Quasi-Periodic Solutions for 1D Nonlinear Wave Equation with a General Nonlinearity

Transcription

1 Quasi-Periodic Solutions for 1D Nonlinear Wave Equation with a General Nonlinearity Zhenguo Liang and Jiangong You Department of Mathematics and Institute of Mathematics Nanjing University, Nanjing , P.R.China Abstract In this paper, one dimensional (1D) wave equation with a general nonlinearity u tt u xx +mu+f(u) = 0, m > 0 under Dirichlet boundary conditions is considered; the nonlinearity f is a real analytic, odd function and f(u) = au 2 r+1 + f 2k+1 u 2k+1, a 0 and r N. It is proved that k r+1 for almost all m > 0 in Lebesgue measure sense, the above equation admits smallamplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills. 1 Statement of the main result In this paper, we are going to study the nonlinear wave equation u tt u xx +mu+f(u) = 0, m > 0 (1.1) on the finite x interval [0, π] with Dirichlet boundary conditions u(t,0) = 0 = u(t,π). (1.2) Here, m > 0 is a real parameter, sometimes referred to as mass, and f is a real analytic, odd function of u of the form f(u) = au 2 r+1 + f 2k+1 u 2k+1, a 0 and r N. (1.3) k r+1 As [24], we study this equation (1.1) as an infinite dimensional hamiltonian system on P = H0 1([0,π]) L2 ([0,π]) with coordinates u and v = u t. Let 2 φ j = π sinjx, λ j = j 2 +m, j 1 The work was supported by the Special Funds for Major State Basic Research Projects of China (973 projects). 1

2 be the basic modes and frequencies of the linear equation u tt = u xx mu with Dirichlet boundary conditions. Then every solution of the linear equation is the superposition of their harmonic oscillations and of the form u(t,x) = j 1 q j (t)φ j (x), q j (t) = I j cos(λ j t+φ 0 j) with amplitudes I j 0 and initial phases φ 0 j. Their combined motion is periodic, quasiperiodic or almost periodic, respectively, depending on whether one, finitely many or infinitely many modes are excited. In particular, for every choice J = {j 1 < j 2 < < j b } N of finitely many modes there is an invariant 2b dimensional linear subspace E J completely foliated into rotational tori with frequencies λ j1,,λ jb : E J = {(u,v) = (q 1 φ j1 + +q b φ jb,p 1 φ j1 + +p b φ jb )} = T J (I), I P b that is where P b = {I R b : I j > 0 for 1 j b} is the positive quadrant in R n and T J (I) = {(u,v) : qj 2 +λ 2 j p 2 j = I j for 1 j b}, using the above representation of u and v. Upon restoration of the nonlinearity f, we show that there exists a Cantor set C P b, a specially chosen index set I = {n 1 < n 2 < < n b } N (see below) and a family of b-tori T I [C] = I C T I (I) E I over C, and a Whitney smooth embedding Φ : T I [C] E I P, such that the restriction of Φ to each T I (I) in the family is an embedding of a rotational b-torus for the nonlinear equation. In [24], The image E I of T I [C] is called a Cantor manifold of rotational b-tori. Theorem 1 (Main Theorem) For almost all m > 0 and each index set I = {n 1 < < n b } with b 2, satisfying sn i n j for any s = 1,2,, r, i < j, i,j {1,,b}, (1.4) the wave equation (1.1) with (1.2) possesses a local, positive-measure, 2b dimensional invariant Cantor manifold E I given by a Whitney smooth embedding Φ : T I [C] E I, which is a higher order perturbation of the inclusion map Φ 0 : E I P restricted to T I [C]. Moreover, the Cantor manifold E I is foliated by real analytic, linearly stable, b dimensional invariant tori carrying quasiperiodic solutions. 2

3 Remark 1.1 The size of Cantor manifold E I is not uniform, but depends on m, b, I and etc.. Remark 1.2 The frequencies of the diophantine tori are also under control. They are ω(i) = (ω n1 (I),,ω nb (I)), where ω nj (I) = λ nj +A j +O(I r+ 1 6 ), j = 1,,b, A j = c r(2 r +1) r +1 I r n j λ r+1 + c r( r +1) n j λ nj i j I r n i λ r n i + 0 p t < r, t=1,,b p 1 + +p b = r O(I p 1 n 1 I p b n b ) and c r = C r+1 2 r+2 (2 r 1)!! 4π r (2 r)!!. Remark 1.3 The similar conclusion holds for 1D nonlinear wave equation with periodic boundary condition where the nonlinearity g(u) is real analytic and u tt u xx +mu+g(u) = 0, m > 0 (1.5) u(t,x+2π) = u(t,x), (1.6) g(u) = au 2 r+1 +O(u 2 r+2 ), a 0 and r N, if one uses the so-called compactness property observed in [16]. For r = 1, the result has been obtained in [22]. The rest of the paper is organized as follows: In section 2 we firstly discuss some known results on nonlinear wave equations, and then present the idea of the proof. Section 3 contains a concluding theorem about 1D nonlinear beam equation with a general nonlinearity under the hinged boundary conditions. In section 4, the hamiltonian function is written in infinitely many coordinates and then put into partial normal form in section 5. In section 6 we introduce an infinite dimensional KAM theorem and the measure estimates are given in section 7. Some lemmata are proved in the Appendix. 2 Discussion and idea of the proof In this section, we will mainly discuss the relations of our results with previous results on 1D nonlinear wave equations with constant potentials. For r > 1, the known results are all about periodic solutions. The first result is due to Walter Craig and C. E. Wayne, who discuss ϕ d nonlinear Klein-Gordon equation u tt u xx +b 2 u u d 1 = 0, d > 4 (2.1) in [15]. They obtain families of periodic solutions for open set of parameters b 2 of full measure when d = 2n(n > 2) under either periodic or Dirichlet boundary conditions. For 3

4 the existences of periodic solutions of 1D completely resonant wave equation(m = 0) with all kinds of nonlinearities, see [1, 3, 4, 5, 18, 23]. All the above cases are about the existences of periodic solutions. There are also many results on the existences of quasi-periodic solutions for r = 1. The first result in this direction is due to A.I.Bobenko and Kuksin [6]. They get the quasi-periodic solutions corresponding to finite dimensional invariant tori of (1.1)+(1.2). Their starting point is to take the equation (1.1) as a perturbed sine-gordon equation. This result is regained by Pöschel [24] by the infinite KAM theory and normal form technique. The existence of quasi-periodic solutions of the equation (1.5)+(1.6) is firstly proved by Bourgain [10] with the famous Craig-Wayne-Bourgain s methods (see [7, 8, 9, 10, 11, 14]). Later Jean Bricmont, Antti Kupiainen and Alain Schenkel [12] give a new proof of this result based on a renormalization group procedure. For the existences of quasi-periodic solutions corresponding to finite dimensional invariant tori of (1.5)+(1.6), see [22]. The remained case is whether there exist quasi-periodic solutions corresponding to finite dimensional invariant tori of (1.1)+(1.2) when r > 1. Theorem 1 actually gives a positive answer towards this problem. The proof follows the main steps of the infinite dimensional KAM theorem. However, as one will see, there exist some technical difficulties when considering 1D wave equation with a general nonlinearity. The first appears when we want to get a partial Birkhoff normal form. For this purpose, we mainly expect that the following inequality holds λ i1 ±λ i2 ± ±λ i2 r +λ i λ j c(m) > 0, (2.2) where i 1,,i 2 r {n 1,,n b }, i j, and n 1,,n b are tangent sites while i, j are normal ones. For the 1D Schrödinger equations with higher order nonlinearities(see [20]), (2.2) is guaranteed by carefully choosing the tangent sites. But this method couldn t be applied to the wave equation as considered in this paper. Comparing with [20], one gets (2.2) from throwing a small set of m. The second technical difficulty lies in the measure estimates. Since the nonlinear term is of higher order, the measure estimate becomes much more complicated. Since it is very technical, the reader is deferred to section 7. 3 A Concluding Theorem The same conclusion holds for 1D beam equation with a general nonlinearity under the hinged boundary conditions u tt +u xxxx +mu+f(u) = 0, m > 0 (3.1) u(0,t) = u xx (0,t) = u(π,t) = u xx (π,t) = 0, (3.2) where f is a real analytic, odd function and f(u) = au 2 r+1 + f 2k+1 u 2k+1, a 0 and r N. k r+1 Here we give the result while omitting the proof. 4

5 Theorem 2 (Main Theorem) For almost all m > 0 and each index set I = {n 1 < < n b } with b 2, satisfying sn i n j for any s = 1,2,, r, i < j, i,j {1,,b}, (3.3) the beam equation (3.1) with (3.2) possesses a local, positive-measure, 2b dimensional invariant Cantor manifold E I given by a Whitney smooth embedding Φ : T I [C] E I, which is a higher order perturbation of the inclusion map Φ 0 : E I P restricted to T I [C]. Moreover, the Cantor manifold E I is foliated by real analytic, linearly stable, b dimensional invariant tori carrying quasiperiodic solutions. Remark 3.1 When r = 1, for any index set J = {n 1 < < n b } and almost all m > 0, our conclusion holds while in [17] a set of positive measure of m is thrown. Remark 3.2 For the completely resonant beam equation u tt +u xxxx +f(u) = 0 (3.4) under the hinged boundary condition (3.2), the result is obtained in [21], where f is a real analytic, odd function and f(u) = au 3 + k 2 f 2k+1 u 2k+1, a 0. Remark 3.3 The similar conclusions hold for 1D nonlinear beam equation u tt +u xxxx +mu+g(u) = 0, m > 0 (3.5) with periodic boundary condition u(t,x+2π) = u(t,x), (3.6) where the nonlinearity g(u) is real analytic and g(u) = au 2 r+1 +O(u 2 r+2 ), a 0 and r N. 4 The hamiltonian setting of wave equations Without losing generality, let a = 1. In the following, we always suppose m I = (0,M ], where M is a fixed large number. Let us rewrite the wave equation (1.1) as follows Equation (4.1) may be rewritten as u tt +Au = f(u), Au u xx +mu, x, t R, (4.1) u(0,t) = 0 = u(π,t), (4.2) u = v, v +Au = f(u), (4.3) 5

6 which, as is 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 + 1 π 2 Au,u + g(u) dx, (4.4) where g is a primitive of f(u) (with respect to the u variable) and, denotes the scalar product in L 2. As in [13], we introduce coordinates q = (q 1,q 2, ),p = (p 1,p 2, ) through the relations u(x) = j 1 q j λj φ j (x), v = j 1 λj p j φ j (x), 0 where φ j = 2/π sinjx for j = 1,2, are the orthonormal eigenfunctions of the operator A with eigenvalues λ 2 j = j2 +m. The coordinates are taken from some Hilbert space H a,ρ of all real valued sequences w = (w 1,w 2, ) with finite norm w 2 a,ρ = j 1 w j 2 j 2a e 2jρ. Below we will assume that a 0 and ρ > 0. We formally obtain the hamiltonian H = Λ+G = 1 λ j (p 2 j +qj 2 )+ 2 j 1 with the lattice hamiltonian equations π 0 g( j 1 q j λj φ j ) dx (4.5) q j = H p j = λ j p j, ṗ j = H q j = λ j q j G q j, (4.6) Rather than discussing the above formal validity, we shall use the following elementary observation: Lemma 4.1 Let Ī be an interval and let t Ī (q(t),p(t)) ( {q j (t)} j 1, {p j (t)} j 1 ) be a real analytic solution of (4.6) for some ρ > 0. Then u(t,x) j 1 q j (t) λj φ j (x), is classical solution of (4.1) that is real analytic on Ī [0,π]. For the proof, see [24]. Next we consider the regularity of the gradient of G. Following Pöschel [24], we have the following Lemma. 6

7 Lemma 4.2 For a > 0 and ρ 0, the gradient G q is real analytic as a map from some neighbourhood of the origin in H a,ρ into H a+1,ρ, with with G q a+1,ρ = O( q 2 r+1 a,ρ ). For the nonlinearity u 2 r+1, we find G = 1 π u 2 r+2 dx = 1 G i1 i 2 r r +2 2 r+2 q i1 q i2 r+2 (4.7) i 1,,i 2 r+2 1 π G i1 i 2 r+2 = φ i1 φ i2 φ i2 r+2 dx. (4.8) λi1 λ i2 r+2 It is not difficult to verify that G i1 i 2 r+2 = 0 unless i 1 ±i 2 ± ±i 2 r+2 = 0, for some combination of plus and minus signs. For simplicity, write Gn t n }{{} t = G nt and Gn j n j n i n i = }{{} 2 r+2 2 r G nj n i, where j i,i,j {1,,b}. Note (1.4), we have by elementary calculation. 0 2 r+1 (2 r +1)!! 2 r G nt = λ r+1 π r nt (2 r +2)!!, G (2 r 1)!! n j n i = λ λ r ni n π r j (2 r)!!. (4.9) 5 Partial Birkhoff normal form Next we transform the hamiltonian (4.5) into some partial Birkhoff normal form so that it may serve for our aim. We first switch to complex coordinates z j = 1 2 (q j +ip j ), z j = 1 2 (q j ip j ), then the hamiltonian (with respect to the symplectic structure i j dz j d z j ) is given by H = j = j λ j z j 2 + λ j z j 2 + π 0 ( z j + z ) j g φ j dx 2λj 1 j ( r +1)2 r+2 i 1 ± ±i 2 r+2 =0 G i1 i 2 r+2 (z i1 + z i1 ) (z i2 r+2 + z i2 r+2 )+O(2 r +4). 1 For simplicity, we will write c =. So we obtain the hamiltonian ( r+1)2 r+2 H = λ j z j 2 +c G i1 i 2 r+2 z i1 z i2 r+2 j i 1 ± ±i 2 r+2 =0 G i1 i 2 r+2 z i1 z i2 z i2 r+2 + +cc2 r+2 1 i 1 ± ±i 2 r+2 =0 +cc2 r+2 k i 1 ± ±i 2 r+2 =0 +cc2 r+2 2 r+2 i 1 ± ±i 2 r+2 =0 G i1 i 2 r+2 z i1 z ik z ik+1 z i2 r+2 + G i1 i 2 r+2 z i1 z i2 z i2 r+2 +O(2 r +4). (5.1) 7

8 We define the index sets, = 0,1,2 and 3. ( = 0,1,2) is the set of index (i 1,,i 2 r+1 ) such that there exist exactly components not in {n 1,n 2,,n b }. 3 is the set of the index (i 1,,i 2 r+1 ) such that there exist at least three components not in {n 1,n 2,,n b }. Define the resonance sets N = {(i 1,i 1,,i r+1,i r+1 )} 0 and M = {(i 1,i 1,,i r+1,i r+1 )} 2. We will transform the hamiltonian (5.1) into a partial Birkhoff form of 2 r + 2 order so that the infinite dimensional KAM Theorem, which will be given in section 6, can be applied. In the following, A(H a,ρ,h a,ρ+1 ) denotes the class of all real analytic maps from some neighbourhood of the origin in H a,ρ into H a,ρ+1. For conveniences, we also denote J = I \ J,where J = J 1 Z0 and Z 0 = Z0 1 Z 2 0 Z 3 0 Z 4 0 Z 5 0. The relative constants τ, γ in J 1 are chosen suitably in section 7. We remark that 1 Z 0 = 0 and J 1 cγ, where c depends on b, n b, M. For the explanations for J 1, see Lemma 8.4 in the Appendix. For the explanations of Z0 i, i = 1,, 5, see Lemma 5.2, Lemma 5.3, Lemma 5.4 and Lemma 7.8. Lemma 5.1 For m J and any given index set {n 1 < n 2 < < n b }, there exists a real analytic, symplectic change of coordinates XF 1 in some neighborhood of the origin that takes the hamiltonian (5.1) into where X G, XĜ, X K A(H a,ρ,h a,ρ+1 ), H X 1 F = Λ+G+Ĝ+K, b G =cc2 r+2 r+1 ( G nj z nj 2 r+2 ) + b k=2 j=1 i 1,,i k {1,,b} i 1 < <i k +c( r +1) 2 C r+1 2 r+2 +c( r +1) 2 C r+1 2 r+2 + b k=2 b i=1 b t=1 i 1,,i k {1,,b} i 1 < <i k j n 1,,n b C l1 lk l 1,,l k, 0<l t < r+1,t=1,,k, l 1 + +l k = r+1 i j i,j {1,,b} i 1 i k Gn i1 n }{{ i1 } 2l 1 G nj n i z nj 2 r z ni 2 j n 1,,n b G ntjz nt 2 r z j 2 l 1,,l k, 0<l t < r, t=1,,k l 1 + +l k = r n ik n ik O(z ni1 2l1 z nik 2l k ) jjz 2l1 ni1 z nik 2l k z j 2, }{{} 2l k (5.2) 1 For a set, we denote by the Lebesgue measure of the set. 8

9 Ĝ = c (i 1,,i 2 r+2 ) 3 i 1 ± ±i 2 r+2 =0 ( C 1 r+1 Cl 1 r C l 2 r l1 C l k r l1 l k 1 ) 2 and c de- K = O(z 2 r+4 a,ρ ) and the constant C l 1 l k pends on r, τ, n b. G i1 i 2 r+2 (z i1 + z i1 ) (z i2 r+2 + z i2 r+2 ), i 1 i k = The proof is obvious from the following Lemmata. Lemma 5.2 For m J, if (i 1,,i 2 r+2 ) 0 and i 1 ± ±i 2 r+2 = 0, then λ j1 + +λ js = λ js λ j2 r+2 if and only if (j 1,,j s ) = (j s+1,,j 2 r+2 ), where (j 1,,j 2 r+2 ) = (i 1,,i 2 r+2 ). For the case λ i1 ± ±λ i2 r+2 0, one has λ i1 ± ±λ i2 r+2 > c(m) > 0. (5.3) Proof: If λ j1 + +λ js = λ js+1 + +λ j2 r+2, from collecting terms, one obtains l 1 λ k1 +l 2 λ k2 + +l p λ kp = 0, (5.4) where 0 l 1 + +l p 2 r +2 and k 1,, k p {n 1,, n b } and l 1 l 2 l p 0 ( otherwise one obtains (j 1,, j s ) = (j s+1,, j 2 r+2 ) ). Now we denote { Z0 1 = m I l } 1λ k1 + +l p λ kp = 0,l 1,,l p {1,,2r +2},. {k 1,k 2,,k p } {1,,n b },1 p b,p Z From Lemma 8.5, we have Z 1 0 = 0. Since m J, then (5.4) can t hold unless (l 1, l 2,, l p ) = (0, 0,, 0). (5.3) is obvious when λ i1 ± ±λ i2 r+2 0. Lemma 5.3 For m J, if (i 1,,i 2 r+2 ) 1 and i 1 ± ±i 2 r+2 = 0, then λ i1 ± ±λ i2 r+2 > c(m) > 0. Proof: one gets We only give a sketch since it is similar with above. After collecting terms, l 1 λ k1 + +l p λ kp +λ i = 0, (5.5) where i is the unique normal site. Since i 1 ± ±i 2 r+2 = 0, we have i (2 r+1)n b. Denote { Z0 2 = m I l } 1λ k1 + +l p λ kp +λ i = 0, l 1,,l p {1,,2r +1},. {k 1,k 2,,k p,i} {1,,(2 r +1)n b }, 1 p b, p Z Similarly, one has Z0 2 = 0. Note m J, one easily gets the conclusion. Lemma 5.4 For m J, if (i 1,,i 2 r+2 ) 2 and i 1 ± ±i 2 r+2 = 0, then λ j1 + +λ js = λ js λ j2 r+2 if and only if (j 1,,j s ) = (j s+1,,j 2 r+2 ), where (j 1,,j 2 r+2 ) = (i 1,,i 2 r+2 ). For the case λ i1 ± ±λ i2 r+2 0, one has λ i1 ± ±λ i2 r+2 c(m) > 0. (5.6) 9

10 Proof: We only need prove the necessary condition. After collecting terms, one obtains l 1 λ k1 +l 2 λ k2 + +l p λ kp ±λ i ±λ j = 0, where i, j are normal sites and the others are tangent ones. conclusions in the following two cases. (i) We only need prove the l 1 λ k1 +l 2 λ k2 + +l p λ kp +λ i +λ j = 0, i j. (5.7) It is obvious that (5.7) can t hold When i > 2 r n 2 b +M. Hence, one has i 2 r n 2 b +M. Denote { Z0 3 = m I } 1λ k1 + +l p λ kp +λ i { +λ j = 0, l 1,,l p } {1,,2r}, {k 1,k 2,,k p,i,j} 1,,[2 r n 2 b +M. ], 1 p b, p Z From Lemma 8.5, one has Z0 3 = 0, since m J, (5.7) can t hold. (ii) Write p = i j. We have l 1 λ k1 +l 2 λ k2 + +l p λ kp +λ i λ j = 0, i > j. (5.8) l 1 λ k1 + +l p λ kp +p +O( 1 ) = 0. (5.9) j It is obvious that (5.8) can t hold when p > 4 rn b M. Therefore p 4 rn b M. (5.10) From (5.8), one gets (l 1,,l p ) 0. Combined with m J, it results in Hence, when M j γ b τ (2 r) 4 r l 1 λ k1 + +l p λ kp +p 2γ b (2 r) τ. 4 r τ, (5.8) can t hold. In other words, when j M (2 r) 4 r, (5.8) can t γ b τ hold true. The left case is that j M (2 r) 4 r and i M (2 r) 4 r +4 rn γ b γ b b M. Similarly, denote l 1 λ + +l Z0 4 k1 p λ +λ kp i λ j = 0, l 1,,l p {1,,2r}, { } = m I τ {k 1,k 2,,k p,i,j} 1,,[ M (2 r) 4 r +4 rn γ b b M ], 1 p b, p Z. From Lemma 8.5, one has Z0 4 = 0. Since m J, (5.8) can t hold. The conclusion of (5.6) is obvious from the above proof. τ 10

11 Now we introduce the symplectic polar and complex coordinates by setting { (ξj +y z j = j )e ix j,j = n 1,n 2,,n b w j, j n 1,n 2,,n b depending on parameters ξ Π = [0,1] b. The precise domain will be specified later. In order to simplify the expression, we substitute ξ nj,j = 1,2,,b by ξ j,j = 1,2,,b. Then one gets i dz j d z j = dx j dy j +i dw j d w j. j 1 j=n 1,n 2,,n b j n 1,n 2,,n b Note (4.9), the new hamiltonian H = Λ+G+Ĝ+K = ω(ξ),y + Ω(ξ)w, w + G+Ĝ+K with frequencies ω(ξ) = α +A(ξ), Ω(ξ) = β +B(ξ), where α = (λ n1,λ n2,,λ nb ), β = (λ i ) i n1,,n b, A j = c r(2 r +1) r +1 A(ξ) = (A j ) b j=1 = (A 1,,A b ), B(ξ) = (B j ) j n1,,n b + c r( r +1) n j λ nj ξ r j λ r+1 i j ξ r i λ r n i + 0 p t < r, t=1,,b p 1 + +p b = r O(ξ p 1 1 ξp b b ), B j =c( r +1) 2 C r+1 2 r+2 + b k=2 b t=1 i 1,,i k {1,,b} i 1 < <i k G ntjξ r t l 1,,l k, 0<l t < r, t=1,,k l 1 + +l k = r C l 1 l k i 1 i k Gn i1 n }{{ i1 } 2l 1 n ik n ik } {{ } 2l k (5.11) l jjξ 1 i 1 ξ l k ik, We obtain c r = C r+1 2 r+2 (2 r 1)!! 4π r (2 r)!!. H = ω(ξ),y + Ω(ξ)w, w +O(y 2 ξ r 1 )+O(w 2 a,ρyξ r 1 )+O(w 3 a,ρξ 2 r 1 2 )+O(ξ r+2 ). Rescaling ξ by ɛ 6 ξ,w, w by ɛ 4 w,ɛ 4 w, and y by ɛ 8 y, one obtains a hamiltonian given by the rescaled hamiltonian H(x,y,w, w,ξ)=ɛ (6 r+8) H(x,ɛ 8 y,ɛ 4 w,ɛ 4 w,ɛ 6 ξ,ɛ) = ω(ξ),y + Ω(ξ)w, w +ɛ P (x,y,w, w,ξ,ɛ), where ω(ξ) = ɛ 6 r α +A(ξ), Ω = ɛ 6 r β +B(ξ), ξ [1,2] b. For simplicity, we rewrite H by H, ω by ω, Ω by Ω and P by P. In the following, we will use the KAM iteration which involves infinite many steps of coordinate transformations to prove the existence of the KAM tori. To make this 11

12 quantitative we introduce the following notations and spaces. Define D(r,s) = {(x,y,z, z) : Imx < s,y < r 2,z a,ρ < r, z a,ρ < r} a complex neighborhood of T b {y = 0} {z = 0} { z = 0}, where denotes the sup-norm for complex vectors. For a p (p 1) order Whitney smooth function F (ξ), define { F =max { F =max sup ξ Π sup ξ Π F,,sup p F } ξ Π ξ p, F ξ,,sup p F } ξ Π ξ p. If F (ξ) is a vector function from ξ to H a, ρ (R n ) which is p order whitney smooth on ξ, define F a, ρ = (F i (ξ) ) i a, ρ ( F R n = max i (F i (ξ) ) ). If F (η,ξ) is a vector function from D Π to H a, ρ, define F a, ρ,d = sup η D F a, ρ. We usually omit D for brevity. To functions F, associate a hamiltonian vector field defined as X F = {F y, F x,if z, if z }. Denote the weighted norm for X F by letting X F r,d(r,s) = F y + 1 r 2 F x + 1 r F z a, ρ + 1 r F z a, ρ. 6 An Infinite Dimensional KAM Theorem Theorem 1 is a direct result of the following Theorem 3 and measure estimates in section 7. Consider small perturbations of an infinite dimensional hamiltonian in the parameter dependent normal form N = ω(ξ),y + Ω(ξ)z, z on a phase space where P a,ρ = T n R n H a,ρ H a,ρ (x,y,z, z), ω j = jd + ɛ t +O(ξ p ), Ω j = jd + ɛ t +O(ξ p ), t, p N, ρ > 0, a 0. Suppose that ω M 1, Ω j M 2 j δ, M 1 + M 2 1. Define M = (M 1 +M 2 ) p. The parameter set Π is [1,2] n. For the hamiltonian H = N + P, there exists n-dimensional, linearly stable torus T n 0 = Tn {0,0,0} with frequencies ω(ξ) when P = 0. Our aim is to prove the persistence of a large portion of this family of linearly stable rotational tori under small perturbations. Suppose that the perturbation P is real analytic in the space variables, C p in ξ, and for each ξ Π its hamiltonian vector field X P = (P y, P x,ip z, ip z ) T defines near T n 0 a real analytic map X P : P a,ρ P a, ρ ( ρ ρ). Without losing generality, suppose ρ ρ δ < d 1. Under the previous assumptions, we have the following theorem. O(ξ p ) means pth order terms in ξ 1,,ξ b 12

13 Theorem 3 Suppose that H = N +P satisfies X P r,d(s,r) γs2(1+µ), (6.1) where γ depends on n,p,τ and M, µ = (p + 1)τ + p + n 2. Then there exists a Cantor set Π ɛ Π, a Whitney smooth family of torus embeddings Φ : T n Π ɛ P a, ρ, and a Whitney smooth map ω : Π ɛ R n, such that for each ξ Π ɛ, the map Φ restricted to T n {ξ} is a real analytic embedding of a rotational torus with frequencies ω (ξ) for the hamiltonian H at ξ. Each embedding is real analytic on Imx < s 2, and Φ Φ 0 r cɛ 1 2, ω ω cɛ, uniformly on that domain and Π ɛ, where Φ 0 is the trivial embedding T n Π T n 0. Moreover, there exist whitney smooth maps ω m and Ω m on Π for m 1 satisfying ω 1 = ω,ω 1 = Ω and ω m ω cɛ, (6.2) Ω m Ω δ cɛ. (6.3) Remark 6.1 Note that in the theorem, we didn t claim that the measure of Π ɛ is positive. For positive measure, one needs further information of the frequencies ω(ξ) and Ω(ξ). We shall come back to this point in Section 7. Since the proof of Theorem 2 is essentially standard, we only state the main step of KAM iteration. The more detailed steps can be found in [25] and other papers. 6.1 Solving the Linearized Equations and KAM Step At each step of KAM iteration, the symplectic coordinate change Φ is obtained as the the time 1-map X t F t=1 of the flow of hamiltonian vector field X F. Its generating function F and some normal correction N to the given normal form N are solutions of the linear equation {F,N}+ N = R, (6.4) where R = 2m+q+ q 2 R kmq qy m z q z q e i k,x,r kmq q = P kmq q, and the coefficients R kmq q depend on ξ such that X R : P a,ρ P a, ρ is real analytic and whitney smooth in ξ. Below we solve the linear equation and estimate the generating function F. Lemma 6.1 Suppose that uniformly on Π + Π, k,ω ɛβ for k 0, (6.5) k,ω +Ω i ɛβ, (6.6) k,ω +Ω i +Ω j ɛβ (i j+1), (6.7) k,ω +Ω i Ω j ɛβ (i j+1), i j, (6.8) 13

14 Then the linear equation has solution F and N, which satisfy [F ] = 0, [ N] = N. Moreover, X N r,d(s,r) X R r,d(s,r),x F r,d(s σ,r) cm ɛ (p+1)β σ µ X R r,d(s,r), (6.9) where = 1+k τ, β will be denoted later. For the proof refer to [25]. Lemma 6.2 If X F r,d(s σ,r) σ, then for any ξ Π +, the flow XF t (,ξ) exists on D(s 2σ, r 2 ) for t 1 and maps D(s 2σ, r 2 ) into D(s σ,r). Moreover, for t 1, X t F id r,d(s 2σ, r 2 ),σdxt F Id r,r,d(s 3σ, r 4 ) cx F r,d(s σ,r), where D is the differentiation operator with respect to (x,y,z, z), id and Id are identity mapping and unit matrix, and the operator norm A(ξ,η) r,r,d(s,r) = sup sup η D(s,r) w 0 A(ξ,η)w a, r w a,r, A r,r = max{a r,r,, p A ξ p r,r}. For the proof see [26]. Below we consider the new perturbation under the sympletic transformation Φ = XF t t=1. Let X P r,d(s,r) ɛ. From the above we have R = 2m+q+ q 2 Thus X R r,d(s,r) X P r,d(s,r) ɛ, and for η 1 8, Since N = 2m+q+ q 2,q= q R kmq q y m z q z q e i k,x. X P R ηr,d(s,4ηr) ηx P r,d(s,r) ηɛ. (6.10) P 0mq q y m z q z q e i k,x, the new normal form is N + = N + N = ω +,y + Ω + z, z. By Lemma 6.1, one has X N r,d(s,r) ɛ. Note that ρ ρ δ, therefore, ω + ω,ω + Ω δ ɛ, (6.11) where Ω δ = max j 1 Ω j j δ. If cmɛ1 β(p+1) σ µ+1 follows that for t 1, 1, by Lemma 6.1 and Lemma 6.2, it 1 σ Xt F id r,d(s 2σ, r 2 ),DXt F Id r,r,d(s 3σ, r 4 ) cmɛ1 (p+1)β σ µ+1. (6.12) 14

15 Under the transformation Φ = XF 1, (N+R) Φ = N ++R +, where R + = { 1 0 (1 t) N } +tr,f XF t. Thus, H Φ = N + +R + +(P R) Φ = N + +P +, where the new perturbation P + = R + +(P R) Φ = (P R) Φ+ 1 0 { R(t),F } X t F dt where R(t) = (1 t) N + tr. Hence, the hamiltonian vector field of the new perturbation is X P+ = (XF 1 ) (X P R ) (Xt F ) [X R(t),X F ]dt. For the estimate of X P+, we need the following lemma. Lemma 6.3 If the hamiltonian vector field W (,ξ) on V = D(s 4σ,2ηr) depends on the parameter ξ Π + with W r,v < +, and Φ = Xt F : U = D(s 5σ,ηr) V, then Φ W = DΦ 1 W Φ and if cmɛ1 (p+1)β 1, we have Φ W η 2 σ µ+1 ηr,u cw ηr,v. For the proof refer to [25]. Now we estimate X P+. By Lemma 6.3, if cmɛ1 (p+1)β η 2 σ µ+1 1, 1 X P+ ηr,d(s 5σ,ηr) cx P R ηr,d(s 4σ,2ηr) +c [X R(t),X F ] ηr,d(s 4σ,2ηr) dt. By Cauchy s inequality and Lemma 6.2, one obtains [X R(t),X F ] ηr,d(s 4σ,2ηr) cmɛ2 (p+1)β η 2 σ µ+1 0 = cmηɛ, where one chooses η 3 = ɛ1 (p+1)β. Combining (6.10) we have σ µ+1 X P+ ηr,d(s 5σ,ηr) cmηɛ. 6.2 Iteration and Proof of Theorem 2 To iterate the KAM step infinitely we must choose suitable sequences. For m 1 set ɛ m+1 = cm(m)ɛ 4 3 m σ 1 3 (1+µ) m 1 3 (p+1)β, σ m+1 = σ m 2, η3 m = ɛ1 (p+1)β m, σ m where β = 1 2(p+1). Furthermore, s m+1 = s m 5σ m, r m+1 = η m r m, M(m) = (M 1 +M 2 +2c(ɛ 1 + +ɛ m 1 )) p, and D m = D(s m,r m ). As initial value fix σ 1 = s Choose ɛ 1 γ 0 σ 2(1+µ) 1, where γ 0 γ, γ is a constant depending on M, p, n, τ. Finally, let K m = K 1 2 m 1 with K 1 = ln 1 ɛ 1, K 0 = 0. 15

16 Lemma 6.4 Suppose H m = N m +P m is given on D m Π m, where N m = ω m (ξ),y + Ω m,z z is a normal form satisfying for any ξ Π m. and k,ω m ɛβ m for k 0, (6.13) k,ω m +Ω m,i ɛβ m, (6.14) k,ω m +Ω m,i +Ω m,j ɛβ m(i j+1), (6.15) k,ω m +Ω m,i Ω m,j ɛβ m(i j+1),i j, (6.16) X Pm r m,d m ɛ m. Then there exists a Whitney smooth family of real analytic symplectic coordinate transformations Φ m+1 : D m+1 Π m D m and a closed subset Π m+1 = Π m \ kl (ɛ m+1 ) k>k m R m+1 of Π m, where and R m+1 kl (ɛ m+1 ) = A m+1 k1 A m+1 k2 A m+1 k3 A m+1 k4, A m+1 k1 ={ξ Π m : k,ω m+1 < ɛβ m+1 }, A m+1 k2 = i A m+1 k3 = i,j B m+1,1 ki = i {ξ Π m : k,ω m+1 +Ω m+1,i < ɛβ m+1 }, B m+1,11 kij = {ξ Π m : k,ω m+1 +Ω m+1,i +Ω m+1,j < ɛβ m+1 (i j+1) }, A i,j k A m+1 k4 = B m+1,12 kij = {ξ Π m : k,ω m+1 +Ω m+1,i Ω m+1,j < ɛβ m+1 (i j+1) }, i j i j such that for H m+1 = H m Φ m+1 = N m+1 +P m+1 the same assumptions are satisfied with m+1 in place of m. Proof: Note the value for p 1,ɛ 1,β and σ 1, one verifies that M m+1 ɛ 1 (p+1)β m+1 σ 1+µ m+1 1 M m ɛm 1 (p+1)β 2 σm 1+µ (6.17) for all m 1. So the smallness condition of the KAM step is satisfied. For the remained proof, see Iterative Lemma in [25]. With (6.11) and (6.12), we also obtain the following estimate. 16

17 Lemma 6.5 For m 1, 1 Φ m+1 id r σ m,d m+1,dφ m+1 I r m,r m,d m+1 cm(m)ɛ1 (p+1)β m m σ m µ+1 (6.18) ω m+1 ω m Π m,ω m+1 Ω m δ,π m cɛ m. (6.19) Proof of Theorem 3. The smallness condition is ɛ 1 γ 0 s2(1+µ) 202(1+µ) 1. (6.20) To apply Lemma 6.4 with m = 1, set s 1 = s,r 1 = r,,n 1 = N,P 1 = P, The smallness condition is satisfied, because γ = γ (1+µ) and ɛ 1 = γs 2(1+µ) 1. X P1 r 1,D(s 1,r 1 ) = X P r,d(r,s) γs2(1+µ) = ɛ 1. The small divisor conditions are satisfied by setting Π 1 = Π \ kl R1 kl (ɛ), where k 0 for A 1 k1, and Π 0 = Π. Then the Iterative Lemma applies. Remark 6.2 For the rescaled hamiltonian H, we fix r = 1. Then X ɛp 1,D(s,1) X ɛp 1,D(1,1) cɛ γs2(1+µ), for ɛ small enough. If fix ρ > 0 and a > 0 arbitrarily, Theorem 3 can be applied to the rescaled hamiltonian. 7 Measure Estimates The whole measure estimates for all the steps are given by the following Lemmata. From section 5, we have p = r, ρ = ρ + 1, β = 1 2( r+1), t = 6 r. Note (4.8) and (5.11), we have δ = 1. For our conveniences, we will extend ω ν and Ω ν defined in Π ν to Π. Before giving all the measure estimates, we outline the main idea of this section in the following. The main trouble lies in estimating the measure of the set k>k ν 1 i j Bν,12 kij. In fact, we have B ν,12 kij B ν,12 k(j+p)j k>k ν 1 i j k>k ν 1 0<p ck j B ν,12 k(j+p)j + k>k ν 1 0<p ck 0<j<j 0 k>k ν 1 0<p ck k>k ν 1 β r j 0 ɛν k τ r 1 r + ɛ β ν ( k τ k j 0 k>k ν 1 j 2 0 ɛt ) 1 r k. Q ν kpj 0 (7.1) The trouble lies in 1 ɛ t, where ɛ = ɛ 0. It s hard for us to choose a suitable j 0. Our method is to pick up a suitable t 0, which satisfies 1 ɛ t < kt 0. (7.2) 17

18 It is obvious that (7.2) holds when ν > N = t 0 ln2 ln 1 ɛ + 2. Now we can choose a suitable j 0 at the cost of choosing a larger τ when ν > N. In order to get the measure estimates for ν > N, we further require τ 2 r 1+t 0. (7.3) At this time, we can t fix the value for t 0. In fact, we have to combine the constraint condition for t 0 from estimating the measure for 1 < ν N. To obtain the measure estimates of these parts, we also require m J and t t 0 > τ +1. (7.4) 4 r Now we fix t 0 = τ 2 r 1 which satisfies (7.3) and (7.4) clearly when τ > 8 r. We remark that the choose of t 0 is not unique. Different choices for t 0 will correspond to different range for τ. The remained is the measure estimates for the first step. In order to get them, we also require m J. We turn to the concrete measure estimates. A couple of Lemmata are needed. The following Lemma has been used many times in this section. For the proof see [27]. Lemma 7.1 Suppose that g(x) is an mth differentiable function on the closure Ī of I, where I R is an interval. Let I h = {xg(x) < h}, h > 0. If for some constant d > 0, g m (x) d for any x I, then I h ch 1 m, where I h denotes the Lebesgue measure of I h and c = 2(2+3+ +m+d 1 ). Lemma 7.2 For k > c > 0, B ν,12 kij [(i j+1)ɛβ ν ] 1 r. Proof: For our conveniences, we write ω and Ω for ω ν and Ω ν. Define v 1 = (1,0,,0) T, and v b = (0,0,,1) T. Define S b 1 = {(x 1,x 2,,x b ) R b : x 1 + x x b = 1}. Write A(m) = (D r v 1 ω,d r v 2 ω,,d r v b ω) T. It is easy to check that A(m) = ( 1) b 1 C (λ n1 λ n2 λ nb ) r+1 0 for any ξ Π, where C is a positive constant depending on r, b. For any (ξ,υ) Π S b 1, A(m)υ 1 c 1 > 0. (7.5) Thus for any (ξ,υ) Π S b 1, there exists an open neighborhood S υ of υ in S b 1, such that for some i, D r v i ω,υ c 1 2b, for any (ξ,υ ) Π S v. Since {Π S v } covers the compact set Π S b 1, there exist finite covers: Π S 1,,Π S k0 such that k 0 Π S i Π S b 1 and for any (ξ,υ) Π S i, D r vω,υ c 1 2b, where v {v 1,v 2,,v b }. Now fix k 0 and suppose k k S i. Then for any ξ Π, i=1 D r vω, k k c 1 > 0. (7.6) 2b 18

19 Define f(ξ) = k,ω +Ω i Ω j. Note f(ξ) D r v = k k k,d r v(ω) + D r v (Ω i Ω j ) k + D r v(ω i Ω i) k + D r v(ω j Ω j ) + k k k,d r v(ω ω). (7.7) We estimate every term in (7.7). From (6.2) and (6.3), one obtains Note k k,d r v(ω ω) D r v(ω ω) cɛ, (7.8) D r v(ω i Ω i) k D r v(ω j Ω j ) k cɛ k 1 k, (7.9) cɛ k 1 k. (7.10) D r v(ω i Ω j ) k c k (7.11) f(ξ) and (7.8), (7.9), (7.10), (7.6), we arrive at D r v k c 1 4b when k c > 0. It is obvious that D r vf(ξ) c 1 4b k > c > 0, when k c > 0. The result now follows with Lemma 7.1. Remark 7.1 If define F (m,x) = A(m)υ 1, (m,υ) [0,M ] S b 1, it is easy to check that F is continuous on [0,M ] S b 1. Therefore c 1 in (7.5) depends on r, b, M. Further, we also know c depends on r, b, M. In the following, we choose t 0 = τ 2 r 1, N = Lemma 7.3 For τ > 2(b+1) r 2 +2, ν > N k>k ν 1 i j B ν,12 t t 0 ln2 ln 1 ɛ r ν 2. kij ɛ β Proof: Write p = i j. We firstly prove k>k ν 1 p ck j B ν,12 k(j+p)j =. (7.12) In fact, we have k,ω ν +Ω ν,i Ω ν,j p+1 k,ω +Ω i Ω j ck p+1 p+1 p+o( 1 j )+ k,λ (p+1)ɛ t ck p+1 1 ɛ t >> 1 19

20 From above, it is clear that (7.12) holds. From (7.12), one has k>k ν 1 i j B ν,12 kij = = k>k ν 1 i>j k>k ν 1 j,p B ν,12 kij B ν,12 k(j+p)j k>k ν 1 0<p ck j B ν,12 k(j+p)j (7.13) In order to get the estimate for (7.13), noticing the following inequality k,ω ν + p ɛ t k,ω ν +Ω ν,i Ω ν,j +Ω ν,i Ω i +Ω ν,j Ω j +Ω i Ω j p ɛ t cɛβ ν p + c j + cp j 2 ɛ t, we define Q ν kpj = {ξ Π : k,ω ν + p ɛ t cɛβ ν p + c j + cp j 2 ɛ t }. It is obvious that B ν,12 k(j+p)j Qν kpj (p > 0) and Qν kpj Qν kpj 0 for j j 0. Therefore (7.13) [( k(j+p)j ) Q ν kpj 0 ] k>k ν 1 0<p ck 0<j<j 0 B ν,12 B ν,12 k(j+p)j + k>k ν 1 0<p ck 0<j<j 0 k>k ν 1 0<p ck k>k ν 1 β r j 0 ɛν k τ r 1 r + ɛ β ν ( k τ k j 0 k>k ν 1 j 2 0 ɛt ) 1 r k. Q ν kpj 0 (7.14) Note ν > N, it is easy to check that Therefore 1 ɛ t < kt 0. (7.14) k>k ν 1 β r j 0 ɛν k τ r 1 r + k>k ν 1 ( ɛ β ν k τ k1+t0 ) 1 r k (7.15) j 0 j

21 Choosing j 0 = k 2 r τ, note τ > 2(b+1) r 2 +2, then β ɛ 2 r ν (7.15) ɛ k>k ν 1 k>k ν 1 β 2 r ν 2. β 2 r ɛν k + τ 2 2 r 1 β k>k ν 1 ( 2 r ɛν k + τ 2 2 r 1 k>k ν 1 Lemma 7.4 For 1 ν N, τ > 2(b+1) r 2 +2, k>ɛ t t 0 i j B ν,12 The proof is similar to Lemma 7.3, we omit it. ɛ β ν k τ 1 + ɛ β 2 r ɛν 2 k τ 2 r r ν 2. kij ɛ β β 2 r ν k τ 2 r + ɛ β r ν ) 1 r k k τ 2 r (7.16) Lemma 7.5 For 2 ν N, m J, τ > max{8 r, 4 3 [ r(b+1)+1]}, ɛ t t i j 0 k>kν 1 B ν,12 kij ɛ β r ν. Proof: We write p = i j. One easily has k,ω ν +Ω ν,i Ω ν,j k,ω ν +Ω i Ω j Ω ν,i Ω i Ω ν,j Ω j Note m J, we get k,λ +p c j ɛ t ck. k,λ +p 2γ b, k 0. k τ 4 r (7.17) For j > ck τ 4 r γ b, one then obtains (7.17) γ b ck. (7.18) ɛ t k τ 4 r In fact γ b ɛ t k τ 4 r ck > 1. (7.19) Firstly, from t > t t 0 ( τ t 4 r +1) and k ɛ t 0, we get ɛ t > ɛ t t 0 ( τ 4 r +1) k τ 4 r

22 Now it is easy to get (7.19) when ɛ is small enough(also depending the constant γ ). On the other hand, we have when p = i j ck. Hence, one gets ɛ t t i j 0 k>kν 1 ɛ β ν (i j+1) B ν,12 kij ɛ β ν k τ 1 ɛ β ν << 1, ɛ t t 0<p ck 0 k>kν 1 j k 4 r τ γ b+2 ( ɛ β ν k ) 1 r k 1+ τ 4 r k>k ν 1 ɛ β r ν. From Lemma 7.4 and Lemma 7.5, we obtain the following Lemma. B ν,12 kij Lemma ν N, m J, τ > τ = max{8 r, 4 3 [ r(b+1)+1], 2(b+1) r2 +2}, k>k ν 1 i j B ν,12 2 r ν 2. kij ɛ β The remained is the measure estimate for the first step. We will consider two cases to get it. From Lemma 7.4 and the same method from Lemma 7.5, we obtain the following Lemma. Lemma 7.7 For m J, τ > τ, k>c i j B 1,12 2 r 2 1, kij ɛ β where τ as above. Lemma 7.8 For m J, then 0 k c i j B 1,12 kij =. Proof: 0<k c i j B 1,12 kij 0<k c i>j B 1,12 kij 0<k c 0<p ck j B 1,12 kij (7.20) 22

23 If j > ck 4 r τ, as Lemma 7.5, we obtain γ b 0<k c 0<p ck j> ck 4 r τ γ b B 1,12 kij =. (7.21) Note Remark 7.1, if j ck 4 r τ γ b prove that for m J, C(M ), one has i = j + p C(M ). In this case we will γ b γ b k,λ +λ i λ j c > 0. (7.22) Denote { Z0 5 = m I l } 1λ k1 + +l p λ kp +λ i { λ j = 0,l 1, },l p {1,,[C(M )]}, {k 1,k 2,,k p,i,j} 1,,[ C(M ).,1 p b,p Z γ b ] From Lemma 8.5, we have Z0 5 = 0. Since m J, then k,λ +λ i λ j 0. It is obvious that (7.22) holds. This follows that B 1,12 kij =, (7.23) 0<k c 0<p ck j ck 4 r τ γ b when ɛ is small enough. From the direct computation, one gets B 1,12 kij =. (7.24) k=0 i j Note (7.21), (7.23) and (7.24), we complete the proof. From the above two Lemmata, we arrive at the following Lemma. Lemma 7.9 For m J and τ > τ, then k 0 i j B 1,12 2 r 2 1. kij ɛ β Since the measure estimates for the remained are obvious, we only give the Lemmata and omit the proofs. Lemma 7.10 For ν > 1 and τ > (b+2) r +1, then k>k ν 1 A ν k3 = Lemma 7.11 For ν > 1 and τ > (b+1) r, then k>k ν 1 A ν k2 = k>k ν 1 i,j k>k ν 1 i B ν,11 kij ɛ β r ν. B ν,1 ki ɛ β r ν. 23

24 Lemma 7.12 For ν > 1 and τ > b r, then Lemma 7.13 If τ > (b+2) r +1, then k 0 Lemma 7.14 If τ > (b+1) r, then k 0 k>k ν 1 A ν k1 ɛ A 1 k3 = k 0 i,j A 1 k2 = k 0 i β r ν. B 1,11 kij ɛ β r 1. B 1,1 ki ɛ β r 1. Lemma 7.15 If τ > b r, then k 0 A 1 k1 ɛ β r 1. Combining with all the Lemmata in this section, we easily get the total measure of the parameter sets Π ɛ, which are thrown in all the steps. Before that, one has to fix some constants. We choose τ > τ = max{τ,4 rb(b+1)}. Then, for m J, β 2 r Π ɛ ɛ 2 1 ɛ β 2 r 2 = ɛ 1 4 r 2 ( r+1), where J cγ and c depends on M, b, n b and ɛ is small enough and depends on all the constants including m. If we choose a series of γ (n) = 1 c2 n, n = 1,, then one gets a series of J (n) choose Ĵ = n for almost all m > 0. correspondingly and J (n) 2 n, n = 1,2,. Write J n = I \J (n). Now, we J n. It is obvious that I \Ĵ = 0. Since M is arbitrary, Theorem 1 is proved 8 Appendix The following two lemmata are born from Lemma 6.8 and Lemma 6.9 of Bambusi [2]. Lemma 8.1 Let m I = (0,M ], if write dλ n1 dm Ā =. d 2 λ n1 dm 2 d b λ n1 dm b dλ n2 dm d 2 λ n2 dm 2. d b λ n2 dm b dλn b dm d2 λ nb dm 2. db λ nb dm b (8.1) we have det(ā) c > 0, where c depends on b, M, n b. 24

25 Lemma 8.2 Let u (1),, u (K) be K independent vectors with u (i) 1 1. Let ω R K be an arbitrary vector, then there exists i [1,,K] such that u (i) ω ω 1det(u (i) ) K 3 2 where det(u (i) ) is the determinant of the matrix formed by the components of the vectors u (i). From the above two Lemmata and the proof of Corollary 6.10 in Bambusi [2], we get the following Lemma. Lemma 8.3 Denote f(m) = k,λ (k 0), then there exists i 0 {1,,b}, so that f (i0) (m) c 0 > 0, where λ = (λ n1,,λ nb ) and c 0 depends on b, n b, M. From Lemma 8.3, one gets the following lemma easily. Lemma 8.4 For τ > 4 rb(b+1), J 1 = k 0,p Z where λ = (λ n1,,λ nb ) and c depends on b, n b, M. {m I : k,λ +p < 2γ b } cγ, (8.2) Lemma 8.5 If f(m) = k 1 λ i1 + + k s λ is, where 0 k k s ß, k 1, k 2,, k s Z, k 1 k 2 k s 0 and i 1,, i s are different with each other and i l = min{i 1,, i s }, we have Number of {m If(m) = 0} < lnß+2 ln (i. l+1) 2 +M i 2 l +M Proof: k τ 4 r For our convenience, write a i = i 2 +m. We have f (n) (m) = c n (k 1 a 1 2 n i 1 where c n = ( 1)n+1 (2n 3)! 2 n 2 (n 2)!2 n. It is easy to check that k l a 1 2 n i l k 1 a 1 2 n ln(ß) i 1 +k 2 a 1 2 n i 2 + +k s a 1 2 n i s ), n 2, + +k l 1 a 1 2 n i l 1 +k l+1 a 1 2 n i l+1 + +k s a 1 2 n i s when n > N 0 = + 1 ln (i l +1)2 +M 2. Therefore, there exists positive integer n 0 such that i 2 l +M f (n0) (m) > 0. For example, we choose n 0 = [N 0 ]+1. This results in at most n 0 m s such that f(m) = 0. The proof is similar with Lemma 2.1 in [27]. We omit it. 25

26 References [1] D. Bambusi, S. Paleari, Families of periodic solutions of resonant PDEs, J. Nonlinear Sci. 11(2001), [2] Bambusi, D., Birkhoff normal form for some nonlinear PDEs, Comm. Math. Physics 234(2003), [3] M. Berti, P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Physics 243(2003), [4] M. Berti, P. Bolle, Multiplicity of periodic solutions of nonlinear wave equations, Nonlinear Analysis, 56/7, , [5] M. Berti, P. Bolle, Cantor families of periodic solutions for completely resonant nonlinear wave equations, preprint, [6] A. I. Bobenko and S. B. Kuksin, The nonlinear Klein-Gordon equation on an interval as a perturbed sine-gordon equation, Comm. Math. Helv. 70(1995), [7] J. Bourgain, Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Annals of Mathematics, 148(1998), [8] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5(1995), [9] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE. International Mathematics Research Notices, 1994, [10] J. Bourgain, Nonlinear Schrödinger equations, Park City Lectures, July, [11] J. Bourgain, Green s function estimates for lattice Schrödinger operators and applications, [12] J. Bricmont, A. Kupiainen and A. Schenkel, renormalization Group and the Melnikov Problem for PDE s, Comm. Math. Physics 221(2001), [13] L.Chierchia, J. You, KAM tori for 1D nonlinear wave equations with periodic boundary condtions, Comm. Math. Physics 211(2000), [14] W. Craig, C. E. Wayne, Newton s method and periodic solutions of nonlinear wave equations, Comm. Pure. Appl. Math. 46(1993), [15] W. Craig, C. E. Wayne, Nonlinear Waves and the KAM Theorem: Nonlinear Degeneracies, Proceedings of the Conference on Nonlinear Waves. [16] J. Geng, J. You, A KAM Theorem for higher dimensional partial differential equations, preprint, [17] J. Geng, J. You, KAM tori of hamiltonian perturbations of 1D linear beam equations, J. Math. Anal. Appl. 277(2003),

27 [18] G.Gentile, V. Mastropietro, M. Procesi, Periodic solutions for completely resonant nonlinear wave equations, to appear on Comm. Math. Physics. [19] S. B. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. 143(1996) [20] Z. Liang, J. You, Quasi-periodic solutions for for 1D Schrödinger equation with higher order nonlinearity, accepted by SIAM, J. Math. Anal., [21] Z. Liang, J. Geng, Quasi-periodic solutions for 1D resonant beam equation, preprint, [22] Z. Liang, J. You, Quasi-periodic solutions for 1D nonlinear wave equation, preprint, [23] B. V. Lidskij, E.I. Shulman, Periodic solutions of the equation u tt u xx + u 3 = 0, Funct. Anal. Appl. 22(1988), [24] J. Pöschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici 71(1996), [25] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Sup. Pisa, CI. Sci. 23(1996), [26] J. Xu, J. You, and Q. Qiu, A KAM theorem of degenerate infinite dimensional Hamiltonian systems(i)(ii), Science in China(Series A), Vol.39 No.4.(1996), [27] J. Xu, J. You, and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degenaracy, Math. Z., 226(1997),