c Γ 2 ɛ 4 3 < c ɛ + (4.36) Thus, there exists a big constant c, independent of iteration steps, such that The KAM step is now completed. X P+ r+,s + = X P+ ā,ρ D 12 η cγ c Γ 2 ηɛ = cɛ +. (4.37) 5 Iteration Lemma and Convergence For any given s, ɛ, r, γ, we define, for all ν 1, the following sequences ν+1 r ν = r(1 2 i ), i=2 ɛ ν = cγν c Γ(r ν 1 r ν ) 2 ɛ 4 3 ν 1. ν+1 γ ν = γ(1 2 i ) i=2 η ν = 1 2 ɛ 1 3 ν, L ν = L ν 1 + ɛ ν 1, s ν = 1 ν 1 2 η ν 1s ν 1 = 2 ν ( ɛ i ) 1 3 s0, i=0 ) 1 d 2τ, K ν = c ( ɛ 1 d 2 ν 1 (γν 1 γ d 2 ν ) 2 D ν = D a,ρ (r ν, s ν ), (5.1) where c is the constant in (4.37). The parameters r 0, ɛ 0, γ 0, L 0, s 0, K 0 are defined respectively to be r, ɛ, γ, L, s, 1. Note that Ψ(r) = is a well defined finite function of r. [Γ(r i 1 r i )] 2( 3 4 )i, i=1 18

19 5.1 Iteration Lemma The preceeding analysis may be summarized as follows. Lemma 5.1 Suppose that ɛ 0 = ɛ(d, d, δ, δ 1, ā a, L, τ, γ) is small enough. Then the following holds for all ν 0. Let N ν = e ν + ω ν (ξ), I + i N A ν i (ξ)z i, z i, be a normal form with parameters ξ satisfying k, ω ν 1 < k τ γ ν, (i k, ω ν I di + A ν i J di ) 1 < ( k τ γ ν ) d (i k, ω ν I di d j + (A ν i J di ) I dj I di (J dj A ν j )) 1 < ( k τ γ ν ) d 2 (5.2) on a closed set O ν of R n for all k 0, i, j Z. Moreover, suppose that ω ν (ξ), A ν i (ξ) are C d 2 smooth and satisfy on O ν (in Whitney s sense). Finally, assume that d 2 (ω ν ω ν 1 ) ξ d ɛ 2 ν 1, d 2 (A ν i Aν 1 i ) ξ d ɛ 2 ν 1 i δ, Then, there is a subset O ν+1 O ν, where R ν+1 kij (γ ν+1) = ξ O ν : X Pν ā,ρ D ν,o ν ɛ ν. O ν+1 = O ν \ k Kν+1 R ν+1 kij (γ ν) k,ω ν+1 1 > k τ γν, ( k,ων I 2m+(A ν+1 i J di ) 1 ( k τ γν ( k,ω ν+1 >I di d j +(A ν+1 j J di ) I dj I di (J dj A ν+1 j )) 1 >( k τ ) d 2 γν with ω ν+1 = ω ν + P0l0 ν, and a symplectic change of variables ) d, or, Φ ν : D ν+1 O ν+1 D ν, (5.3) such that H ν+1 = H ν Φ ν, defined on D ν+1 O ν+1, has the form satisfying H ν+1 = e ν+1 + ω ν+1, I + i N A ν+1 i z i, z i + P ν+1, (5.4) max l (ω ν+1 (ξ) ω ν (ξ)) l d 2 ξ l ɛ ν, max l d2 l (A ν+1 i (ξ) A ν i ) ξ l ɛ ν i δ, (5.5) X Pν+1 ā,ρ D ν+1,o ν+1 ɛ ν+1. (5.6) 19

20 5.2 Convergence Suppose that the assumptions of Theorem 1 are satisfied. To apply the iteration lemma with ν = 0, recall that ɛ 0 = ɛ, γ 0 = γ, s 0 = s, L 0 = L, N 0 = N, P 0 = P, O 0 = ξ O : k,ω 1 < k τ γ, ( k,ω I d i +A i J di ) 1 <( k τ γ ) d, or ( k,ω I di d j +(A i J di ) I dj I di (J dj A j )) 1 <( k τ γ ) d 2, (with ɛ and γ small enough). Inductively, we obtain the following sequences O ν+1 O ν, Ψ ν = Φ 1 Φ ν : D ν+1 O ν D 0, ν 0, H Ψ ν = H ν+1 = N ν+1 + P ν+1. Let O γ = ν=0 O ν. As in [16], thanks to Lemma 4.4, we may conclude that N ν, Ψ ν, DΨ ν, ω ν+1 converge uniformly on D O γ = D( 1 2 r, 0, 0) O γ with N = e + ω, I + A z, z = e + ω, I + i N A i z i, z i, Since ɛ ν+1 = cγ c ν Γ(r ν r ν+1 )ɛ ν (cγ c Ψ(r)ɛ) ( 4 3 )ν. It follows that ɛ ν+1 0 provided ɛ is sufficiently small. Let φ t H be the flow of X H. Since H Ψ ν = H ν+1, we have that φ t H Ψ ν = Ψ ν φ t H ν+1. (5.7) The convergence of Ψ ν, DΨ ν, ω ν+1, X Hν+1 implies that one can take limit in (5.7) so as to get φ t H Ψ = Ψ φ t H, (5.8) on D( 1 2 r, 0, 0) O γ, with Ψ : D( 1 2 r, 0, 0) O γ P a,ρ R d. From (5.8) it follows that φ t H(Ψ (T d {ξ})) = Ψ φ t N (T d {ξ}) = Ψ (T d {ξ}), for ξ O γ. This means that Ψ (T d {ω}) is an embedded torus invariant for the original perturbed Hamiltonian system at ξ O γ. We remark here the frequencies ω (ξ) associated to Ψ (T d {ξ}) is slightly different from ξ. The normal behaviour of the invariant torus is governed by the matrix A i = ν N Aν i. 20

21 6 Measure Estimates At each KAM step, we have to exclude the following resonant set of ξ s: R ν = (R ν k Rν ki Rν kij ), k >K ν,i,j the sets R ν k, Rν ki, Rν kij being, respectively, where {ξ O ν : k, ω ν 1 > k τ γ ν }, {ξ O ν : M 1 1 > ( k τ γ ν ) d}, and {ω O ν : M 1 2 > ( k τ γ ν ) d 2 }, (6.1) M 1 = k, ω ν I di + A ν i J di M 2 = k, ω ν I di d j + (A ν j J dj ) I di I dj (J di A ν i ).. (6.2) In the set {ξ O : M(ω) 1 > C} are included also the ξ s for which M is not invertible. Remind that ω ν (ξ) = ξ + ν 1 j=0 P j 000 (ξ) with18 P j 000 (ξ) C d 2 ɛ, Aν i = A i + 2 ν R0,ν ii with ν R0,ν ii = O(ɛi δ ). Lemma 6.1 There is a constant K 0 such that, for any i, j, and k > K 0, meas(r ν k Rν ki Rν kij ) < c γ k τ 1. Proof. As it is well known meas (R ν k ) γ ν k τ. The set R ν ki is empty if i > const k, while, if i const k, from Lemmata 7.6, 7.7 there follows that meas(r ν ki ) < c γ ν k τ 1. We now give a detailed proof for the most complicated estimate, i.e., the estimate on the measure of the set R ν kij. Note that the main part of M 2 is diagonal 19. In fact M 2 can be rewritten as M 2 A ij + B ν ij, with A ij = k, ω ν+1 I di d j + λ j Diag(I dj /2, I dj /2) I di λ i I dj Diag ( I di /2, I di /2). (6.3) The matrix A ij is diagonal with entries λ kij = k, ω ν ± λ i ± λ j in the diagonal, where λ i, λ j are given in (2.10) and ± sign depends on the position. Bij ν is a matrix of size O(i δ + j δ ) since A ν i = A i + B i + O(i δ ) = A i + O(i δ ) by (2.11) and (4.32). 18 Recall (4.32), (5.5). 19 Recall (2.10), (6.2). 21

22 In the rest of the proof we drop in the notation the indices i, j since they are fixed. Now either all λ kij k or there are some diagonal elements λ kij > k. We first consider the latter case. By permuting rows and columns, we can find two non-singular matrices Q 1, Q 2 with elements 1 or 0 such that ( ) ( ) Q 1 (A + B ν A11 0 B11 B12 )Q 2 = + (6.4) 0 A 22 B 21 B22 where A 11, A 22 are diagonal matrices and A 11 contains all diagonal elements λ kij which are bigger than k. Moreover, defining Q 3, Q 4, D as and we have ( Q 3 = I B 21 (A 11 + B 11 ) 1 0 I ) ( I (A11 +, Q 4 = B 11 ) 1 B12 0 I D = A 22 + B 22 B 21 (A 11 + B 11 ) 1 B12 = A 22 + O(i δ + j δ ), (6.5) Q 3 Q 1 ( A + B ν+1 ) Q 2 Q 4 = For ξ O such that D is invertible, we have ( A11 + B D (A + B ν ) 1 = Q 2 Q 4 ( (A11 + B 11 ) D 1 ) ),. (6.6) ) Q 3 Q 1. (6.7) Since the norm of Q 1, Q 2, Q 3, Q 4, (A 11 + B 11 ) 1 are uniformly bounded, it follows from (6.7) that {ξ O ν : (A + B ν ) 1 > ( k τ γ ν ) d 2 } {ξ O ν : D 1 > c ( k τ γ ν ) d 2 }. (6.8) If all λ kij < c k we simply take D = A + B ν. Since all elements in D are of size O( k ), by Lemma 7.6 in the Appendix, we have {ξ O ν : D 1 > c ( k τ ) d 2 } {ξ O ν : det D < c ( γ ν k τ 1 ) d 2 }. (6.9) Let N denote the dimension of D (which is not bigger than d 2 ). Since D = A 22 + O(i δ + j δ ), the N th order derivative of det D with respective to some ξ i is bounded away from zero by 1 2d k N (provided that k is bigger enough). From (6.8), (6.9) and Lemma 7.7, it follows that This proves the lemma. meas R ν kij = meas {ξ O ν : (A + B ν ) 1 > ( k τ γ ν ) d 2 } meas{ξ O ν : det D < c ( k τ 1 ) d 2 } < c ( k γ ν d 2 ) τ 1 γ ν γ N < c. (6.10) k τ 1 γ ν 22

23 Lemma 6.2 If i > c k, then R ν 1 ki = ; If max{i, j} > c k b 1, i j for b > 1 or i j > const k for b = 1, then R ν kij = where the constant c depends on the diameter of O. Proof. As above, we only consider the most complicated case, i.e., the case of R ν kij. Notice that max{i, j} > const k 1 b 1 for b > 1 or i j > const k for b = 1 implies λ i ± λ j = (j b i b )(1 + O(i δ + j δ )) 1 2 j i (ib 1 + j b 1 )(1 + O(i δ + j δ )) const k. (6.11) It follows that A ij defined in (6.3) is invertible and (A ij ) 1 < k 1. By Neumann series, we have (A ij + B ν ij ) 1 < 2 k 1 for large k (say k > K 0 ), i.e, R ν kij =. Lemma 6.3 For b 1, we have meas( ν 0 R ν ) = meas ν, k >K ν,i,j (R ν k Rν ki Rν kij ) < c γ δ 1+δ. Proof. The measure estimates for R 0 comes from our assumption (2.12). We then consider the estimate meas( R ν kij ), ν which is the most complicate one. k >K ν i,j Let us consider separately the case b > 1 and the case b = 1. We first consider b > 1. By Lemmata 6.1, 6.2, if k > K 0 and i j, we have meas ( i j R kij ) = meas ( 1 i j;i,j<c k b 1 R k ij) < c k 2 b 1 γ k τ 1 = γ. (6.12) k τ 1 2 b 1 For i = j. As in Lemma 6.1, we can find Q 1, Q 2 so that (6.4) holds with the diagonal elements of A 11 being < k, ω ν > ±2λ i and A 22 =< k, ω ν > I. Repeating the arguments in Lemma 6.1, we get (6.9) and γ ν R ν kii {ξ : det D < c ( k τ 1 ) d 2 } = {ξ : k, ω ν + O(i δ ) < c ( k τ 1 ) d 2 } γ {ξ : k, ω ν < c ( k τ i δ )} Qk ii. (6.13) γ ν 23

24 Since Q k ii Qk i 0 i 0 for i i 0, using (6.10), we find that meas ( i R kii ) i<i 0 R kii + Q k i 0 i 0 < c ( i 0γ k τ for any i 0. Following Pöschel ([16]), we choose i 0 = ( k τ 1 γ ) 1 1+δ, so that meas ( i i δ 0 γ R kii < c ( k τ 1 ) δ 1+δ (6.14) Let τ > max{d δ b 1, (d + 1) δ + 1}. As in (6.12), (6.14), we find meas ( The quantity meas( ν meas( ν 1 k >K ν i,j +meas ( k >Kν R ν kij (γ ν)) = meas ( i k >K ν i j R ν kii (γ ν+1)) < c Kν 1 γ δ 1+δ. k >K ν i,j Rν kij ) is then bounded by k >K ν i,j R ν kij (γ ν)) < c γ δ 1+δ ν 0 K 1 ν ) R ν kij (γ ν)) < c γ δ 1+δ, (6.15) provided τ > max{d δ b 1, (d + 1) δ + 1}. This concludes the proof for b > 1. Consider now b = 1. Without loss of generality, we assume j i and j = i + m. Note that Lemma 6.2 implies R k ij = for m > C k. Following the scheme of the above proof, we find R kij = R ki,i+m = where k,i,j k,i,m k,m<c k ( k,m<c k i R ki,i+m i<i 0 R ki0,i 0 +m Q ki0,i 0 +m). (6.16) γ Q ki0,i 0 +m = {ξ : k, ω ν + m < c ( k τ )}. i δ Again, taking i 1+δ 0 = k τ 1 γ, we have, for fixed k, i,j R k ij < c As in the case b > 1, we have that meas( ν τ > (d + 1) 1+δ δ + 1. m<c k ( i 0γ k + i δ τ 1 0 ) γ < c k ( k τ 1 ) δ 1+δ. (6.17) δ k >K ν i,j Rν kij ) is bounded by O(γ 1+δ ) if Remark In (6.13), det D = k, ω + O(i δ ) (guaranteed by the regularity property) is crucial for the proof. But it is not necessary for the periodic solution case, i.e., d = 1. since R ν k,i+m,i = if, i > c > k 1 are sufficiently large. 24

25 7 Appendix Proof of Proposition 3.1 From the hypotheses there follows that the eigenfuctions φ n are analytic (respectively, smooth) and bounded with, in particular, sup( φ n + φ n ) const µ n. R Thus, the sum defining u(t, x) is uniformly convergent in I [0, 2π]. Since G = 1 f( q k φ k )φ n, q n λn λk k one has q n const e nρ e nρ n a, q n const λ n n a const e nρ n a 1, q n const e nρ n a+1. Thus (if a is big enough, in the smooth case) u(t, x) is a C 2 function and u tt + Au = q n φ n + q n Aφ n λ λn = ( f(u)φ n )φ n = f(u), where in the last equality we used the fact that f(u) is a smooth periodic function. Lemma 7.1 F G D(r,s) F D(r,s) G D(r,s). (7.1) Proof. Since (F G) klp = l F k k,l l,p p G k l p. we have that F G D(r,s) = sup (F G) klp y l z α e k r D klp sup F k k,l l,p p G k l p y l z α e k r D and the proof is finished. Lemma 7.2 (Cauchy inequalities) klp l = F D(r,s) G D( r,s) (7.2) F θi D(r σ,s) cσ 1 F D(r,s), and F I D(r, 1 2 s) 2 1 s 2 F D(r,s), F zn D(r, 1 2 s) e nρ 2na F s D(r,s) 25

26 Let {, } is Poisson bracket of smooth functions {F, G} = ( F G F G ) + F G, ij di, (7.3) θ i I i I i θ i z i z i i N where J di are standard symplectic matrix in R d i. Lemma 7.3 If then X F r,s < ɛ, X G r,s < ɛ, X {F,G} r σ,ηs < c σ 1 η 2 ɛ ɛ, η 1. Proof. Note that d dz n {F, G} = F θzn, G I + F θ, G Izn F Izn, G θ F I, G θzn + i N( F zi z n, J di G zi + F zi, J di G zi z n ) (7.4) Since F θzn, G I D(r σ,s) < c σ 1 F zn G y F θ, G Izn D(r σ, 1 2 s) < c s 2 F θ G zn F Izn, G θ D(r, 1 2 s) < c s 2 F zn G θ F I, G θzn D(r σ,s) < c σ 1 F I G zn F zi z n, J di G zi D(r, 1 2 s) < c s 1 F zn G zi i a e iρ F zi z n, J di G zi D(r, 1 2 s) < c s 1 F zn G zi i a e iρ (7.5) it follows from the definition of the weighted norm(see (2.6)), that X {F,G} r σ,ηs < c σ 1 η 2 ɛ ɛ. In particular, if η ɛ 1 3, ɛ, ɛ ɛ, we have X {F,G} r σ,ηs ɛ 4 3. Lemma 7.4 Let O be a compact set in R d for which (4.2) holds. Suppose that f(ξ) and ω(ξ) are C m Whitney-smooth function in ξ O with CW m norm bounded by L. Then is C m Whitney-smooth in O with 20 g(ξ) f(ξ) k, ω(ξ) g O < c γ c k c L, Proof. The proof follows directly from the definition of the Whitney s differentiability. 20 Recall the definition in (2.4). 26

27 A Similar lemma for matrices holds: Lemma 7.5 Let O be a compact set in R d for which (4.2) holds. Suppose that B(ξ), A i (ξ) are C m Whitney-smooth matrices and ω(ξ) is a Withney-smooth function in ξ O bounded by L. Then C(ξ) = BM 1, is C m Whitney-smooth with F O < c γ c k c L, where M stands for either k, ω I di +A i J di if B is (d i d i )-matrix, or k, ω I di d j +(A i J di ) I dj I di (J dj A j ) if B is (d i d j d i d j )-matrix, For a N N matrix M = (a ij ), we denote by M its determinant. Consider M as a linear operator on (R N, ) where x = x i. Let M be its operator norm and recall that M is equivalent to the norm max a ij ; thus disregarding a constant (depending only on dimensions) we will simply denote M = max a ij. Lemma 7.6 Let M be a N N non-singular matrix with M < c k, then {ω : M 1 > h} {ω : det M < c k N 1 } h Proof. First, note that if M is a nonsingular N N matrix with elements bounded by m ij m, its inverse is M 1 = 1 M adjm so that M 1 < c mn 1 detm with a constant depending on N. In particular, if m = const k, DetM > k N 1 h then This proofs the lemma. M 1 < c h. In order to estimate the measure of R ν+1, we need the following lemma, which has been proven in [19] [21]. A similar estimate is also used by Bourgain [4]. Lemma 7.7 Suppose that g(u) is a C m function on the closure Ī, where I R1 is a finite interval. Let I h = {u : g(u) < h}, h > 0. If for some constant d > 0, g (m) (u) d for all u I, then meas (I h ) ch 1 m where c = 2( m + d 1 ). For the proof of Lemma 3.1, we need the following Lemma 7.8 e n j r+ρ j Ce ρ n, q j e n j r+ n ρ C q ρ j Z if ρ < r, q Z ρ where C depends on r ρ. j,n Z 27

28 Lemma 7.9 (1 + n j ) K j a < c n a, q j (1 + n j ) k n a C q a j Z j,n Z if K > a + 1, q Z a,ρ=0 where C depends on K a 1. The proofs of the above two lemmata are elementary and we omit them. A direct proof of Lemma 3.1. It is clearly enough to consider the case of f(u) equal to a monomial u N+1 for some N 1. From (3.10), one can see that the regularity of G implies the regularity of G. In the following, we shall give the proof for G. Suppose that the potential V (x) is analytic in Imx < r (respectively, belongs to Sobolev space H K ) then the eigenfunctions are analytic in Imx < r (respectively, belong to H K+2 ). If we let φ i (x) = a n i ei n,x then (see, e.g.,[7]) Recall that where a n i < c e i n r respectively a n i < c (1 + n i K 2 ). G(q) = i 0,,i N C i0 i N C i0 i N = φ i0 φ in dx = T 1 q i0 q in λi0 λ in N ( n 0 +n 1 + +n N =0 s=0 a ns i s ), with a ns i s < c e is ns r (respectively, a ns i s < c (1 + n s i s K 2 ). In what follows, we assume either a = 0, ρ > 0 or a > 0, ρ = 0. Since G qj = (N + 1) q q i1 in λj λ i1 λ in it follows that i 1,,i N C ji1 i N G q a+ 1 2,ρ = G q 0 + G qj j a+ 1 2 e jρ j 1 < c < c < c < c j,i 1,,i N, n 0 + +n N =0 j,i 1,,i N ; n 0 + +n N =0 i 1,,i N ; n 0 + +n N =0 ( i 1,,i N ; s=1 n 1,,n N a n 0 j ja e j ρ ( N s=1 a ns i s q is ) ( N ) (1 + j n 0 ) N j a e j ρ n 0 j r (1 + n s i s ) K 2 e ns is r q is s=1 ( N ) n 0 a e n 0 ρ (1 + n s i s ) K 2 e ns is r q is s=1 ( N n s ) a e P N ) N s=1 ns ρ (1 + n s i s ) K 2 e ns is r q is s=1 28

29 < c < c < c i 1,,i N ; n 1,,n N i 1,,i N ( N s=1 ( N ) (1 + n s i s ) K 2 n s a e ns is r+ ns ρ q is s=1 ( N ) i s a e is ρ q is s=1 i s i s a e is ρ q is ) < c q N a,ρ. (7.6) References [1] Arnold, V. I., Proof of A. N. Kolmogorov s theorem on the preservation of quasi periodic motions under small perturbations of the Hamiltonian, Usp. Math. USSR, 18, 1963, [2] Arnol d, V. I., Small denominators and problems of stability of motions in classical and celestial mechanics, Russ. Math. Surv., 18:6, 1963, [3] Bourgain, J., Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, International Mathematics Research Notices, 1994, [4] Bourgain, J., Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Annals of Mathematics, to appear, [5] Bourgain, J., Nonlinear Schrödinger equations, to appear in IAS-Park City Lectures(AMS). [6] Craig, W., KAM theory in infinite dimensions, Lectures in Applied Mathematics, 31, [7] Craig, W., and Wayne, C.E., Newton s method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math., 46, 1993, [8] Eliasson, L.H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm. Sup. Pisa 15, 1988, [9] Fröhlich J., and Spencer, T, Absence of diffusion in the Anderson tight binding model for large disorder or lower energy, Comm. Math. Phy., 88, 1983, [10] Kuksin, S.B., Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Springer, Berlin, 1556, [11] Kuksin, S.B., Pöschel, J., Invariant cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation, Annals of Mathematics, 142,

30 [12] Lancaster, P., Theory of Matrices, Academic Press LTD, New York and London, [13] Melnikov, V.K., On some cases of the conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Mathematics Doklady, 6, 1965, [14] Moser, J., Convergent series expansions for quasiperiodic motions, Math. Ann., 169(1), 1967, [15] Pöschel,J., Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici, 71, 1996, [16] Pöschel, J., A KAM-Theorem for some Nonlinear Partial Differential Equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23, 1996, [17] Wayne, C.E., Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory, Comm. Math. Phys., 127, 1990, [18] Whitney, H., Analytical extension of differentiable functions defined on closed set Trans. A. M. S., 36, 1934, [19] Xu, J., You, J., Qiu, Q., Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z 226, 1997, [20] You, J., Perturbation of lower dimensional tori for Hamiltonian systems, to appear in J. Differential Equations, [21] You, J., A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192, 1998,