Journal of Differential Equations

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1 J. Differential Equations 5 (0) Contents lists available at SciVerse ScienceDirect Journal of Differential Equations Quasi-periodic solutions for D wave equation with higher order nonlinearity Meina Gao a,, Jianjun Liu b a School of Science, Shanghai Second Polytechnic University, Shanghai 009, PR China b School of Mathematical Science, Fudan University, Shanghai 0033, PR China article info abstract Article history: Received 8 February 0 Revised August 0 Available online 9 October 0 In this paper, one-dimensional (D) nonlinear wave equation u tt u xx + mu + u 5 = 0 on the finite x-interval [0,π] with Dirichlet boundary conditions is considered. It is proved that there are many -dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form. 0 Elsevier Inc. All rights reserved.. Introduction and main results.. Introduction The existence of time-periodic solutions for the nonlinear wave equation u tt u xx + V (x)u + f (u) = 0, f (u) = f k u k (.) k 3 has been investigated by many authors. See [,6,7,9,,] and the references therein, for example. A wide variety of methods have been brought to bear on the problem, ranging from bifurcation theory (see [] for example), to variational techniques, pioneered by Rabinowitz [], to ideas which exploit the hamiltonian structure of the problem. Recently, combining variational methods, Lyapunov Schmidt * Corresponding author. addresses: mngao@sf.sspu.cn (M. Gao), @fudan.edu.cn (J. Liu) /$ see front matter 0 Elsevier Inc. All rights reserved. doi:0.06/j.jde

2 M. Gao, J. Liu / J. Differential Equations 5 (0) decomposition and Nash Moser implicit function theory, Berti and Procesi [5], Gentile, Mastropietro and Procesi [3] obtained small amplitude periodic solutions for completely resonant wave equation (.) with V (x) 0. Looking for quasi-periodic solutions for hamiltonian partial differential equations (PDEs) such as (.), one inevitably encounters the so-called small divisor problem. The KAM (Kolmogorov Arnold Moser) theory for PDEs originated by Kuksin [5 7] and Wayne [5] provides a very powerful tool to deal with this problem. Roughly speaking, the KAM-machinery is to re-formulate the hamiltonian PDE into a perturbation of a non-degenerate, partially integrable system, for which parameters need to be introduced in order to adjust frequencies to overcome small divisor problem. One way of introducing parameters into (.) is to considered parameterized potentials, that is, V = V (x; ξ), where ξ is an n-dimensional parameter. For example, Kuksin [7] showed that there are many quasi-periodic solutions for (.) for most (in the sense of Lebesgue measure) parameters ξ s. See also [8,0,5] for examples. For nonlinear wave equation with a prescribed potential V = V (x), owing to the absence of exterior parameters, one needs to find out some suitable Birkhoff normal form, and then extract parameters by amplitude-frequency modulation. In this aspect, [8] is the pioneer work, where nonlinear Schrödinger equation with constant potential is studied by Kuksin and Pöschel, and Birkhoff normal form of order four is used to extract parameters. For nonlinear wave equation, see Pöschel [] for V (x) m with m > 0, Yuan [7] for V (x) m with < m < 0, Yuan [8] for the completely resonant case V (x) 0, and Yuan [9] for a prescribed non-constant potential. In this paper, we consider the nonlinear wave equation (.) subject to Dirichlet boundary conditions with the prescribed potential V (x) m and the nonlinearity f (u) = u 5, that is, { utt u xx + mu + u 5 = 0, (t, x) R [0,π], u(t, 0) = 0 = u(t,π), (.) where m is real and positive. We will show that (.) admits small amplitude quasi-periodic solutions corresponding to -dimensional invariant tori of an associated infinite-dimensional dynamical system, using the KAM scheme. The method allows one to study more general nonlinear terms than the quintic term u 5 in (.). See Remark for the details. By and large, we will search for some suitable Birkhoff normal form of (.) so as to introduce parameters into (.), while Bambusi [] discovered special quasi-periodic solutions for the nonlinear wave equation { utt u xx + mu + u p = 0, p, u(t, 0) = 0 = u(t,π) (.3) by taking m > 0 as the parameter and avoiding Birkhoff normal form reductions. More precisely, in our paper, we will follow the procedure in [], where the existence of quasi-periodic solutions for the nonlinear wave equation u tt u xx + mu + u 3 + f k u k = 0 (.) k 5 is proved. In [], firstly the equation is written as an infinite-dimensional hamiltonian system, then the hamiltonian is put into its partial Birkhoff normal form of order, finally the Cantor manifold

3 68 M. Gao, J. Liu / J. Differential Equations 5 (0) theorem in [8] is used to obtain the result. During this procedure, there are two important points: the first one is to transform the hamiltonian into its partial Birkhoff normal form, while the second one is to find out an appropriate KAM theorem that can be applied to the normal form. However, the two points are different for (.). First of all, in order to obtain the Birkhoff normal form in [], the following proposition (Lemma in []) is essentially important: If i, j, k, l are non-zero integers, such that i ± j ± k ± l = 0,but(i, j, k,l) (p, p, q, q), then λ i + λ j + λ k + λ l cm, N + m 3 N = min( i, j, k, l ) with some absolute constant c, where λ j = sgn j j + m, j Z \{0}. While (.) lies outside the validity range of the above proposition, because it does not have the cubic term u 3. Moreover, observing that the nonlinearity in (.) is the quintic term u 5, we would like to eliminate the non-normal form terms and get the Birkhoff normal form of order 6. To this end, similarly to Lemma in [], we have to estimate divisors of the form λ j + λ j + λ j3 + λ j + λ j5 + λ j6, j, j, j 3, j, j 5, j 6 Z \{0}, (.5) under the condition j ± j ± j 3 ± j ± j 5 ± j 6 = 0, ( j, j, j 3, j, j 5, j 6 ) (p, p, q, q, r, r). But unfortunately, it s hard to give (.5) a uniform bound from below even if at least members of { j,..., j 6 } take values in a finite index set J N. Therefore,wehavetochoosesomesuitable index set J. In [], the index set J ={j < < j n } N, n fulfills min j i+ j i n. i<n Moreover, for D Schrödinger equation with the nonlinearity u u iu t u xx + mu + u u = 0, (.6) the authors also consider some admissible index set J. See Liang and You [9] for (.6) with Dirichlet boundary conditions and J. Geng and Y. Yi [] for (.6) with periodic boundary conditions. And for D Schrödinger equation with higher order nonlinearity u p u iu t u xx + mu + u p u = 0, p 3 (.7) under periodic boundary conditions, Z. Liang [0] supposed that the index set J ={j < j } satisfies j > pj > 0. Unfortunately, due to the lack of super-linear growth of eigenvalues λ j, the methods in [9,,0] fail for (.), because their methods crucially depend on the spectral asymptotics μ j j. In our settings, we pick J ={n,n } with n =, n 0, and thus get the estimates of some divisors in (.5) λ j + +λ j6 m, (j,..., j 6 ) ( 0 ) \ N. (.8) n 3 This estimate is crucial in this paper. See Lemma. for the details. Thus with the help of Lemma., we can eliminate the non-normal form terms with the index ( j,..., j 6 ) ( 0 ) \ N, and

4 M. Gao, J. Liu / J. Differential Equations 5 (0) get a partial Birkhoff normal form of order 6. See Theorem. for the details. Due to the particular structure of wave equation, the 6-order partial Birkhoff normal form is not enough here. In the following we explain this in a rough manner. Seeing the partial normal form (.5) in Theorem., Ĝ is the remaining 6-order non-normal form part with Ĝ = O( z 3 ẑ 3 ), and K is the at least 0-order part with K = O( z 0 ).Hereẑ = (z j ) j N\{n,n }. Assume z =O(r β ), ẑ =O(r) with 0 β to be determined. Then after introducing coordinate transformation in (3.5), the parameters ξ =O(r β ) and the hamiltonian vector fields are estimated by XĜ = O(r 3β+ ), X K = O(r 0β ). For the smallness condition (.5) in KAM theorem being true, we need α O ( r min(3β+,0β )). (.9) On the other hand, from (.7) in KAM theorem, the measure of the excluding set of parameters is O(r β κ α κ+ χ/ ), where κ =, 0 χ <. Unfortunately, by direct calculation, we know κ min(3β +, 0β ) < β (.0) κ + χ/ κ for every 0 β. This leads to O(r β α κ+ χ/ )>O(r 8β ), which means that, all the parameters are excluded and thus the KAM iteration procedure becomes invalid! Therefore, some higher order terms must to be eliminated. Thus, we have to find out a 0-order partial Birkhoff normal form in order to apply the KAM theorem. However, it is very difficult to calculate the coefficients of all terms of order 0. Fortunately, we only need to eliminate some 0-order terms without normal components, which is just K in (.57). In Lemma.3, we prove the estimates of the divisors λ j + +λ j0, (j,..., j 0 ) 0 \ N, j ± ± j 0 = 0. Then by using this lemma, we obtain an appropriate partial Birkhoff normal form of order 0. See Theorem. for the details. In the following we roughly explain why the KAM iteration procedure is valid now. Seeing the partial normal form (.60) in Theorem., Ĝ is the same as above, ˆK is the remaining 0-order non-normal form part with ˆK = O( z 9 ẑ ), and T is the at least -order part with T = O( z ). Thus, the hamiltonian vector fields are estimated by X ˆK = O(r 9β ), X T = O(r β ). In contrary to (.0), we know κ min(3β +, 9β, β ) > β (.) κ + χ/ for (3 + χ/) <β<(3 χ/), which means the domain of parameters is sufficiently large to obtain a good control of small divisors. Actually, we simply take χ = /5, β = /3 in our proof. Now we are in a position to search for a KAM theorem applicable to our partial Birkhoff normal form in Theorem.. We will use the KAM theorem in [3] with some modifications. The difference lies in the measure estimates of the resonant sets in the first iteration step, which are defined in terms of the unperturbed frequencies. In [3], the author used Theorem D to obtain the measure estimates for nonlinear wave equation. While formulating Theorem D, the author assumed that the unperturbed frequencies are affine functions of the parameters, which corresponds to the case for the nonlinear wave equation (.). Considering (.), we find out that the unperturbed frequencies are polynomials of the parameters of order. Therefore, we cannot use the result of Theorem D directly. Our idea to deal with this problem is to suppose that the conclusion of Theorem D, that is, (.6) is fulfilled in our KAM theorem (Theorem. in Section ). While applying to (.), we will prove that (.6) holds true. So we are able to apply Theorem. to (.) and get quasi-periodic solutions of it.

5 70 M. Gao, J. Liu / J. Differential Equations 5 (0) Main results We study (.) as an infinite-dimensional hamiltonian system on the phase space P = H 0 ([0, π]) L ([0, π]) with coordinates u and v = u t, where H 0 ([0, π]), L ([0, π]) are the usual Sobolev spaces. The hamiltonian for (.) is then H = v, v + Au, u + π u 6 dx, (.) 6 0 where A = d /dx + m and, denotes the usual scalar product in L. The hamiltonian equations of motions are u t = H v = v, v t = H u = Au u5. (.3) The quasi-periodic solutions of (.) to be constructed are of small amplitude. Thus, in first approximation the high order term u 5 may be considered as a small perturbation of the linear equation u tt u xx + mu = 0. The latter is of course well understood and has plenty of quasi-periodic solutions. To be more precise, let φ j = π sin jx, λ j = j + m, j =,,... be the basic modes and frequencies of the linear system u tt u xx + mu = 0 with Dirichlet boundary conditions. Then every solution is the superposition of their harmonic oscillations and of the form u(t, x) = j q j (t)φ j (x), q j (t) = I j cos ( λ j t + ϕ 0 j ). Their combined motions are periodic, quasi-periodic or almost-periodic, respectively, depending on whether one, finitely many or infinitely many modes are excited. In particular, for every choice J ={n < n } N of -modes there is an invariant -dimensional linear subspace E J that is completely foliated into rotational tori with frequencies λ n, λ n : E J = { (u, v) = (q φ n + q φ n, p φ n + p φ n ) } = where P ={I R : I b > 0, b =, } is the positive quadrant in R and T J (I) = { (u, v): q b + λ n b p b = I b, b =, }, I P T J (I), using the above representation of u and v. Upon restoring the nonlinearity u 5, E J with their quasiperiodic solutions will not persist in their entirety due to the modes and the strong perturbing effect of u 5 for large amplitudes. However, there does persist a Cantor subfamily of rotational -torus which are only slightly deformed. More exactly, we have the following theorem:

6 M. Gao, J. Liu / J. Differential Equations 5 (0) Theorem.. Considering D nonlinear wave equation (.), assume 0 < m {n < n } satisfies and the index set J = n =, n 0. Then, there is a set C in P with positive Lebesgue measure, a family of -tori [ T J C ] = T J (I) E J I C over C, a Lipschitz continuous embedding [ Φ : T J C ] P, which is a higher order perturbation of the inclusion map Φ 0 : E J P restricted to T J [C ], such that the restriction of Φ to each T J (I) in the family is an embedding of a rotational invariant -torus for the nonlinear wave equation (.). Remark. For the nonlinear wave equation { utt u xx + mu + g(u) = 0, (t, x) R [0,π], u(t, 0) = 0 = u(t,π), where the nonlinearity g(u) = g 5 u 5 + k 7 g ku k with g 5 0 is real analytic in u, the results of the above theorem can be proved in the same way.. The hamiltonian and partial Birkhoff normal form To rewrite (.) as a hamiltonian in infinitely many coordinates we make the ansatz u = S q = j q j λ j φ j, v = j λ j p j φ j. The coordinates are taken from some Hilbert space l a,p of all real valued sequences w = (w, w,...) with finite norm w a,p = j w j j p e aj. Below we will assume that a 0 and p >. We then obtain the hamiltonian with equations of motions H = Λ + G = j ( λ j p j + ) q j + 6 π 0 (S q) 6 dx (.) q j = H p j = λ j p j, ṗ j = H q j = λ j q j G q j. (.)

7 7 M. Gao, J. Liu / J. Differential Equations 5 (0) These are the hamiltonian equations of motions with respect to the standard symplectic structure j dq j dp j on l a,p l a,p. The same as Lemma 3 in [], we know the gradient G q is real analytic as a map from some neighborhood of the origin in l a,p into l a,p+,with Thus the associated hamiltonian vector field X G = ( G G ) p j q j q j p j j G q a,p+ = O ( q 5 a,p). (.3) defines a real analytic map from some neighborhood of the origin in l a,p l a,p into l a,p+ l a,p+. Note that G(q) = π (S q) 6 dx = G j j q j q j6 (.) j,..., j 6 0 with π G j j 6 = φ j φ j6 dx. (.5) λ j λ j6 It is not difficult to verify that G j j 6 = 0unless j ± ± j 6 = 0 for some combination of plus and minus signs. Thus, only a codimension one set of coefficients is actually different from zero, and the sum extends only over j ± ± j 6 = 0. In particular, we have 8 π G jjkkll = π 3 sin jxsin kxsin lxdx λ j λ k λ l = 0 0 π λ j λ k λ l ( + δ jk + δ jl + δ kl δ j+k,l δ j+l,k δ k+l, j ) (.6) by elementary calculation. In the rest of this section we transform the hamiltonian (.) into some partial Birkhoff normal form of order 0 so that it happens, in a sufficiently small neighborhood of the origin, as a small perturbation of some nonlinear integrable system. For the rest of this paper we introduce complex coordinates z j = (q j + ip j ), z j = (q j ip j ), j. Inserting them into (.), we obtain a real analytic hamiltonian H = Λ + G = j = j λ j z j + 6 π ( 0 λ j z j + 8 j j,..., j 6 N z j + z j λ j φ j ) 6 dx G j j 6 (z j + z j ) (z j6 + z j6 ) (.7)

8 M. Gao, J. Liu / J. Differential Equations 5 (0) on the now complex Hilbert space l a,p with symplectic structure i j dz j d z j. Real analytic means that H is a function of z and z, real analytic in the real and imaginary part of z. Conveniently introducing z j := z j for j, then H in (.7) is written as H = Λ + G (.8) with Λ = j λ j z j z j, (.9) G = 8 j,..., j 6 Z G j j 6 z j z j6, (.0) where G j j 6 := G j j 6 for j,..., j 6 Z := Z \{0}. Define the normal form set N = { ( j,..., j 6 ) Z 6 : There exists a 6-permutation τ such that j τ () = j τ (), j τ (3) = j τ (), j τ (5) = j τ (6) }. Define the following index sets l = { ( j,..., j 6 ) Z 6 :Thereareexactlyl components not in {±n, ±n } } for l = 0,,, and 3 = { ( j,..., j 6 ) Z 6 : There are at least 3 components not in {±n, ±n } }. Split G in (.0) into three parts: G = Ḡ + G + Ĝ, (.) where Ḡ is the normal form part of G with ( j,..., j 6 ) ( 0 ) N : Ḡ = 8 ( j,..., j 6 ) ( 0 ) N G j j 6 z j z j6 = 5 ( Gn n n n n n z n 6 + G n n n n n n z n 6) + 5 ( Gn n n n n n z n z n + G n n n n n n z n z n ) j n,n ( Gn n n n jj z n z j + G n n n n jj z n z j ) j n,n G n n n n jj z n z n z j, (.)

9 7 M. Gao, J. Liu / J. Differential Equations 5 (0) G is the non-normal form part of G with ( j,..., j 6 ) ( 0 ) \ N : G = 8 ( j,..., j 6 ) ( 0 )\N G j j 6 z j z j6, (.3) and Ĝ is the part of G with ( j,..., j 6 ) 3 : Ĝ = G j j 8 6 z j z j6. (.) ( j,..., j 6 ) 3 We will eliminate G by a symplectic coordinate transformation X F, which is the time--map of the flow of a hamiltonian vector X F given by a hamiltonian F = F j j 6 z j z j6 (.5) ( j,..., j 6 ) Z 6 with coefficients { G j j 6 if j j 6 = 8 λ j + +λ j6, for ( j,..., j 6 ) ( 0 ) \ N, 0, otherwise. (.6) Here λ j := sgn j λ j for j Z.Thenformallywehave {Λ, F }+ G = 0, (.7) where {, } is Poisson bracket with respect to the symplectic structure i j dz j dz j. Thus expanding at t = 0 and using Taylor s formula we formally get H X F = H Xt F t= = Λ +{Λ, F }+ ( t) { {Λ, F }, } F X t F dt + G + {G, F } X t F dt 0 0 = Λ + Ḡ + Ĝ + {Ḡ + t G + Ĝ, F } X t F dt. (.8) 0 Now we need to show the correctness of the definition (.6) and establish the regularity of the vector field X F. To this end, we prove that the divisors λ j + +λ j6 are away from zero: Lemma.. Suppose n =,n 0 and 0 < m.thenfor( j,..., j 6 ) ( 0 ) \ N,wehave λ j + +λ j6 m. (.9) n 3

10 M. Gao, J. Liu / J. Differential Equations 5 (0) Proof. This lemma is equivalent to prove that, for j,..., j 6 N and (σ,...,σ 6 ) {, } 6, if (σ j,...,σ 6 j 6 ) ( 0 ) \ N,thenwehave 6 σ l λ jl m. n 3 (.0) l= We firstly consider the case 6 l= σ l j l 0. In view of λ j = j + ( j + m ) m j = j +, j N (.) j + m + j and 0 < m,wehave σ l λ jl σ l j l m l= l= l= 3m j + m + j, (.) l l which is larger than m. Therefore, in the following, we assume n 3 6 σ l j l = 0. (.3) l= Introduce the function f (t) = t m + m t = t + m +, t (.) which is positive, monotone decreasing and convex for t 0. Thus, by (.3) and (.), we have 6 σ l λ jl = l= 6 σ l (λ jl j l ) = l= 6 σ l f ( j l ). (.5) We secondly consider the case σ k j k + σ l j l = 0 for some k,l 6. Without loss of generality, assuming k = 5, l = 6, then we have l= σ l j l = 0 and l= min( j, j, j 3, j ) n. Thus by using Lemma in [], we get 6 m σ l λ jl = σ l λ jl, (n + ) + m 3 (.6) l= l= which is larger than m. Therefore, in the following, we assume n 3 σ k j k + σ l j l 0, k,l 6. (.7)

11 76 M. Gao, J. Liu / J. Differential Equations 5 (0) Nowouraimistoprove(.0)for(σ j,...,σ 6 j 6 ) ( 0 ) \ N with (.3), (.7). It is obvious ( 0 ) \ N = ( 0 \ N ) ( \ N ) and no element in 0 \ N fulfills (.3). In the remaining proof, we consider and \ N respectively. For (σ j,...,σ 6 j 6 ), denote the unique index different with n, n in { j,..., j 6 } as a. Then from (.3) and (.7), we get a = rn + (5 r)n, 0 r 5 (.8) or a = (5 r)n rn, r. (.9) If (.8) is true with 0 r, then we know f (n ) f (a). In this case, in view of (.5), we get 6 σ l λ jl = rf(n ) + (5 r) f (n ) f (a) l= rf(n ) + ( r) f (n ) f (n ). (.30) If (.8) is true with r = 5, then we know f (n ) f (a). In this case, in view of (.5), we get 6 σ l λ jl = 5 f (n ) f (a) f (n ) f (n ). (.3) l= If (.9) is true, then in view of (.5), we get, for r, 6 σ l λ jl = rf(n ) (5 r) f (n ) + f (a) l= f (n ) f (n ) f (n ). (.3) From (.30), (.3), (.3) and f (n ) m, we know (.0) holds true for (σ n 3 j,...,σ 6 j 6 ) with (.3), (.7). For (σ j,...,σ 6 j 6 ) \ N,denotea, b the two indices different with n, n in { j,..., j 6 }. Without of generality, we assume a b. Then from (.3) and (.7), we get b a = rn + ( r)n, 0 r (.33) or b a = ( r)n rn, r 3 (.3) or a + b = rn + ( r)n, 0 r (.35)

12 M. Gao, J. Liu / J. Differential Equations 5 (0) or a + b = ( r)n rn, r 3. (.36) In the following, we will give the proof in six cases respectively. Case : Suppose (.33) is true. Then in view of (.5), we get, for 0 r, 6 σ l λ jl = rf(n ) + ( r) f (n ) + f (a) f (b) l= rf(n ) + ( r) f (n ) = f (n ). Case : Suppose (.3) is true. Then in view of (.5), we get, for r 3, 6 σ l λ jl = rf(n ) ( r) f (n ) f (a) + f (b). (.37) l= Observing a n,wehavea, and thus f (a) f (). Additionally by f (b) 0, we get Calculating directly, for n =, n 0 and 0 < m,wehave f () = f (a) f (b) f (). (.38) m m > + m m, Thus, f () = f (n ) = m + m + m, m m n + m + n n 0 m. f () f () f (n )>0. (.39) Therefore, by (.38) and (.39), we get (.37) rf(n ) ( r) f (n ) f () f (n ) 3 f (n ) f () f (n ). (.0) Case 3: Suppose (.35) is true with r = 0, i.e. a + b = n. (.) We prove this case in three subcases:

13 78 M. Gao, J. Liu / J. Differential Equations 5 (0) Subcase 3.: a n. Then in view of (.) and (.5), we get 6 σ l λ jl = f (a) + f (b) f (n ) l= ( ) n f + f (n ) f (n ) m (n /) + + m m (n ) + m m n m 3m = (n ) + m n + 6m( n + 6m + n )n ( m n + (m/6) 8 6m ) n ( m n + m 8 ) 0 m. (.) 6 n + m Subcase 3.: n < a < n. By Taylor s formula, we have, for j, λ j = j + m = j + m j m, 8 j + θm 3 (.3) where 0 <θ< depends on j. Thus, we have 6 σ l λ jl = λ n λ a λ b l= ( m = n a ) ( ) m b 8 n + θ 3, m a + θ m 3 b + θ 3 m 3 (.) where 0 <θ,θ,θ 3 <. Since n, a, b are integers and a + b = n,weknow ab n. Otherwise, we have ab n = 0, and further a = ( 3 )n, b = ( + 3 )n, which is impossible. Thus, n a b = ab n (a + b) n ab = ab n n ab n ab. (.5) On the other hand, n + θ 3 m a + θ m 3 b + θ 3 m 3 a + 3 b 3 n + 3 m a 3 3 n. + 3 m (.6)

14 M. Gao, J. Liu / J. Differential Equations 5 (0) Thus, from (.) (.6), we get 6 σ l λ jl m ( ) n ab m 8 a 3 3 n + 3 m l= ( n b ) + = m m 3m a 6a 8 n + 3 m m ( ) a n (n ) / 3m + 6(n /) 8 n + 3 m 3m = 8 n. + 3 (.7) m Subcase 3.3: a n.inviewofa + b = n and our assumption a b, weknowb n.thusin view of (.) and (.5), we get 6 σ l λ jl = f (n ) f (a) f (b) l= ( ) n f (n ) f f (n ) m m m n + m n n m. (.8) n + m Case : Suppose (.35) is true with r 3. Noting f (t) is monotone decreasing and convex for t 0, we have Thus, in view of (.5), we get, for r 3, f (a) + f (b) f (n ) + f (a + b n ) f (n ) + f (n ). 6 σ l λ jl = rf(n ) + ( r) f (n ) f (a) f (b) l= Case 5: Suppose (.35) is true with r =, i.e. (r ) f (n ) + (3 r) f (n ) = f (n ). (.9) a + b = n, (.50) where a = b = is the only possible situation. Thus, in view of (.5), we get

15 80 M. Gao, J. Liu / J. Differential Equations 5 (0) σ l λ jl = f (n ) f () l= f (n ). (.5) Case 6: Suppose (.36) is true. Then in view of (.5), we get, for r 3, 6 σ l λ jl = rf(n ) ( r) f (n ) + f (a) + f (b) l= f (n ) 3 f (n ) f (n ). (.5) It is easy to check that all the right hands of (.37), (.0), (.), (.7), (.8), (.9), (.5) and (.5) are larger than m. Hence (.0) holds true for (σ n 3 j,...,σ 6 j 6 ) \ N with (.3), (.7). This completes the proof of this lemma. In view of (.5) and the above lemma, in the same way as [], the regularity of the vector field X F could be easily established: where A (l a,p b X F A ( l a,p b,l a,p+ ) b, (.53),l a,p+ ) denotes the class of all real analytic maps from some neighborhood of the ori- b into l a,p+, and l a,p denotes the Hilbert space of all bi-infinite sequences with finite norm b b gin in l a,p b q a,p = q 0 + j q j j p e j a. Therefore, in view of (.8), we obtain the following theorem: Theorem.. Suppose n =, n 0 and 0 < m. Then by the symplectic change of coordinates Γ := X F, which is real analytic in some neighborhood of the origin in la,p, the hamiltonian H = Λ + G b in (.8) is taken into where Λ is in (.9), Ḡisin(.), Ĝisin(.), and K = Moreover, XḠ,XĜ,X K A (l a,p,l a,p+ ). b b By simple calculation we have { K = Ḡ + Ĝ + } G, F + H Γ = Λ + Ḡ + Ĝ + K, (.5) 0 0 {Ḡ + t G + Ĝ, F } X t F dt. (.55) {{ ( t)(ḡ + Ĝ) + ( ) } } t G, F, F X t F dt, (.56) where the first term is order 0 and the second term is at least order. In order to obtain a partial Birkhoff normal form of order 0, we need another real analytic, symplectic coordinate change. To this end, define the normal form set

16 M. Gao, J. Liu / J. Differential Equations 5 (0) N = { ( j,..., j 0 ) Z 0 : There exists a 0-permutation τ such that } j τ () = j τ (), j τ (3) = j τ (), j τ (5) = j τ (6), j τ (7) = j τ (8), j τ (9) = j τ (0), and the following index sets 0 = { ( j,..., j 0 ) Z 0 : All the components are in {±n, ±n } }, = { ( j,..., j 0 ) Z 0 : There are at least components not in {±n, ±n } }. Split the first term of K in (.56) into three parts: { Ḡ + Ĝ + } G, F = K + K + ˆK, (.57) where K is the normal form part with ( j,..., j 0 ) 0 N, K is the non-normal form part with ( j,..., j 0 ) 0 \ N, and ˆK is the part with ( j,..., j 0 ). The same procedure as eliminating G, we will eliminate K by another symplectic coordinate transformation. Similarly, a lemma about the divisors λ j + +λ j0 is needed: Lemma.3. Suppose n =,n 0 and 0 < m.thenfor( j,..., j 0 ) 0 \ N satisfying j ± ± j 0 = 0, wehave λ j + +λ j0. (.58) Proof. In view of j,..., j 0 {±n, ±n }, n =, n 0, thus from j ± ± j 0 = 0 we know the number of {±n } in { j,..., j 0 } is even. Hence this lemma is equivalent to prove that, for nonnegative integers α, ᾱ, β, β with α + ᾱ + β + β = 0, β + β even, and α ᾱ + β β 0, we have αλ n + ᾱλ n + βλ n + βλ n. (.59) If β β = 0, then the left hand of (.59) is not less than λ ; otherwise, the left hand of (.59) is not less than λ n 8λ. Thus the inequality (.59) follows from the fact that both λ and λ n 8λ are larger than. This completes the proof of this lemma. Therefore, we can further obtain the following result: Theorem.. Suppose n =,n 0 and 0 < m. Then by another symplectic change of coordinates Γ, which is real analytic in some neighborhood of the origin in l a,p, the hamiltonian H Γ b in (.5) is taken into H Γ Γ = Λ + Ḡ + Ĝ + K + ˆK + T, (.60) where K isoftheform K = K 0 z n 0 + K z n 8 z n + K z n 6 z n + K 3 z n z n 6 + K z n z n 8 + K 5 z n 0 (.6) with coefficients K 0,...,K 5 real and depending only on n,n and m, and

17 8 M. Gao, J. Liu / J. Differential Equations 5 (0) ẑ = (z j ) j N\{n,n }. Moreover, X K,X,XˆK T A (l a,p,l a,p+ ). b b 3. Proof of Theorem. ˆK =O ( z 9 a,p ẑ a,p ), (.6) T =O ( z a,p), (.63) By the last section there exist two real analytic, symplectic changes of coordinates Γ, Γ, which take H into H Γ Γ = Λ + Ḡ + Ĝ + K + ˆK + T, where Λ is in (.9), Ḡ is in (.), Ĝ is in (.), K is in (.6), ˆK is in (.6), T is in (.63). Letting I = ( z j : j N), thenwehave Λ = j λ j I j, (3.) Ḡ = 5 ( 5I 3 π + 5 λ 3 + 7I I n λ λ n + 7I In λ λn + 5I 3 n λ 3 n ) ( G jj I + ) G n n jji I n + G n n n n jjin I j, (3.) j,n K = K 0 I 5 + K I I n + K I 3 I n + K 3 I I3 n + K I I n + K 5 I 5 n. (3.3) Moreover, we know ( ) ( ) ( Ĝ =O z a,p 3 ẑ 3 a,p, ˆK =O z 9 a,p ẑ a,p, T =O z a,p). (3.) Step : New coordinates. We introduce symplectic polar and real coordinates (x, y, u, v) by setting z nb = ξ b + y be ix b, b =,, z j = (u j + iv j ), j,n, (3.5) depending on the parameter ξ = (ξ,ξ ) R +.Thenwehave i dz j d z j = dx b dy b + du j dv j j j,n b=, and I = ξ + y, I n = ξ + y, I j = (u j + v j ) for j,n. Up to a constant depending only on ξ, the normal form Λ + Ḡ + K becomes with tangential frequencies ω(ξ), y + Ω(ξ),u + v + Q ω (ξ) = λ ξ 8π λ 3 + π λ λ ξ ξ + 5 ξ n 8π λ λn + 5K 0 ξ + K ξ 3 ξ + 3K ξ ξ + K 3 ξ ξ 3 + K ξ, (3.6)

18 normal frequencies and remainder M. Gao, J. Liu / J. Differential Equations 5 (0) ω (ξ) = λ n + 8π λ λ ξ + ξ n π λ λn ξ + 5 ξ 8π λn 3 + K ξ + K ξ 3 ξ + 3K 3ξ ξ + K ξ ξ 3 + 5K 5ξ, (3.7) Ω j (ξ) = λ j + 5 ( G jj ξ + G n n jjξ ξ + ) G n n n n jjξ, (3.8) The total hamiltonian H = N + P with Q = O ( y ) + O ( y u + v ). (3.9) N = ω(ξ), y + Ω(ξ),u + v, (3.0) P = Q + Ĝ + ˆK + T. (3.) Now let r > 0 and consider the phase space domain D(, r): Im x <, y < r, u a,p + v a,p < r, (3.) and the parameter domain { Π = ξ = (ξ,ξ ): r 3 ξ,ξ 6 } 5 r 3. (3.3) Step : Checking assumption A of Theorem.. By (3.6) we know ω = 5 + ξ 8π λ 3 ω ξ = = 5 + 8π λ 3 5 = 8π λ λ ξ n 5 8π λ λ ξ n 5 8π λ λ ξ n ξ ξ + 0K 0ξ + 6K ξ ξ + 3K ξ + K 3 ξ ξ 3 + O( r 3 ), (3.) 5 8π λ λ ξ ξ n 8π λ λn K ξ 3 ξ + 3K ξ + 3K 3 ξ ξ + K ξ ξ ( ) 8π λ λn O r 3. (3.5) In view of λ n > 8λ and 5 6 ξ ξ 6, from (3.), (3.5) we have, for r small enough, 5 Similarly, by (3.7), we know 5 ω, 8π λ 3 ξ π λ 3 (3.6) 5 π λ λ ω 7 n ξ π λ λ. n (3.7)

19 8 M. Gao, J. Liu / J. Differential Equations 5 (0) and furthermore we have ω 5 = ξ 8π λ λ + n ω ξ = 5 ξ 8π λ λn ξ + O( r 3 ), (3.8) 5 ξ 8π λ λn ξ ( ) 8π λn 3 O r 3, (3.9) 5 8π λ λ ω 3 n ξ π λ λ, n (3.0) 5 ω 7. π λ λn ξ π λ λn (3.) Thus, from (3.6), (3.7), (3.0) and (3.), we can obtain the estimate of the Lipschitz semi-norm ω L Π ω ξ + ω Π ξ Π { ω max ξ, Π 7 + π λ 3 π λ λ n 5 π λ 3 ω ξ } { ω + max Π ξ, Π ω ξ } Π (3.) and the estimate of the Jacobi determinant ( ) ω det = ω ω ω ω ξ ξ ξ ξ ξ π λ 3 π λ λn π λ λ n 8π λ λ n 9 =. 8π λ (3.3) λ n From ( ξ ω ) = ( ω ξ ) we know ( ξ ξ ω ω ξ ξ ω ω ) ( ω ξ = ω ξ ω ξ ω ξ ) ( ω /det ξ ). (3.) Thus, from (3.6), (3.7), (3.0), (3.) and (3.3), we get ξ ω 6π λ 3, (3.5) ξ 56 ω π λ 9 λ n, (3.6) ξ 5 ω π λ 9 λ n, (3.7) ξ 3 π λ λn 9. (3.8) ω

20 M. Gao, J. Liu / J. Differential Equations 5 (0) In the same way, we can obtain the estimate of the Lipschitz semi-norm ω L ω(π) ξ ω + ξ Π ω 5 π λ Π 9 λ n + 3 π λ λn 9 5π λ λn. (3.9) In view of (3.) and (3.9), we know the map ξ ω(ξ) between Π and its image is a homeomorphism which is Lipschitz continuous in both directions. Finally, there obviously holds l,ω(ξ) 0 for all integer vectors l Z with l. Step 3: Checking assumption B of Theorem.. By (3.8) we know Ω j = λ j, ˆΩ j = 5 ( G jj ξ + G n n jjξ ξ + ) G n n n n jjξ, j,n. (3.30) From (.6) we know 5 π G jj = λ λ, j =, 3 π λ λ, j,,n, j 3, j = n π G n n jj = λ λ n λ ±, j, j,n π λ λ n λ,n ±, j 5, j = n π G n n n n jj = λn λ, n 3, j,n π λn λ, n. j (3.3) (3.3) (3.33) We can easily see that ˆΩ is a Lipschitz map from Π to l with lp the space of all complex sequences with finite norm w p = sup j w j j p, and by calculation we can get ˆΩ L,Π < 0. π λ (3.3) In view of (3.), (3.9) and (3.3), denoting M = and L = 5π λ π λ 3 π λ λn, the assumptions (.) in Theorem. are satisfied. Finally, observing that λ j = j + m = j + m j + O( j 3 ) and λ j = j + O( j 3 ), we know the assumption (.) in Theorem. is satisfied with κ =. Step : Checking assumption C and smallness condition (.5) of Theorem.. Observing (3.) for the perturbation P, it can be easily checked that P is real analytic in the space coordinates and Lipschitz in the parameters, and for each ξ Π its hamiltonian vector field X P is an analytic map from P a,p to P a, p with p = p +. In the following we check the smallness condition (.5). In view of (3.9), we have Q =O ( r ). (3.35) In view of (3.) and ξ =O(r 3 ), wehave Ĝ =O (( r 3 ) 3r 3 ) = O ( r ), (3.36) (( ) ˆK =O r 3 9r ) ( ) = O r, (3.37) (( ) T =O r ) ( ) 3 = O r 3. (3.38)

21 86 M. Gao, J. Liu / J. Differential Equations 5 (0) From (3.), (3.35) (3.38), we know P =O(r ) and thus X P r,d(s,r) Π = O ( r ). (3.39) Since X P is real analytic in ξ,wehave X P r,d(s,r) Π L = O( r r ) ( ) 3 = O r 3. (3.0) We choose α = r 5 8 γ, (3.) where γ is taken from the KAM theorem. It s obvious that when r is small enough, ε := X P r,d(s,r) Π + α M X P L r,d(s,r) Π = O( r ) γα, (3.) which is just the smallness condition (.5). Till now there only remains the assumption (.6) of Theorem.. Step 5: Checking assumption (.6) of Theorem.. For convenience, we introduce ζ = (ζ,ζ ) Θ as parameters by letting ζ = ξ, ζ = ξ, where { } Θ = ζ = (ζ,ζ ): r 6 3 ζ,ζ 5 r 3. (3.3) In view of (.3), denoting { R kl (α) = ζ Θ: k,ω(ζ ) + l,ω(ζ) < α l }, (3.) A k then to prove the assumption (.6) is equivalent to prove R kl ( ˆα) cr κ 3 α (κ+ χ/), (3.5) (k,l) X where ˆα = α κ+ χ κ+ χ/,0 χ < and c is a positive constant. We only need to give the proof of the most difficult case that l has two non-zero components of opposite sign. In this case, rewrite R kl (α) in (3.) as { R kij (α) = ζ Θ: k,ω(ζ ) + Ω i (ζ ) Ω j (ζ ) < α i j A k }, (3.6) where k Z and i, j N \{,n }, i j. In view of (.), it is sufficient to prove R kij ( ˆα) cr κ 3 α (κ+ χ/), (3.7) 0< k <K, 0<i+ j<l where K, L are defined in the KAM theorem and here they satisfy

22 M. Gao, J. Liu / J. Differential Equations 5 (0) K = 6LM < 75λ n, (3.8) L = 36 ( ω Π + ) LM/β < 9600λ 3 n. (3.9) From (3.6) (3.8), we have, for 0 < k < K,0< i + j < L, k,ω(ζ ) + Ωi (ζ ) Ω j (ζ ) = k λ + k λ n + λ i λ j ( 5k + 8π λ 3 ( 5k + + 5k 8π λ λ + 5G ii 5G ) jj ζ n π λ λ + 5k + n π λ λn 5G n n ii 5G n n jj ( 5k + + 5k + 5G n n n n ii 5G n n n n jj 8π λ λn 8π λn 3 ) ζ ζ ) ζ + O ( r 8 3 ). (3.50) Denoting C =, then if for every 0< k < K 3π λn 3, 0< i + j < L at least one of the following inequalities holds: 5k 8π λ 3 5k + 5k 8π λ λn 8π λn 3 then for r small enough, either + 5k 8π λ λ + 5G ii n + 5G n n n n ii 5G jj 5G n n n n jj ( k, ω(ζ ) +Ω ζ i (ζ ) Ω j (ζ )) or C, (3.5) C, (3.5) ( k, ω(ζ ) +Ω ζ i (ζ ) Ω j (ζ )) is larger than C. Thus, by using Lemma 3. at the end of this section and noting that k, i, j can be bounded by a positive constant depending only on λ n,weget R kij ( ˆα) ( + )( ) / i j ˆα diam Θ = ( O ˆα / r /3). (3.53) C A k Since the number of (k, i, j) satisfying 0 < k < K, 0< i + j < L can be bounded by a positive constant depending only on λ n, we finally get 0< k <K, 0<i+ j<l R kij ( ˆα) = O( ˆα / r /3) = ( O α κ+ χ ) (κ+ χ/) r 3, (3.5) which is less than the right hand of (3.7) by the fact χ <. Therefore, till now the only remaining task is to prove that at least one of (3.5) and (3.5) holds. Supposing this not true, then we have 0k λ + 8k 8k λ + 0k λ n + π λ λ (G ii G jj ) 6π λ C <, n 5 0λ n + π λn (G n n n n ii G n n n n jj) 6π λn C 5 = 0λ n. (3.55)

23 88 M. Gao, J. Liu / J. Differential Equations 5 (0) We discuss (3.55) in four different cases in the following. We also mention that k Z,0< k < K and i, j N \{,n }, i j, 0< i + j < L. Case : Both i and j are not in {,,n, n }. Then (3.55) becomes 0k λ + 8k λ n + 8 8k λ + 0k λ n λ i λ, j 0λ n 8 λ. i 0λ n λ j (3.56) Eliminating λ i λ,weget j and eliminating k λ n,weget 8k 8k λ λ, (3.57) n 5λ n k λ λ 7. (3.58) i 5λ n λ j However, if k = 0thenk 0 and then (3.57) is impossible; otherwise, k 0 and then (3.58) is impossible by noticing k λ λ and 7 λ i 7 λ j < 7 λ. 3 Case : i = and j is not in {,,n, n }. Then (3.55) becomes 0k λ + 8k λ n + 5 8k λ + 0k λ n λ λ, j 0λ n 8 λ. 0λ n λ j (3.59) Eliminating k λ n,weget k λ λ 7. (3.60) 5λ n λ j However, if k = 0 then (3.60) is impossible by noticing 87 λ 7 λ j > 5 λ ;otherwise,k 0 and then (3.60) is impossible by noticing k λ λ and 87 λ 7 λ j < 87 λ. In the same way, we can prove the case: j = and i is not in {,,n, n }. Case 3: i = n and j is not in {,,n, n }. Then (3.55) becomes Eliminating λ j,weget 0k λ + 8k λ n + 8 8k λ + 0k λ n + 5 8k 8k 3 λ λ n 8 λ n λ, j 0λ n 8 λ. n 0λ n λ n λ j (3.6), (3.6) 5λ n

24 M. Gao, J. Liu / J. Differential Equations 5 (0) and eliminating k λ n,weget k λ λ 7. (3.63) n 5λ n λ j However, if k = 0thenk 0 and then (3.6) is impossible; otherwise, k 0 and then (3.63) is impossible. In the same way, we can prove the case: j = n and i is not in {,,n, n }. Case : i = and j = n. Then (3.55) becomes 0k λ + 8k λ n + 5 λ 8 λ n 8k λ + 0k λ n + 8 λ 5 λ n, 0λ n. 0λ n (3.6) Eliminating k λ n,weget k λ λ λ n 7. (3.65) 5λ n However, if k = 0 then (3.65) is impossible by noticing 87 λ λ 5 n > λ ;otherwise,k 0 and then (3.65) is impossible by noticing k λ λ and 87 λ 7 λ j < 87 λ. In the same way, we can prove the case: j = and i = n. Till now, all the assumptions in the KAM theorem have been checked. Taking χ = /5, in view of (.7), the measure of the excluding set of parameters is O ( r 3 α κ κ+ χ/ ) = O ( r 3 r ), which is of higher order than Π =O(r 8 3 ). This means that, when r is small enough, the rotational tori persist for most of ξ Π. Thus Theorem. follows from Theorem. in the next section. Finally, we mention that, here the remaining set of parameters doesn t have full density measure at the origin, since even the considered total parameter domain Π in (3.3) doesn t have full density measure at the origin either. Lemma 3.. Suppose that g(x) is a -th differentiable function on the closure ĪofI,whereI R is an interval. Let I h ={x: g(x) < h, x I},h> 0.IfonI, d g(x) dx d > 0, where d is a constant, then I h (+d )h. Proof. This is a special case of Lemma. in [6].. A KAM theorem In this section, we introduce a KAM theorem in [3]. We consider small perturbations of a family of linear integrable hamiltonians on a phase space N = j n ω j (ξ)y j + Ω j (ξ) ( u j + ) v j, n <, j P a,p = T n R n l a,p l a,p (x, y, u, v)

25 90 M. Gao, J. Liu / J. Differential Equations 5 (0) with symplectic structure j n dx j dy j + j du j dv j. The frequencies ω = (ω,...,ω n ) and Ω = (Ω,Ω,...) depend on n-parameters ξ Π R n, with Π a closed bounded set of positive Lebesgue measure. For each ξ there is an invariant n-torus T 0 = T n {0, 0, 0} with frequencies ω(ξ). In its normal space described by the uv-coordinates the origin is an elliptic fixed point with characteristic frequencies Ω(ξ). The aim is to prove the persistence of a large portion of this family linearly stable rotational tori under small perturbations H = N + P of N. To this end the following assumptions are made. Assumption A: Nondegeneracy. The map ξ ω(ξ) between Π and its image is a homeomorphism which is Lipschitz continuous in both directions. Moreover, l,ω(ξ) 0 onπ forallintegervectorsl Z with l. Assumption B: Spectral Asymptotics. There exists δ<0 such that Ω j (ξ) = j + + O ( j δ), where the dots stand for fixed lower order terms in j, allowing also negative exponents. More precisely, there exists a fixed, parameter-independent sequence Ω with Ω j = j + such that ˆΩ j = Ω j Ω j give rise to a Lipschitz map ˆΩ : Π l δ, with l p the space of all complex sequences with finite norm w p = sup j w j j p. Assumption C: Regularity. The perturbation P is real analytic in the space coordinates and Lipschitz in the parameters, and for each ξ Π its hamiltonian vector field X P = (P y, P x, P v, P u ) T defines near T 0 a real analytic map X P : P a,p P a, p, p > p. To make this more precise we introduce complex neighborhoods D(s, r): Im x < s, y < r, u a,p + v a,p < r of T 0 and weighted norms (x, y, u, v) r = (x, y, u, v) p,r = x + r y + r u a, p + r v a, p, where is the sup-norm for complex vectors. Then we assume that the hamiltonian vector field X P is real analytic on D(s, r) for some s and r uniformly in ξ with finite norm X P r,d(s,r) = sup D(s,r) X P r, and that the same holds for its Lipschitz semi-norm X P L r = sup ξ ζ ξζ X P r, ξ ζ where ξζ X P = X P (,ξ) X P (,ζ), and where the supremum is taken over Π. To state the KAM theorem we also assume that ω L Π + ˆΩ L δ,π M <, ω L L <, (.) ω(π)

26 M. Gao, J. Liu / J. Differential Equations 5 (0) where the Lipschitz semi-norms are defined analogously to X P L r such that.letκ > 0 be the largest exponent Ω i Ω j = + ( O j κ), i > j, (.) i j uniformly on Π. Without loss of generality, we can assume that δ κ by increasing δ if necessary. Moreover, we introduce { R kl (α) = ξ Π: k,ω(ξ) + l,ω(ξ) < α l }, (.3) A k where l =max(, jl j ), A k = + k τ, τ (n + 3) δ δ. Finally let X = { (k,l) Z n Z :0< k < K, 0 < l σ < L }, (.) where K = 6LM, σ = min(, δ), l σ = l j j σ, L = 36( ω Π + )LM/β with β the largest positive constant such that l,ω 7 β l for every l. Theorem.. Suppose H = N + P satisfies assumptions A, B, C, and ε = X P r,d(s,r) Π + α M X P r,d(s,r) Π L γα, (.5) where 0 < α is another parameter, and γ depends on n, τ, s. Then there exists a Cantor set Π α Π, a Lipschitz continuous family of torus embeddings Φ : T n Π α P a, p, and a Lipschitz continuous map ω : Π α R n,suchthatforeachξ Π α 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 Im x < s/,and Φ Φ 0 r + α M Φ Φ 0 L r cε α, ω ω + α M ω ω L cε, uniformly on that domain and Π α,whereπ 0 : T n Π T 0 is the trivial embedding, and c γ depends on the same parameter as γ. Moreover, denoting ˆα = α 3w χ, w= κ+ χ with χ any fixed number in 0 χ < min( p p, ), then if (k,l) X R kl ( ˆα) c ρ n α κ κ+ χ/ (.6) for all sufficiently small α,whereρ = diam Π, and the constant c depends on χ and p p, then we have κ Π \ Π α c ρ n α κ+ χ/, (.7) where the constant c also depends on χ and p p.

27 9 M. Gao, J. Liu / J. Differential Equations 5 (0) Proof. This is a combination of Theorem A and Theorem D in [3] with some modifications. First, compared with d in [3], this theorem only concerns the case d = sincethecased > is not relevant for the application to the wave equations. Second, the following two assumptions in [3] are removed: () the unperturbed frequencies satisfy { ξ: k,ω(ξ) + l,ω(ξ) = } 0 = 0 for integer vectors (k,l) Z n Z with l ; () the unperturbed frequencies are affine functions of the parameters. These two assumptions are only used to estimate the measure for (k,l) X. Thus, instead of them, in this theorem the direct assumption (.6) is sufficient. Acknowledgments The authors are very grateful to X. Yuan for his invaluable help and to the referees for their invaluable suggestions. Jianjun Liu is supported by China Postdoctoral Science Foundation (No ), Shanghai Postdoctoral Science Foundation (No. R000) and China Postdoctoral Special Science Foundation (No. 0036). References [] D. Bambusi, On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity (999) [] M. Berti, P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. 3 (003) [3] M. Berti, P. Bolle, Cantor families of periodic solutions for wave equations via a variational principle, Adv. Math. 7 (008) [] M. Berti, P. Bolle, M. Procesi, An abstract Nash Moser theorem with parameters and applications to PDEs, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (00) [5] M. Berti, M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, Comm. Partial Differential Equations 3 (006) [6] H. Brézis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. Soc. 8 (983) [7] H. Brézis, L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math. 3 (978) 30. [8] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and application to nonlinear PDE, Int. Math. Res. Not. IMRN (99) [9] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (995) [0] L. Chierchia, J. You, KAM tori for D nonlinear wave equations under periodic boundary conditions, Comm. Math. Phys. (000) [] W. Criag, C.E. Wayne, Newton s method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math. 6 (993) [] J. Geng, Y. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations 33 (007) 5 5. [3] G. Gentile, V. Mastropietro, M. Procesi, Periodic solutions for completely resonant nonlinear wave equations, Comm. Math. Phys. 6 (006) [] J.K. Hale, Periodic solutions for a class of hyperbolic equations containing a small parameter, Arch. Ration. Mech. Anal. 3 (967) [5] S.B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen. (3) (987) 37; English translation in: Funct. Anal. Appl. (987) [6] S.B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat. 5 (988) 63; English translation in: Math. USSR Izv. 3 () (989) [7] S.B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Springer-Verlag, Berlin, 993. [8] S.B. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. 3 (996) [9] Z. Liang, J. You, Quasi-periodic solutions for D Schrödinger equations with higher order nonlinearity, SIAM J. Math. Anal. 36 (005) [0] Z. Liang, Quasi-periodic solutions for D Schrödinger equations with the nonlinearity u p u, J. Differential Equations (008) [] B.V. Lidskij, E. Shulman, Periodic solutions of the equation u tt u xx + u 3 = 0, Funct. Anal. Appl. (988) [] J. Pöschel, Quasi-periodic solutions for nonlinear wave equations, Comment. Math. Helv. 7 (996) [3] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Super. Pisa 3 (996) 9 8. [] P. Rabinowitz, Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 3 (978) 3 68.

28 M. Gao, J. Liu / J. Differential Equations 5 (0) [5] C.E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 7 (990) [6] J. Xu, J. You, Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 6 (997) [7] X. Yuan, Invariant manifold of hyperbolic elliptic type for nonlinear wave equation, Int. J. Math. Sci. 8 (003) 36. [8] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations 95 (006) 30. [9] X. Yuan, Invariant tori of nonlinear wave equations with a given potential, Discrete Contin. Dyn. Syst. 6 (3) (006)