KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions

Available online at www.sciencedirect.com

ScienceDirect J. Differential Equations 256 (2014) 1627–1652 www.elsevier.com/locate/jde

KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions ✩ Jianjun Liu a , Xiaoping Yuan b,∗ a School of Mathematical Sciences, Sichuan University, Chengdu 610065, PR China b School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China

Received 18 July 2013; revised 13 November 2013 Available online 7 December 2013

Abstract This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditions     iut + uxx + i f |u|2 u x = 0,

x ∈ T := R/2πZ,

where f is real analytic in some neighborhood of the origin in C, f (0) = 0, and f  (0) = 0. We show the above equation possesses Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields. © 2013 Elsevier Inc. All rights reserved.

1. Introduction and main results In this paper, we consider the derivative nonlinear Schrödinger equation with periodic boundary conditions     iut + uxx + i f |u|2 u x = 0,

x ∈ T,

(1.1)

where f is real analytic in some neighborhood of the origin in C, f (0) = 0, and f  (0) = 0. As in [20], we may assume f  (0) is positive for convenience; furthermore, by rescaling u ✩

Supported by NSFC 11301358 and NSFC 11271076 and NSFC 11121101.

* Corresponding author.

E-mail addresses: [email protected] (J. Liu), [email protected] (X. Yuan). 0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.11.007

1628

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

appropriately, we may assume f  (0) = 1. For example, if f is the identity function, i.e., f (z) = z, then (1.1) reduces to   iut + uxx + i |u|2 u x = 0,

(1.2)

which appears in various physical applications and has been widely studied in the literature. We study (1.1) as a Hamiltonian system on some suitable phase space P; for example, we may take H02 (T), the usual Sobolev space on T with vanishing average. Under the standard inner product on L2 (T), (1.1) can be written in the form ∂u d ∂H =− ∂t dx ∂ u¯

(1.3)

with Hamiltonian  H = −i

 ux u¯ dx +

T

  g |u|2 dx,

(1.4)

T

z where g(z) = 0 f (ζ ) dζ . Our aim is to construct time quasi-periodic solutions of small amplitude. To begin in first approximation, we consider the linear equation iut + uxx = 0 with periodic boundary conditions and vanishing average. If we let 1 ej (x) = √ eij x , 2π

λj = j 2 ,

j ∈ Z¯ := Z \ {0}

be the basic modes and frequencies, then every solution of the linear equation is of the form u(t, x) =



qj (t)ej (x),

qj (t) = qj0 eiλj t .

j ∈Z¯

In particular, for every index set ¯ J = {j1 < j2 < · · · < jn } ⊂ Z, there is an invariant linear subspace EJ of complex dimension n which is completely foliated into rotational tori with frequencies λj1 , . . . , λjn :   E J = u = q 1 e j1 + · · · + qn e jn : q ∈ C n = TJ (I ), I ∈Pn

where Pn = {I ∈ Rn : Ib > 0, 1  b  n} is the positive quadrant in Rn and   TJ (I ) = u = q1 ej1 + · · · + qn ejn : |qb |2 = Ib , 1  b  n .

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1629

Upon restoration of the nonlinear i(f (|u|2 )u)x , we show, in a sufficiently small neighborhood of the origin, a large Cantor subfamily of rotational n-tori persist with slight deformation. A result of similar form was previously obtained in [20] by Kuksin and Pöschel for the nonlinear Schrödinger equation with Dirichlet boundary conditions   iut = uxx − mu − f |u|2 u,

u(t, 0) = 0 = u(t, π),

where m is real, f is real analytic in some neighborhood of the origin in C, f (0) = 0, and f  (0) = 0. For convenience, we keep fidelity with the notation and terminology from [20]. The following is our result for (1.1): ¯ there exist: Theorem 1.1. For any positive integer n and index set J = {j1 < j2 < · · · < jn } ⊂ Z, (1) a Cantor set C ⊂ Pn with full density at the origin; of the inclu(2) a Lipschitz embedding Ψ : TJ [C ] → P, which is a higher order perturbation

sion mapping Ψ0 : EJ → P restricted to TJ [C ], where TJ [C ] := I ∈C TJ (I ) ⊂ EJ is a family of n-tori over C , such that the image EJ := Ψ (TJ [C ]) is a Cantor manifold of diophantine n-tori for the derivative nonlinear Schrödinger equation (1.1). Moreover, the restriction of Ψ to each torus TJ (I ), I ∈ C is smooth, and EJ has a tangent space at the origin equal to EJ . Following [20], we prove the above theorem by using infinite dimensional KAM theory. In that paper, a KAM theorem for bounded perturbation vector field is used; however, since the nonlinearity of (1.1) contains a spatial derivative, a suitable KAM theorem for unbounded perturbation vector field is required for our result. Historically, KAM theory for partial differential equations with bounded perturbations was earlier established in the initial work [15,26] by Kuksin and Wayne, and then was more widely investigated by many authors [3,7,8,10–14,16,17,24,25,27,28]. (We cannot list all papers in this field.) In contrast, there are fewer results of KAM theory for partial differential equations with unbounded perturbations. The first KAM theorem for unbounded perturbations is due to Kuksin [18,19]. In [18], under the assumption 0 < d˜ < d − 1 (d is the order of the linear vector field and d˜ is the order of the perturbation vector field), a suitable estimate, which is now called Kuksin’s lemma, is proved for the small-denominator equation with large variable coefficient. In [19], by using this estimate, a KAM theorem with 0 < d˜ < d − 1 is established to prove the persistence of the finite-gap solutions of the KdV equation, alongside its hierarchy subject to periodic boundary conditions. See also [21] by Kappeler and Pöschel. Another example of a KAM theorem based on Kuksin’s lemma for unbounded linear Hamiltonian perturbation can be found in Bambusi and Graffi [2] where the spectrum property is investigated for a class of time dependent linear Schrödinger equations. Recently, KAM theory for unbounded perturbations has been extended to the limiting case 0 < d˜ = d − 1. In [22,23], the small-denominator equation with large variable coefficient is suitably estimated for 0 < d˜  d − 1, and consequently the corresponding KAM theorems for infinite dimensional Hamiltonian systems are established. As a result, the spectrum property can be analyzed for a larger class of time dependent linear Schrödinger equations including the quantum Duffing oscillator, and quasi-periodic solutions can be obtained for a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed

1630

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

Benjamin–Ono equation with periodic boundary conditions. By using the estimate in [22] to solve homological equations, Zhang, Gao and Yuan [29] establish an infinite dimensional KAM theorem for reversible systems with unbounded perturbations, which can then be applied for another class of derivative nonlinear Schrödinger equations. Since both Kuksin’s lemma in [18] and the estimate in [22] are only valid for scalar homolog ical equations, the normal frequency Ωj is required to be simple; that is, Ωj = 1. This implies the range of applications of the previous KAM theorems for unbounded perturbations pertain to those PDEs with simple frequencies. This precludes the derivative nonlinear Schrödinger equa tion (1.1) with periodic boundary conditions, since the multiplicity Ωj = 2. Fortunately, this difficulty can be avoided since the nonlinear i(f (|u|2 )u)x does not contain the space variable x explicitly, so that momentum is conserved for (1.1); consequently, passing to Fourier coefficients, the corresponding Hamiltonian consists of monomials qn1 q¯n2 qn3 q¯n4 · · · qn2r−1 q¯n2r with n1 − n2 + n3 − n4 + · · · + n2r−1 − n2r = 0.

(1.5)

After introducing action-angle coordinates for tangential variables, the monomials of the Hamiltonian take the form eik1 x1 +···+ikn xn y1m1 · · · ynmn



l

l¯

zjj z¯ jj ,

(1.6)

(lj − l¯j )j = 0.

(1.7)

¯ j ∈Z\J

and (1.5) becomes −

 1bn

k b jb +

 ¯ j ∈Z\J

In p. 2 of [9], under the assumption |n1 |  |n2 |  · · ·  |n2r |, Bourgain observed that with the restriction (1.5) together with near resonance, one may control |n1 |+|n2 | by |n3 |+|n4 |+· · ·+|n2r | unless n1 = n2 . Consider the corresponding lower order terms among (1.6) which would be eliminated in the KAM iteration scheme. For example, take the most difficult terms: eik·x zi z¯ j , k ∈ Zn , i, j ∈ Z¯ \ J , where k · x := k1 x1 + · · · + kn xn . Roughly speaking, Bourgain’s observation means that for every nearly resonant term eik·x zi z¯ j under the restriction (1.7), |i| + |j | is controlled by |k| unless i = j . Hence, for a fixed k, all the nearly resonant terms except eik·x zj z¯ j are finite in number and so can be eliminated by digging out parameters for finite number of small-divisor conditions. As a result, only eik·x zj z¯ j are left as normal form terms. The homological equations are then scalar, and the estimate in [22] for small-denominator equation with large variable coefficient still works. Of course, to support the KAM iteration procedure, we must verify that (1.7) (i.e. momentum conservation) is preserved for the Hamiltonian at every KAM step; thus, it is necessary to prove that (1.7) persists under Poisson bracket and solving homological equations. The persistence under Poisson bracket can be directly checked. Since the homological equations are not of constant coefficients, the persistence under solving homological equations is not obvious. Subsection 4.2 contains our method. If the nonlinear term contains the space variable x explicitly so that (1.5) is not true, then both eik·x zj z¯ j and eik·x zj z¯ −j remain as normal form terms. The homological equations are not scalar and so more investigation is needed about the existence of KAM tori.

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1631

In the remainder of the paper, we say a Hamiltonian H (q, q) ¯ satisfies (1.5) if the indices of its monomials satisfy (1.5), and a Hamiltonian H (x, y, z, z¯ ) satisfies (1.7) if the indices of its monomials satisfy (1.7). Recently, the existence of quasi-periodic solutions for forced KdV equations and derivative nonlinear wave equations with periodic boundary conditions was studied in [1,4–6] by Baldi, Berti, Biasco and Procesi. In [1], the existence of quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations is attained. In [4], the authors study Hamiltonian derivative wave equations, and a key aspect of their method is to prove first order asymptotic expansions of the perturbed normal frequencies. In [5,6], they discuss a class of reversible derivative wave equations, and solve the problem of x-dependence by introducing weighted norms penalizing high-momentum terms and a symmetrization procedure on the space of even solutions. We now lay out an outline of the present paper. In Section 2, we give a KAM theorem (Theorem 2.1) which can be viewed as a modified version of Theorem 1.1 in [23]. In our theorem, the ¯ \ J , and the perturbation P is required to satisfy (1.7); index set of normal variables becomes Z moreover, the assumptions δ  1, 4LM2  C1J and d˜  1 are added in the assumption (C), in (D), and in the assumption (2.7), respectively. Section 3 contains the proof of Theorem 1.1. We first transform the Hamiltonian into a partial Birkhoff normal form up to order four, while making sure (1.5) is preserved for the new Hamiltonian; consequently, after introducing action-angle coordinates for tangential variables, the new Hamiltonian satisfies (1.7). Extracting parameters by amplitude-frequency modulation and using the KAM theorem, Theorem 1.1 is finally achieved. Here we mention [20,25] as the pioneering work extracting parameters from Birkhoff normal forms of Hamiltonian partial differential equations. In Section 4, the KAM theorem is proved. The derivation of the homological equations is covered in Subsection 4.1. Note that both the normal form (2.1) and the symplectic structure (2.2) carry the symbols σj ’s. As a result, the homological equations will be the same as in [23] after replacing “j  1” by “j ∈ Z∗ ” and adding a restriction (1.7) with reference to (4.6)–(4.12). Compared with the proof of Theorem 1.1 in [23], the proof of Theorem 2.1 follows a parallel course except for two essential differences: (1) the persistence of (1.7) under KAM iteration; (2) the measure estimate. These two aspects will be discussed in detail while the other parts of the proof will be omitted. The persistence of (1.7) under KAM iteration has been mentioned above. We now give a brief introduction for the measure estimate. Due to the double multiplicity of normal frequencies, the essential difference compared with [23] is to estimate the resonant set 

 ξ ∈ Π : k, ω(ξ ) + Ωj (ξ ) − Ω−j (ξ ) is small  for k ∈ Zn \ {0}, j ∈ Z∗ with the restriction − 1bn kb jb + 2j = 0. Clearly, we hope the Lebesgue measure of the set is small. To that end, we need to verify that k, ω(ξ ) + Ωj (ξ ) − Ω−j (ξ ) is twisted with respect to ξ ∈ Π , equivalently, twisted with respect to ω ∈ ω(Π); that is, we need to show     lip () :=  k, ω + Ωj ξ(ω) − Ω−j ξ(ω) ω(Π ) > 0, where ω(ξ(ω)) = ω. At the ν-th KAM step, because of the modification of frequencies from unbounded perturbation, we have

1632

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

  ˜  lip |Ωj |ω(Π ) = O |j |δ + O |j |d . ˜ So there exists a constant C˜ 1 By a small trick used in Subsection 4.1 below, we can take δ = d. such that ˜

()  |k| − C˜ 1 |j |d .  From − 1bn kb jb + 2j = 0, there exists a constant C˜ 2 such that |k|  C˜ 2 |j | from which it follows that ˜ ()  C˜ 2 |j | − C˜ 1 |j |d .

To verify the twist condition () > 0, we then require d˜  1, C˜ 2 > C˜ 1 , which are just the additional assumptions δ  1, d˜  1, 4LM2  C1J in the KAM theorem. For the derivative nonlinear Schrödinger equation (1.1) these conditions fortunately hold. 2. A KAM theorem Denote Z∗ := Z¯ \ J . For a  0 and p  0, define the Hilbert space a,p of all complex sequences z = (zj )j ∈Z∗ with z2a,p =



e2a|j | |j |2p |zj |2 < ∞.

j ∈Z∗

In the whole of this paper the parameter a is fixed. For convenience, we keep fidelity with the notation and definitions from [21]. Consider small perturbations H = N + P of an infinite dimensional Hamiltonian in the parameter dependent normal form 

N=

σjb ωb (ξ )yb +



σj Ωj (ξ )zj z¯ j

(2.1)

j ∈Z∗

1bn

on the phase space P a,p = Tn × Rn × a,p × a,p (x, y, z, z¯ ) with the symplectic structure  1bn

σjb dyb ∧ dxb − i



σj dzj ∧ d z¯ j ,

(2.2)

j ∈Z∗

where σj = 1 for j > 0 and σj = −1 for j < 0. The tangential frequencies ω = (ω1 , . . . , ωn ) and normal frequencies Ω = (Ωj )j ∈Z∗ are real vectors depending on parameters ξ ∈ Π ⊂ Rn , Π a closed bounded set of positive Lebesgue measure, and roughly Ωj (ξ ) = |j |d + · · · .

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1633

The perturbation term P is real analytic in the space coordinates and Lipschitz in the parameters. Additionally we assume P satisfies (1.7). Moreover, for each ξ ∈ Π its Hamiltonian vector field T  XP = (σjb Pyb )1bn , −(σjb Pxb )1bn , −i(σj Pz¯ j )j ∈Z∗ , i(σj Pzj )j ∈Z∗ defines near T0 := Tn × {y = 0} × {z = 0} × {¯z = 0} a real analytic map XP : P a,p → P a,q , where ˜ p − q = d. a,p

Denote by PC the complexification of P a,p . For s, r > 0, we introduce the complex a,p T0 -neighborhoods in PC D(s, r):

|Im x| < s, |y| < r 2 , za,p < r, ¯za,p < r,

(2.3)

¯ ∈P and weighted norm for W = (X, Y, Z, Z) C

a,q

W r,a,q = |X| +

¯ a,q Z |Y | Za,q + + , 2 r r r

where | · | denotes the sup-norm for complex vectors. Furthermore, for a map W : D(s, r) × Π → a,q PC , for example, the Hamiltonian vector field XP , we define the norms W r,a,q,D(s,r)×Π =

sup

W r,a,q ,

D(s,r)×Π lip

W r,a,q,D(s,r)×Π =

sup

ξ,ζ ∈Π, ξ =ζ

ξ ζ W r,a,q , |ξ − ζ | D(s,r) sup

where ξ ζ W = W (·; ξ ) − W (·; ζ ). In a completely analogous manner, the Lipschitz semi-norms of the frequencies ω and Ω are defined as lip

|ω|Π =

sup

ξ,ζ ∈Π, ξ =ζ

|ξ ζ ω| , |ξ − ζ |

lip

|Ω|−δ,Π =

sup

sup

ξ,ζ ∈Π, ξ =ζ j ∈Z∗

|j |−δ |ξ ζ Ωj | |ξ − ζ |

(2.4)

for any real number δ. Theorem 2.1. Suppose the normal form N described above satisfies the following assumptions: (A) The map ξ → ω(ξ ) between Π and its image is a homeomorphism which is Lipschitz conlip tinuous in both directions, i.e. there exist positive constants M1 and L such that |ω|Π  M1 lip and |ω−1 |ω(Π )  L;

1634

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

(B) There exists d > 1 such that   |Ωi − Ωj |  m|i|d − |j |d 

(2.5)

for all i, j ∈ Z∗ ∪ {0} uniformly on Π with some constant m > 0. Here Ω0 = 0; Ω (ξ ) (C) There exists δ  min{1, d − 1} such that the functions ξ → jj δ are uniformly Lipschitz lip

on Π for j ∈ Z∗ , i.e. there exists a positive constant M2 such that |Ω|−δ,Π  M2 ; (D) We additionally assume   m 1 , , 4LM2  min E CJ

(2.6)

where E = |ω|Π := supξ ∈Π |ω(ξ )| and CJ := max1bn |jb |. Set M = M1 + M2 . Then for every β > 0, there exists a positive constant γ , depending only on n, d, δ, m, the frequencies ω and Ω, s > 0 and β, such that for every perturbation term P described above with d˜ = p − q  min{1, d − 1}

(2.7)

and  := XP r,a,q,D(s,r)×Π +

α lip XP r,a,q,D(s,r)×Π  (αγ )1+β M

(2.8)

for some r > 0 and 0 < α < 1, there exist: (1) a Cantor set Πα ⊂ Π with |Π \ Πα |  c1 ρ n−1 α,

(2.9)

where | · | denotes Lebesgue measure, ρ := diam Π represents the diameter of Π , and c1 > 0 is a constant depends on n, J , ω and Ω; (2) a Lipschitz family of smooth torus embeddings Φ : Tn × Πα → P a,p satisfying: for every non-negative integer multi-index k = (k1 , . . . , kn ),   k ∂ (Φ − Φ0 ) x

where ∂xk :=

∂ |k| k ∂x1 1 ···∂xnkn

r,a,p,Tn ×Πα

+

 1 α ∂ k (Φ − Φ0 )lip 1+β /α, n ×Π  c2  x r,a,p,T α M

(2.10)

with |k| := |k1 | + · · · + |kn |, Φ 0 : T n × Π → T0 ,

(x, ξ ) → (x, 0, 0, 0)

is the trivial embedding for each ξ , and c2 is a positive constant which depends on k and the same parameters as γ ;

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1635

(3) a Lipschitz map φ : Πα → Rn with |φ − ω|Πα +

α lip |φ − ω|Πα  c3 , M

(2.11)

where c3 is a positive constant which depends on the same parameters as γ , such that for each ξ ∈ Πα the map Φ restricted to Tn × {ξ } is a smooth embedding of a rotational torus with frequencies φ(ξ ) for the perturbed Hamiltonian H at ξ . In other words,   t → Φ θ + tφ(ξ ), ξ ,

t ∈R

is a smooth quasi-periodic solution for the Hamiltonian H evaluated at ξ for every θ ∈ Tn and ξ ∈ Πα . 3. Proof of Theorem 1.1 In the first subsection, we write the derivative nonlinear Schrödinger equation (1.1) into Hamiltonian form of infinitely many coordinates, and then transform it into a partial Birkhoff normal form up to order four. In the second subsection, we prove Theorem 1.1 by using Theorem 2.1. 3.1. Birkhoff normal form We introduce for any a  0 and p˜ > 3/2 the phase space a,p˜

H0

   2 u(j = u ∈ L2 (T): u(0) ˆ = 0, u2a,p˜ = ˆ ) |j |2p˜ e2aj < ∞ j ∈Z¯

of complex-valued functions on T, where 2π u(j ˆ )=

u(x)e−j (x) dx, 0

1 ej (x) = √ eij x . 2π

To write (1.3) in infinitely many coordinates, we make the ansatz u(t, x) =



γj qj (t)ej (x),

(3.1)

j ∈Z¯

√ where γj = |j |. The coordinates are taken from the Hilbert space ¯a,p of all complex-valued sequences q = (qj )j ∈Z¯ with q2a,p =

 j ∈Z¯

|qj |2 |j |2p e2aj < ∞,

(3.2)

1636

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

¯ is abbreviated as “= 0” where p = p˜ + 12 . In the following, for convenience the notation “∈ Z” or omitted. Now (1.3) can be rewritten as 

∂H q˙j = −iσj , ∂ q¯j

σj =

1, −1,

j  1, j  −1

(3.3)

with the Hamiltonian H = Λ + G + K,

(3.4)

where Λ=



σj j 2 |qj |2 ,

(3.5)

j =0

G=



1 4π

γj γk γl γm qj q¯k ql q¯m ,

(3.6)

j,k,l,m=0 j −k+l−m=0

  |K| = O q6a,p .

(3.7)

Now consider the 4-order term G. The normal form part of G is (3.6) with j = k or j = m, that is, 1  2 1 j |qj |4 + 4π 2π j =0

 j,l,j −l=0

 2 1  2 1  4 2 |j l||qj | |ql | = − j |qj | + |j ||qj | . 4π 2π 2

2

j =0

Observe that Eq. (1.1) has a conservation comes B +

C2

2π ,



2 T |u| dx,

that is,

(3.8)

j =0



2 j =0 |j ||qj |

= C. Thus, (3.8) be-

where B =−

1  2 j |qj |4 . 4π

(3.9)

j =0

Fix a positive integer N . Define the index sets    = (j, k, l, m): j, k, l, m = 0, j − k + l − m = 0, j = k, m ,   1 = (j, k, l, m) ∈ : there are at least 2 components in {±1, . . . , ±N } . Split the non-normal form part of G into two parts: Q1 = Q2 =

1 4π

1 4π



γj γk γl γm qj q¯k ql q¯m ,

(3.10)

(j,k,l,m)∈1

 (j,k,l,m)∈\1

γj γk γl γm qj q¯k ql q¯m .

(3.11)

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1637

Then, up to a constant, the Hamiltonian is written as H = Λ + B + Q1 + Q2 + K.

(3.12)

Obviously the functions B, Q1 , Q2 , K are analytic in ¯a,p with real value, and their gradients ∂q¯ B, ∂q¯ Q1 , ∂q¯ Q2 , ∂q¯ K are analytic maps from ¯a,p into ¯a,p−1 with   ∂q¯ Ba,p−1 , ∂q¯ Q1 a,p−1 , ∂q¯ Q2 a,p−1 = O q3a,p ,   ∂q¯ Ka,p−1 = O q5a,p .

(3.13)

In this subsection, the symplectic structure is −i



σj dqj ∧ d q¯j ,

(3.14)

j =0

and the corresponding Poisson bracket for two Hamiltonians H , F is {H, F } = −i



 σj

j =0

 ∂H ∂F ∂H ∂F − . ∂qj ∂ q¯j ∂ q¯j ∂qj

(3.15)

The following result can be directly checked: Lemma 3.1. If both Hamiltonians H and F satisfy (1.5), then their Poisson bracket {H, F } also satisfies (1.5). In the following lemma, we will eliminate Q1 in the Hamiltonian and thus get a partial Birkhoff normal form up to order four. Lemma 3.2. There exists a real analytic symplectic coordinate transformation Ψ defined in a neighborhood of the origin of ¯a,p , which transforms the above Hamiltonian H into a partial Birkhoff normal form up to order four. That is, H ◦ Ψ = Λ + B + Q2 + R

(3.16)

  ∂q¯ Ra,p−1 = O q5a,p .

(3.17)

with

Moreover, the new Hamiltonian H ◦ Ψ also satisfies (1.5). Proof. Define F =



j,k,l,m=0 Fj klm qj q¯k ql q¯m

 Fj klm =

by

γj γk γl γm i 4π j 2 −k 2 +l 2 −m2 ,

for (j, k, l, m) ∈ 1 ,

0,

otherwise.

(3.18)

1638

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

Then we have {Λ, F } + Q1 = 0.

(3.19)

Let Ψ = XF1 be the time-1-map of the flow of the Hamiltonian vector field XF , and then  H ◦ Ψ = H ◦ XFt t=1 = Λ + {Λ, F } + B + Q1 + Q2 1 +

  (1 − t) {Λ, F }, F ◦ XFt dt +

0

1 {B + Q1 + Q2 , F } ◦ XFt dt + K ◦ XF1 0

= Λ + B + Q2 + R,

(3.20)

where 1 R=

{B + tQ1 + Q2 , F } ◦ XFt dt + K ◦ XF1 .

(3.21)

0

The m-th element of ∂q¯ F reads explicitly  ∂F = 2Fj klm qj q¯k ql . ∂ q¯m

(3.22)

j,k,l

¯ j − k + l − m = 0 we have Now consider (j, k, l, m) ∈ 1 . For j, k, l, m ∈ Z, j 2 − k 2 + l 2 − m2 = −2(m − j )(m − l).

(3.23)

Observe that m = j, l. Hence, if |m|  2N , then   2(m − j )(m − l)  2  |m| ; N

(3.24)

if |m| > 2N , at least one of j , l being in {±1, . . . , ±N }, then     2(m − j )(m − l)  2 |m| − N  |m| . N

(3.25)

We conclude form (3.23)–(3.25) the inequality  2  j − k 2 + l 2 − m2   |m| . N

(3.26)

Thus,   ∂F   ∂ q¯

   N  2πγ m m

 j −k+l=m

γj γk γl |qj ||q¯k ||ql | =

N gm , 2πγm

(3.27)

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1639

 where gm stands for the sum j −k+l=m γj γk γl |qj ||q¯k ||ql |. Obviously, g = (gj )j =0 could be written as the three-fold convolution: g = w ∗ v ∗ w, where w = (wj )j =0 , wj = γj |qj | and v = (vj )j =0 , vj = γj |q¯−j |. Thus, for any r > 1, N cN cN gr− 1  wr− 1 vr− 1 wr− 1 = q3r , 2 2 2 2 2π 2π 2π

∂q¯ F r 

(3.28)

where c > 0 depends only on r. This establishes the regularity of the vector field XF with (3.17) as follows. Observing that all of Λ, B, Q1 , Q2 , K, F satisfy (1.5), hence the new Hamiltonian H ◦ Ψ also satisfies (1.5) by using Lemma 3.1. This completes the proof of Lemma 3.2. 2 3.2. Using the KAM theorem ¯ define N := max(|j1 |, . . . , |jn |). Then by For a given index set J = {j1 < j2 < · · · < jn } ⊂ Z, the transformation Ψ in Lemma 3.2, we get a new Hamiltonian, still denoted by H , H = Λ + B + Q2 + R,

(3.29)

which is analytic in some neighborhood U of the origin of ¯a,p with Λ in (3.5), B in (3.9), Q2 in (3.11), R satisfying (3.17). Moreover, H (q, q) ¯ satisfies (1.5). Introduce new symplectic coordinates (x, y, z, z¯ ) by setting 

  qjb = ξb + yb e−ixb , q¯jb = ξb + yb eixb , ¯ \ J, qj = zj , q¯j = z¯ j , j ∈ Z∗ = Z

b = 1, . . . , n,

(3.30)

where ξ = (ξ1 , . . . , ξn ) ∈ Rn+ . Then, 

Λ=

σj j 2 |zj |2 ,

(3.31)

j ∈Z∗

1bn

B =−



σjb jb2 (ξb + yb ) +

1  2 1  2 jb (ξb + yb )2 − j |zj |4 , 4π 4π

(3.32)

j ∈Z∗

1bn

and thus the new Hamiltonian, still denoted by H , up to a constant depending only on ξ , is given by H=

 1bn

σ jb ω b y b +



σj Ωj zj z¯ j + Q˜ + Q2 + R

(3.33)

j ∈Z∗

with the symplectic structure  1bn

where

σjb dyb ∧ dxb − i

 j ∈Z∗

σj dqj ∧ d q¯j ,

(3.34)

1640

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

ωb = jb2 −

σjb jb2 ξb , 2π

(3.35)

Ωj = j 2 ,

(3.36)

1  2 1  2 2 jb y b − j |zj |4 . Q˜ = − 4π 4π

(3.37)

j ∈Z∗

1bn

Moreover, H (x, y, z, z¯ ) satisfies  (1.7).  Denote the normal form 1bn σjb ωb yb + j ∈Z∗ σj Ωj zj z¯ j by N and the perturbation Q˜ + Q2 + R by P . Firstly consider the Hamiltonian H = N + P on D(s, r) × Ξr , where the phase space domain D(s, r) is defined in (2.3) and the parameter domain   Ξr = ξ ∈ Rn+ : |ξ |  r 6/5 .

(3.38)

  XQ˜ r,a,p−1,D(s,r)×Ξr = O r 2 .

(3.39)

In view of (3.37), we have

In view of (3.11), Q2 is at least 3-order about z, z¯ , and hence     XQ2 r,a,p−1,D(s,r)×Ξr = O r 3/5 r 3 r −2 = O r 8/5 .

(3.40)

In view of (3.17), R is at least 6-order of q, and hence 6     XR r,a,p−1,D(s,r)×Ξr = O r 3/5 r −2 = O r 8/5 .

(3.41)

We conclude from (3.39), (3.40), (3.41)   XP r,a,p−1,D(s,r)×Ξr = O r 8/5 .

(3.42)

Πr := U−α Ξr ,

(3.43)

Define

where U−ρ Ξ is the subset of all points in Ξ with boundary distance greater than ρ, and α will be chosen as a function of r later. Since XP is analytic in ξ ,   lip XP r,a,p−1,D(s,r)×Πr = O r 8/5 /α .

(3.44)

Now study the Hamiltonian H = N + P on D(s, r) × Πr by the KAM theorem. In view of (3.35), the assumption (A) is fulfilled with M1 :=

1 max j 2 , 2π 1bn b

L := 2π max jb−2 . 1bn

(3.45)

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1641

In view of (3.36), the assumption (B) is fulfilled with d = 2, m = 1 and the assumption (C) is fulfilled with δ = 1, M2 a positive constant arbitrarily small. Furthermore, (2.6) in the assumption (D) is fulfilled by choosing M2 small enough. Choose α = r 7/5 γ −1 ,

β = 1/15,

(3.46)

where γ is taken from the KAM theorem. Set M := M1 + M2 , which only depends on the set J . Observe that when r is small enough,  := XP r,a,p−1,D(s,r)×Π +

  α lip XP r,a,p−1,D(s,r)×Π = O r 8/5  (αγ )1+β , M

(3.47)

which is just the smallness condition (2.8). Applying Theorem 2.1, we obtain a Cantor set Πr− ⊂ Πr with        Πr \ Π −  = O r 65 n−1 α = O r 65 n+ 15 , r

(3.48)

a Lipschitz family of smooth torus embeddings Φr : Tn × Πr− → P a,p , and a Lipschitz frequency map φr : Πr− → Rn , such that for each ξ ∈ Πr− the map Φr restricted to Tn × {ξ } is a smooth embedding of a rotational torus with frequencies φr (ξ ) for the perturbed Hamiltonian H at ξ . Moreover, for every non-negative integer multi-index k = (k1 , . . . , kn ),  k     1   α ∂ (Φr − Φ0 ) ∂ k (Φr − Φ0 )lip = O  1+β /α = O r 1/10 , − + n x x r,a,p,T ×Πr r,a,p,Tn ×Πr− M (3.49)  8/5  α lip |φr − ω|Πr− + |φr − ω|Π − = O() = O r . (3.50) r M The Cantor set Πr− by itself is not dense at the origin. To obtain such a set, following [20], we take the union of a suitable sequence of subsets of Πr− . Set Rr = Ξr \ Ξ 2r and Cr = Πr− ∩ (U−α Rr ). In view of Rr \ Cr ⊂ (Rr \ (U−α Rr )) ∪ (Πr \ Πr− ), we have      6 1 |Rr \ Cr |  Rr \ (U−α Rr ) + Πr \ Πr−  = O r 5 n+ 5 . Set rj =

r0 2j

(3.51)

, j  0, and define C :=



Crj .

j 0

Choose r0 small enough such that all the Crj are nonempty. Observe that Crj are disjoint and even have a pairwise positive distance to each other. In view of (3.51), we get    6 1     1  6 n+ 1  n+   |Ξrj \ C | =  (Rrl \ Crl ) = O rl5 5 = O rj5 5 = O rj5 |Ξrj |, lj

lj

(3.52)

1642

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

and hence |C ∩ Ξrj | |Ξrj |

 1 = 1 − O rj5 → 1 as j → ∞,

(3.53)

which indicates the Cantor set C has full density at the origin. Define the embedding Φ : Tn × C → P a,p and the frequency map φ : C → Rn by piecing together the corresponding definitions on each component. We will estimate Φ − Φ0 and φ − ω on C ∩ Ξrj . In view of (3.49), we get   k ∂ (Φ − Φ0 ) x

rj ,a,p,Tn ×(C ∩Ξrj )

   1/10  ,  sup∂xk (Φ − Φ0 )r ,a,p,Tn ×C = O rj lj

l

rl

(3.54)

and for ξ, ζ ∈ Crj ,  1/10  α(rj ) ξ ζ ∂xk (Φr − Φ0 )rj ,a,p,Tn . = O rj M |ξ − ζ |

(3.55)

On the other hand, |ξ − ζ |  α(rj ) for ξ ∈ Crj , ζ ∈ Crl , l > j , and thus α(rj ) ξ ζ ∂xk (Φr − Φ0 )rj ,a,p,Tn M |ξ − ζ |     1  ∂ k (Φ − Φ0 ) + ∂xk (Φ − Φ0 )r ,a,p,Tn ×C  x rj ,a,p,Tn ×Crj rl j M  1/10  . = O rj

(3.56)

We conclude from (3.55), (3.56)   1/10  α(rj )  ∂ k (Φr − Φ0 )lip . n ×(C ∩Ξ ) = O rj x r ,a,p,T rj j M

(3.57)

In view of (3.50), in the same way, we conclude |φr − ω|C ∩Ξrj +

 8/5  α(rj ) lip |φr − ω|C ∩Ξr = O rj . j M

(3.58)

The embedding Φ describes the invariant tori in terms of the relative actions y. Define Ψ := Φ + Tξ , where Tξ (x, ξ ) = (0, ξ, 0, 0). Then Ψ gives the same tori in terms of the absolute actions I = ξ + y. Since Ψ − Ψ0 = Φ − Φ0 , where Ψ0 is the trivial embedding (x, I ) → (x, I, 0, 0), the above estimates are preserved. This finally completes the proof of Theorem 1.1. 4. Proof of the KAM theorem ˜ decreasConsider the conditions δ  min{1, d − 1} and d˜ = p − q  min{1, d − 1}. If δ > d, ˜ ˜ ing q such that δ = d, then for the new q, the inequality (2.8) still holds true; if δ < d, increasing δ ˜ then for the new δ, the assumption (C) still holds true. Thus, without loss of gensuch that δ = d, erality we assume δ = d˜  min{1, d − 1} in the following.

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1643

In this section, the Poisson bracket {H, F } for two Hamiltonians H , F is defined with respect to the symplectic structure (2.2), i.e., {H, F } =



 σ jb

1bn

∂H ∂F ∂H ∂F − ∂xb ∂yb ∂yb ∂xb

 −i



 σj

j ∈Z∗

 ∂H ∂F ∂H ∂F − . ∂zj ∂ z¯ j ∂ z¯ j ∂zj

(4.1)

4.1. Derivation of homological equations The proof of Theorem 2.1 employs the rapidly converging iteration scheme of Newton type to deal with small-divisor problems introduced by Kolmogorov, involving infinite sequence of coordinate transformations. At the ν-th step of the scheme, the Hamiltonian Hν = Nν + Pν is considered, where Nν is a generalized normal form Nν =



σjb ων,b (ξ )yb +



σj Ων,j (x, ξ )zj z¯ j ,

j ∈Z∗

1bn

Pν is a small perturbation, and Hν satisfies (1.7). A transformation Φν is set up so that Hν+1 = Hν ◦ Φν = Nν+1 + Pν+1 , where Nν+1 is another generalized normal form, Pν+1 is a much smaller perturbation, and Hν+1 still satisfies (1.7). We drop the index ν of Hν , Nν , Pν , ων , Ων , Φν and shorten the index ν + 1 as +. For a function u on Tn , let  1 [u] = u(x) dx. (2π)n Tn

Let R be 2-order Taylor polynomial truncation of P , that is,











R = R x + R y , y + R z , z + R z¯ , z¯ + R zz z, z + R z¯ z¯ z¯ , z¯ + R z¯z z, z¯ ,

(4.2)

where · , · is formal product for two column vectors and R x , R y , R z , R z¯ , R zz , R z¯ z¯ , R z¯z depend on x and ξ . Denote by JRK the part of R in generalized normal form as follows       JRK = R x + R y , y + diag R z¯z z, z¯ , where diag(R z¯z ) is the diagonal of R z¯z . In the following, the term [R x ] will be omitted since it does not affect the dynamics. The coordinate transformation Φ is obtained as the time-1-map XFt |t=1 of a Hamiltonian vector field XF , where F is of the same form as R:











F = F x + F y , y + F z , z + F z¯ , z¯ + F zz z, z + F z¯ z¯ z¯ , z¯ + F z¯z z, z¯ ,

(4.3)

1644

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

and JF K = 0. Then we have H+ = H ◦ Φ = (N + R) ◦ XF1 + (P − R) ◦ XF1 1 = N + {N, F } + R +

  (1 − t){N, F } + R, F ◦ XFt dt + (P − R) ◦ XF1 .

(4.4)

0

Denote ∂ω =



∂ 1bn ωb ∂xb ,

Λ = diag(Ωj : j ∈ Z∗ ). Then by directly calculating, we get







{N, F } = −∂ω F x − ∂ω F y , y − ∂ω F z − iΛF z , z − ∂ω F z¯ + iΛF z¯ , z¯     − ∂ω F zz − iΛF zz − iF zz Λ z, z − ∂ω F z¯ z¯ + iΛF z¯ z¯ + iF z¯ z¯ Λ z¯ , z¯       ∂Ωj y z¯z z¯z z¯z − ∂ω F + iΛF − iF Λ z, z¯ + σj F zj z¯ j . ∂xb b j ∈Z∗

(4.5)

1bn

We wish to find the function F such that R − JRK is eliminated. To this end, F x , F y , F z , F z¯ , F zz , F z¯ z¯ and F z¯z should satisfy the homological equations: ∂ω F x = R x ,

(4.6)

 y

(4.7)

= Rjz ,

j ∈ Z∗ ,

(4.8)

∂ω Fjz¯ + iΩj Fjz¯ = Rjz¯ ,

j ∈ Z∗ ,

(4.9)



∂ω F y = R y − R , ∂ω Fjz

− iΩj Fjz

∂ω Fijzz − i(Ωi + Ωj )Fijzz = Rijzz ,

i, j ∈ Z∗ ,

(4.10)

∂ω Fijz¯ z¯ + i(Ωi + Ωj )Fijz¯ z¯ = Rijz¯ z¯ ,

i, j ∈ Z∗ ,

(4.11)

∂ω Fijz¯z + i(Ωi − Ωj )Fijz¯z = Rijz¯z ,

i, j ∈ Z∗ , i = j.

(4.12)

The homological equations (4.6)–(4.12) above are the in [23] after replacsame as (3.13)–(3.19)  ing “j  1” by “j ∈ Z∗ ” and adding a restriction − 1bn kb jb + j ∈Z∗ l˜j j = 0. Compared with the proof of Theorem 1.1 in [23], the proof of Theorem 2.1 follows a parallel course except for two essential differences: (1) the persistence of (1.7) at every KAM step; (2) the measure estimate for 

˜ k ∈ Zn \ {0}, l˜ ∈ NZ∗ , |l| ˜  2, − X := (k, l):

 1bn

k b jb +



 ˜lj j = 0 .

j ∈Z∗

Therefore, we discuss these two aspects in detail while omit the other parts of proof.

(4.13)

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1645

4.2. Checking (1.7) for the new Hamiltonian The initial Hamiltonian in Theorem 2.1 satisfies (1.7). By induction, we only need to deduce H+ in (4.4) satisfying (1.7) from H satisfying (1.7). The following result can be directly checked: Lemma 4.1. If both Hamiltonians H and F satisfy (1.7), then their Poisson bracket {H, F } also satisfies (1.7). Thus, it’s sufficient to prove F in (4.3) satisfying (1.7). Definition 4.2. We say an analytic function f (x), x ∈ Tn belongs to I (m), m an integer, if the indices of its monomials eik·x , k ∈ Zn satisfy 

kb jb = m.

(4.14)

1bn

Then, it’s sufficient to verify the following lemma: Lemma 4.3. The coefficients of F in (4.3), which are determined by the homological equations (4.6)–(4.12), satisfy: y

F x ∈ I (0),

Fb ∈ I (0),

Fjz ∈ I (j ),

Fjz¯ ∈ I (−j ),

Fijzz ∈ I (i + j ),

Fijz¯ z¯ ∈ I (−i − j ),

1  b  n,

(4.15)

j ∈ Z∗ ,

(4.16)

Fijz¯z ∈ I (i − j ),

i, j ∈ Z∗ .

(4.17)

Proof. By the assumption of induction, H satisfies (1.7), and hence N , R satisfy (1.7), that means Ωj ∈ I (0),

j ∈ Z∗ ,

y

R x ∈ I (0),

Rb ∈ I (0),

Rjz ∈ I (j ),

Rjz¯ ∈ I (−j ),

Rijzz ∈ I (i + j ),

Rijz¯ z¯ ∈ I (−i − j ),

(4.18)

1  b  n,

(4.19)

j ∈ Z∗ ,

(4.20)

Rijz¯z ∈ I (i − j ),

i, j ∈ Z∗ .

(4.21)

The homological equations (4.6), (4.7) can be directly solved by comparing Fourier coefficients, and hence (4.15) follows from (4.19); the homological equations (4.8)–(4.12) are solved by using Lemma 4.4 and Lemma 4.5 below, and thus (4.16), (4.17) follow from (4.18), (4.20), (4.21). 2 The following lemma is Theorem 1.4 in [22] with some modifications: add an assumption μ ∈ I (0), and thecorresponding small-divisor condition (4.23) below is only required for k ∈ Zn \ {0} with 1bn kb jb = 0; add an assumption p ∈ I (m), and the corresponding small-divisor condition (4.24) below is only required for k ∈ Zn with 1bn kb jb = m; add a conclusion u ∈ I (m).

1646

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

Lemma 4.4. Consider the first order partial differential equation −i∂ω u + λu + μ(x)u = p(x),

x ∈ Tn ,

(4.22)

for the unknown function u defined on the torus Tn , where ω = (ω1 , . . . , ωn ) ∈ Rn and λ ∈ C. Assume: (1) There are constants α, γ˜ > 0 and τ > n such that |k · ω| 

α , |k|τ

|k · ω + λ| 



k ∈ Zn \ {0} with

kb jb = 0,

(4.23)

1bn

α γ˜ , 1 + |k|τ



k ∈ Zn with

kb jb = m.

(4.24)

1bn

(2) μ : D(s) → C is real analytic (here ‘real’ means μ(Tn ) ⊂ R) with μ ∈ I (0) and [μ] = 0. Moreover, assume there is constant C > 0 such that |μ|s,τ +1 :=



|μˆ k ||k|τ +1 e|k|s  C γ˜ ,

(4.25)

k∈Zn

 where μˆ k := (2π)−n Tn μ(x)e−ik·x dx is the k-Fourier coefficient of μ. (3) p(x) is analytic in x ∈ D(s) with p ∈ I (m). Then (4.22) has a unique solution u(x) which is defined in a narrower domain D(s − σ ) with 0 < σ < s, and which satisfies     u(x)  c(n, τ ) e2C γ˜ s/α sup p(x) n+τ α γ ˜ σ x∈D(s−σ ) x∈D(s) sup

(4.26)

for 0 < σ < min{1, s}, where the constant c(n, τ ) = (6e + 6)n [1 + ( 3τe )τ ]. Moreover, u ∈ I (m). Proof. Following the proof of Theorem 1.4 in [22], let U (x) =

 k∈Zn \{0}

μˆ k ik·x e . ik · ω

(4.27)

 Observe that μ ∈ I (0) means μˆ k = 0 unless 1bn kb jb = 0, which implies the sum of  the right hand of (4.27) is actually for k ∈ Zn \ {0} with 1bn kb jb = 0. Thus, in view of (4.23), (4.25), U (x) is well defined in the domain D(s) with U ∈ I (0). Set u = e−iU v, p = e−iU g, and then (4.22) becomes −i∂ω v + λv = g.

(4.28)

iU From U ∈ I (0) we have  e ∈ I (0); together with p ∈ I (m), we conclude g ∈ I (m), which means gˆ k = 0 unless 1bn kb jb = m. Under the small-divisor condition (4.24), by comparing

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1647

Fourier coefficients (4.28) can be solved with v ∈ I (m); together with e−iU ∈ I (0), we get u ∈ I (m). The proof of (4.26) is the same as in [22]. 2 For any positive number K, we introduce a truncation operator ΓK as follows: (ΓK f )(x) :=



fˆk eik·x ,

∀f : Tn → C,

|k|K

where fˆk is the k-Fourier coefficient of f . The following lemma is Lemma 2.6 in [22] with an additional conclusion u ∈ I (m) under the additional assumptions μ ∈ I (0), p ∈ I (m). Lemma 4.5. Consider the first order partial differential equation with the truncation operator ΓK −i∂ω u + λu + ΓK (μu) = ΓK p,

x ∈ Tn ,

(4.29)

for the unknown function u defined on the torus Tn , where ω ∈ Rn , 0 = λ ∈ C, and 0 < 2K|ω|  |λ|. Assume that μ is real analytic in x ∈ D(s) with μ ∈ I (0), 

|μˆ k |e|k|s 

k∈Zn

|λ| 4ι

(4.30)

for some constant ι  1, and assume p(x) is analytic in x ∈ D(s) with p ∈ I (m). Then (4.29) has a unique solution u(x) with u = ΓK u and     u(x)  c(n) sup p(x), n |λ|σ x∈D(s) x∈D(s−σ )

(4.31)

    (1 − ΓK )(μu)(x)  c(n) e−9Kσ/10 sup p(x) n ισ x∈D(s−σ ) x∈D(s)

(4.32)

sup

sup

for 0 < σ < s, where the constant c(n) = 4(20e + 20)n . Moreover, u ∈ I (m). Proof. The existence, uniqueness and related estimates (4.31), (4.32) for the solution of Eq. (4.29) have been proved in [22]; thus, we only need to verify u ∈ I (m) here. Split u into two parts, u = v + w, such that v ∈ I (m), w ∈ I (m), i.e. wˆ k = 0 for 1bn kb jb = m. From μ ∈ I (0), we have −i∂ω v + λv + ΓK (μv) ∈ I (m),

−i∂ω w + λw + ΓK (μw) ∈ I (m).

Since ΓK p ∈ I (m), we obtain −i∂ω w + λw + ΓK (μw) = 0, which is Eq. (4.29) with the right hand vanishing. Thus, by the uniqueness of the solution, we conclude w = 0. This completes the proof of this lemma. 2

1648

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

4.3. Measure estimate We only need to give the measure estimate corresponding to the homological equations (4.12), the most difficult case that l˜ has two non-zero components of opposite sign. Following [23], at the ν-th KAM step (ν  0), on the set Πν of parameters ξ , the tangential frequencies ων are Lipschitz continuous in both directions with |ων |Πν  Eν ,

lip

|ων |Πν  M1,ν ,

 −1 lip ω  ν ω (Π )  Lν , ν

ν

(4.33)

and the normal frequencies Ω¯ ν are Lipschitz continuous with     Ω¯ ν,i (ξ ) − Ω¯ ν,j (ξ )  mν |i|d − |j |d ,

|Ω¯ ν |−δ,Πν  M2,ν , lip

i, j ∈ Z∗ ,

(4.34)

where the iteration parameters Eν =

 E 10 − 2−ν , 9 M2,ν

  M1  L Lν = 10 − 2−ν , 10 − 2−ν , 9 9   M2  m −ν −ν mν = = 10 − 2 , 9+2 . 9 10 M1,ν =

The excluding set of parameters corresponding to the homological equations (4.12) is

Ξαν :=

Rνkij ,

(4.35)

 |k|>Jν , i=j, − 1bn kb jb +i−j =0

where    

||i|d − |j |d | Rνkij = ξ ∈ Πν :  k, ων (ξ ) + Ω¯ ν,i (ξ ) − Ω¯ ν,j (ξ ) < αν , |k|τ

i, j ∈ Z∗ , i = ±j, (4.36)



Rνk(−j )j

  

|j |δ = ξ ∈ Πν :  k, ων (ξ ) + Ω¯ ν,−j (ξ ) − Ω¯ ν,j (ξ ) < αν τ , |k|

−j, j ∈ Z∗ ,

(4.37)

with the iteration parameters  α αν = 9 + 2−ν , 10

J0 = 0,

−κ ν−1 /(τ +1) Jν = γ0 ,

  4 1 β 1 ν  1, κ = − min , . 3 3 1+β 4

It’s sufficient to prove the following lemma: Lemma 4.6. If γ0 is sufficiently small and τ  n + 1 + 2/(d − 1), then    ν  Ξα   cρ n−1 α,  ν0

where c > 0 is a constant depending on n, d, E, L, M2 , m and CJ .

(4.38)

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1649

Proof. To estimate the measure of Ξαν , we introduce the perturbed frequencies ζ = ων (ξ ) as ˙ ν = ων (Rν ) parameters over the domain Z := ων (Πν ) and consider the resonance zones R kij kij in Z. Regarding Ω¯ ν as function of ζ , then from (4.33), (4.34), we have 10 10 lip E, |ξ |Z  Lν  L, 9 9  d    9 |Ω¯ ν,i − Ω¯ ν,j |Z  mν |i| − |j |d   m|i|d − |j |d , 10  2 10 lip |Ω¯ ν,j |Z  Lν M2,ν |j |δ  L0 M2 |j |δ . 9 |ζ |  Eν 

˙ ν with k ∈ Zn \ {0}, i, j ∈ Z∗ , i = j , − Now consider a fixed R kij

(4.39) (4.40) (4.41)



1bn kb jb + i − j = 0. 10Eν 9mν d d d Case 1: − |j | | > 9mν |k|. We have | k, ζ | < 10 ||i| − |j | |; together with (4.40) and ˙ ν is empty. αν  m10ν , we conclude R kij 1 9 3m δ δ n ν Case 2: 0 < ||i|d − |j |d |  10E 9mν |k|. We have |k|  2 ( 10 ) E (|i| + |j | ). Fix w1 ∈ {−1, 1} such that |k| = k · w1 , and write ζ = aw1 + w2 with w1 ⊥w2 . As a function of a, for t > s,

||i|d

k, ζ |ts = |k|(t − s),  2     Ω¯ ν,i − Ω¯ ν,j  t  10 LM2 |i|δ + |j |δ (t − s). s 9

(4.42) (4.43)

Then,    2 t  δ  10 δ  ¯ ¯ k, ζ + Ων,i − Ων,j s  |k| 1 − LM2 |i| + |j | /|k| (t − s) 9   5  10 ELM2  |k| 1 − 2 (t − s) 9 m



 by using the assumption 4LM2 

m E

1 |k|(t − s), 10

(4.44)

in Theorem 2.1 and the fact (9/10)6 > 1/2. Hence we get

 n+2  ν  10 ||i|d − |j |d | E √ α 10 n−1 R ˙   10 (diam Z) αν ( nM2 ρ)n−1 τ , kij τ |k| |k| 9 m |k| by using ||i|d − |j |d | 10Eν   |k| 9mν diam Z 

√



10 9

3

E , m

nM2,ν diam Πν 

αν  α,

10 √ nM2 ρ. 9

(4.45)

1650

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

Going back to the original parameter domain Πν by the inverse frequency map ω−1 , we conclude  ν  R   c1 ρ n−1 α , kij |k|τ

(4.46)

√ n−1 n where c1 = 10(10/9)2n+2 (E/m)(  nM2 ) L . Case 3: i = −j . We have − 1bn kb jb − 2j = 0, and thus 1 1 |j |  |k| max |jb | = |k|CJ . 2 1bn 2

(4.47)

Fix w1 ∈ {−1, 1}n such that |k| = k · w1 , and write ζ = aw1 + w2 with w1 ⊥w2 . As a function of a, for t > s, 2 2 |j |(t − s)  |j |δ (t − s), CJ CJ  2 t 10  ¯ − Ων,j | s  2 LM2 |j |δ (t − s). 9

k, ζ |ts = |k|(t − s)  

|Ω¯ ν,−j

(4.48) (4.49)

Then, 

t k, ζ + Ω¯ ν,−j − Ω¯ ν,j s 



by using the assumption 4LM2 

 2  2 10 1 −2 LM2 |j |δ (t − s)  |j |δ (t − s), CJ 9 CJ

1 CJ

(4.50)

in Theorem 2.1. Hence we get

n−1   CJ |j |δ 1 10 √ n−1  (diam Z) α  C nM ρ α τ. ν J 2 k(−j )j δ τ |j | |k| 9 |k|

 ν R ˙

(4.51)

Going back to the original parameter domain Πν by the inverse frequency map ω−1 , we conclude  ν R

k(−j )j

   c ρ n−1 α , 1 |k|τ

(4.52)

√ where c1 = CJ (10/9)2n−1 ( nM2 )n−1 Ln . For any fixed k ∈ Zn \ {0} with |k| > Jν , we discuss the number of nonempty Rνkij , i, j ∈ Z∗ ,  i = j , − 1bn kb jb + i − j = 0. In view of Case 1 and Case 2, such that Rνkij is nonempty, the i, j with i = ±j must satisfy   |i|d−1 + |j |d−1  2|i|d − |j |d   2(10/9)3 (E/m)|k|.

(4.53)

Hence the number is no more than 4(2(10/9)3 (E/m)|k|)2/(d−1) .  1 In view of Case 3, if ± 2 1bn kb jb ∈ Z∗ , then j is uniquely determined by − 1bn kb × jb − 2j = 0; otherwise, there is no such Rνk(−j )j at all. Hence the number is at most 1.

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

1651

Now from (4.46), (4.52) and the corresponding number discussed above, we have    

 

 i,j ∈Z∗ , i=j, − 1bn kb jb +i−j =0

Rνkij   c2 ρ n−1

α |k|

2 τ − d−1

,

(4.54)

where c2 = 4(2(10/9)3 (E/m))2/(d−1) c1 + c1 . Further, since τ  n + 1 + 2/(d − 1), we have  ν Ξ   c2 c3 ρ n−1 α

α , 1 + Jν

(4.55)

where c3 > 0 depends only on n. The sum of the latter inequality over all ν converges, and we finally obtain the estimate of Lemma 4.6. 2 Acknowledgments The authors are very grateful to Alexander Wires for his invaluable help and to the referee and the reviewers for their invaluable suggestions. References [1] P. Baldi, M. Berti, R. Montalto, KAM theory for quasi-linear and fully nonlinear perturbations of KdV, preprint, 2012. [2] D. Bambusi, S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys. 219 (2001) 465–480. [3] M. Berti, L. Biasco, Branching of Cantor manifolds of elliptic tori and applications to PDEs, Comm. Math. Phys. 305 (2011) 741–796. [4] M. Berti, L. Biasco, M. Procesi, KAM theory for the Hamiltonian DNLW, Ann. Sci. Ec. Norm. Super. 46 (2013) 301–373. [5] M. Berti, L. Biasco, M. Procesi, Existence and stability of quasi-periodic solutions of reversible derivative wave equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 24 (2013) 1–16. [6] M. Berti, L. Biasco, M. Procesi, KAM theory for the reversible derivative wave equation, preprint, 2012. [7] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations for 2D linear Schrödinger equation, Ann. of Math. 148 (1998) 363–439. [8] J. Bourgain, Green’s Function Estimates for Lattice Schrödinger Operators and Applications, Ann. of Math. Stud., vol. 158, Princeton University Press, Princeton, NJ, 2005. [9] J. Bourgain, On invariant tori of full dimension for 1D periodic NLS, J. Funct. Anal. 229 (2005) 62–94. [10] L. Chierchia, J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211 (2000) 497–525. [11] W. Craig, C. Wayne, Newton’s method and periodic solutions of nonlinear wave equation, Comm. Pure Appl. Math. 46 (1993) 1409–1501. [12] H.L. Eliasson, S.B. Kuksin, KAM for the non-linear Schrödinger equation, Ann. of Math. 172 (2010) 371–435. [13] J. Geng, J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys. 262 (2006) 343–372. [14] B. Grebert, L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys. 307 (2011) 383–427. [15] S.B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen. 21 (3) (1987) 22–37; English translation in: Funct. Anal. Appl. 21 (1987) 192–205. [16] S.B. Kuksin, Perturbations of quasiperiodic solutions of infinite-dimensional Hamiltonian systems, Izv. Akad. Nauk SSSR, Ser. Mat. 52 (1988) 41–63; English translation in: Math. USSR Izv. 32 (1) (1989) 39–62. [17] S.B. Kuksin, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math., vol. 1556, Springer-Verlag, Berlin, 1993.

1652

J. Liu, X. Yuan / J. Differential Equations 256 (2014) 1627–1652

[18] S.B. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys. (J. Appl. Math. Phys.) 48 (1997) 262–271. [19] S.B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Univ. Press, Oxford, 2000. [20] S.B. Kuksin, J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. 143 (1996) 147–179. [21] T. Kappeler, J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, Heidelberg, 2003. [22] J. Liu, X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large variable coefficient, Comm. Pure Appl. Math. 63 (2010) 1145–1172. [23] J. Liu, X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys. 307 (2011) 629–673. [24] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Sc. Norm. Super. Pisa 23 (1996) 119–148. [25] J. Pöschel, Quasi-periodic solutions for nonlinear wave equations, Comment. Math. Helv. 71 (1996) 269–296. [26] C.E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990) 479–528. [27] X. Yuan, Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension, J. Differential Equations 195 (2003) 230–242. [28] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations 230 (2006) 213–274. [29] J. Zhang, M. Gao, X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity 24 (2011) 1198–2118.