An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrodinger equation systems

0.23 MB·

by  Shidi Zhou

An abstract infinite dimensional KAM theorem with
application to nonlinear higher dimensional Schr¨odinger
7
1 equation systems
∗
0
2

n
a
Shidi Zhou
J
0 Departmentof Mathematics,Nanjing University,Nanjing 210093, P.R.China
2 Email: [email protected]

]
S
D
. Abstract
h
t
a In this paper we consider nonlinear Schr¨odinger systems with periodic boundary
m condition in high dimension. We establish an abstract infinite dimensional KAM
[ theoremandapply itto the nonlinearSchr¨odingerequationsystems with realFourier

Multiplier. By establishing a block-diagonal normal form, We prove the existence
1
of a class of Whitney smooth small amplitude quasi-periodic solutions corresponding
v
7 to finite dimensional invariant tori of an associated infinite dimensional dynamical
2 system.
7
5 Mathematics Subject Classification: Primary 37K55; 35B10
0
Keywords: KAM theory; Hamiltonian systems; Schro¨dinger equation system; Birkhoff
.
1
normal form; To¨plitz-Lipschitz property
0
7
1
1 Introduction
:
v
i
X Inthis paper,weconsider aclass of higherdimensionalnonlinear Schro¨dingerequation
r systems with real Fourier Multiplier under periodic boundary condition:
a
i∂ u = ∆ u+M u+∂ G(u2, v 2)
− t − x ξ u¯ | | | | (1.1)
i∂ v = ∆ v+M v+∂ G(u2, v 2)
( t x σ v¯
− − | | | |
where u = u(t,x),v = v(t,x),t R,x Td,d 2. G = G(a,b) is a real analytic function
∈ ∈ ≥
defined in a neighbourhood of the origin in R2 with G(0,0) = ∂ G(0,0) = ∂ G(0,0) = 0,
a b
which means that the Taylor series of G with respect to (a,b) should start from the
second order term (a classical example of the nonlinearity is u2 v 2 ). M ,M are two
ξ σ
| | | |
real Fourier Multipliers, which supply artificial parameters (defined in section 2). By a
KAMalgorithm,weprovethatafterremovingasmallmeasureofparameters,theequation
system (1.1) admits a class of small-amplitude quasi-periodic solutions under sufficiently
small nonlinear perturbations.
∗This work is partially supported by NSFCgrant 11271180.
1
KAM theory has been a fundamental tool for years in studying Hamiltonian PDEs by
constructingaclassofinvarianttorus. Startingfromthepioneeringwork[22]byKuksinin
1987, Newtonian scheme was developed by [7, 22, 34] and has showed its power in doing
Hamiltonian PDEs in one-dimensional. The main idea is to construct a local normal
form of the solution of the equation, which is fundmental in the study of the dynamical
properties of the initial equation, and then carry out an infinite iteration process. Thus
a class of invariant torus and corresponding quasi-periodic solutions are got. For related
works, see [6, 13, 14, 20, 21, 23, 24, 25, 26, 27, 28, 35].
Hamiltonian PDEs in high dimension have also attracted great interests. The KAM
method working on one-dimensional PDEs are not effective enough here, due to the mul-
tiplicity of eigenvalues of the linear operator, which causes terrible resonance between
two eigenvalues λ = n 2 +o(1) and λ = m 2 +o(1) if n = m ( means l2 norm
n m
| | | | | | | | |·|
here). This is a big obstacle in realizing the second Melnikov condition at each KAM
iterative step. The first breakthrough comes from Bourgain’s work [4] in 1998, in which
a class of two-frequency quasi-periodic solution of two-dimensional nonlinear Schro¨dinger
equations was got. In this paper, Bourgain introduced the famous multi-scale analysis
method, which avoided the cumbersome second Melnikov condition. Later in [5], Bour-
gain improved his method and got the small-amplitude quasi-periodic solutions of high
dimensional Schro¨dinger equations and wave equations. Following his idea and method,
abundant works have been done, see [1, 2, 3, 33].
Although multi-scale analysis has great advantages, its drawbacks couldn’t beignored.
For example, we can’t see the local Birkhoff normal form of the equation, and the linear
stability of the solution is also unavailable. Thus a KAM approach is also expected in
dealing with high-dimensional Hamiltonian PDEs. The first work comes from Geng and
You [15] in 2006, in which the quasi-periodic solutions of beam equation and nonlocal
smooth Schro¨dinger equation were got. In [15] the nonlinearity of the equation should
be independent of the spatial variable x, which implies that the Hamiltonian satisfies the
important property “zero-momentum” (condition (A4) in [15]). The multiple eigenvalues
of linear operator are avoided by making use of ”zero-momentum” condition and measure
estimateisconductedbythehelpofregularityofequation. Amuchmoredifficultquestion:
nonlinear Schro¨dinger equation defined on Td,d 2, with convolutional type potential
≥
and nonlinearity dependent on spatial variable x, was solve by Eliasson and Kuksin [9] in
2010. In this milestone-style paper, to work with the multiple eigenvalues, they studied
the elaborate distribution of integers points on a sphereand according to this they got the
normalformofblock-diagonal, witheach blockbecominglargerandlargeralongtheKAM
iteration. Besides,theydevelopedaveryimportantproperty“Lipschitz-Domain”todothe
measureestimate,duetotheabsenceofregularityoftheequation. Followingtheirideaand
method, Geng , Xu and You [12] got the quasi-periodic solutions of completely resonant
Schro¨dingerequationonT2,byconstructingsomeappropriatetangentialsitesonZ2. Later
in [29, 30], C.Procesi and M.Procesi got the same result of [12] in arbitrary dimension. In
[29, 30] their had a very ingenious choice of tangential sites through the method of graph
theory. For other works on PDEs in high dimension, see [8, 10, 11, 16, 17, 18, 31, 32].
Although there has been rich work about Hamiltonian equations, until now little is
known about Hamiltonian equation sets, i.e. two coupled equations. In [19] Grebert,
Paturel and Thomann contructed the beating solutions of Schro¨dinger equation system
in one-dimension and got the growth of Sobolev norms of the solutions. However, quasi-
2
periodicsolutionscorrespondingtofinitedimensionalinvarianttorusisstillunknown. Our
presentpaperisworking onthenonlinear Schro¨dinger system withreal Fourier Multiplier.
A more interesting question is about the completely resonant equation system, i.e. no
artificial parameters are imposed :
i∂ u= ∆ u+ v 2u
t x
− − | | (1.2)
i∂ v = ∆ v+ u2v
( t x
− − | |
This equation system will be dealt with in our forthcoming paper.
Now let’s state our main theorem:
Theorem 1 Let S = i , ,i Zd,S˜ = t , ,t Zd,0 S S˜;d 2;b,˜b 2;.
{ 1 ··· b} ⊆ { 1 ··· ˜b} ⊆ ∈ ≥ ≥
There exists a Cantor set Rb+˜b of positive measure, s.t. (ξ,σ) , the nonlin-
T
C ⊆ ∀ ∈ C
ear Schro¨dinger equation system (1.1) admits a class of small amplitude quasi-periodic
solutions of the form:
b
u(t,x) = cjeiωjtφij +O(|ξ|32 +|σ|32) ωj = |ij|2+O(|ξ|+|σ|)
 j=1 (1.3)
 v(t,x) = P˜b c˜jeiω˜jtφtj +O(|ξ|32 +|σ|32) ω˜j = |tj|2+O(|ξ|+|σ|)
j=1
 P
The rest of the paper is organized as follows: In section 2 we introduce some notations
andstatetheabstractKAMtheorem; Insection3wedealwiththenormalform;Insection
4, we conduct one step of KAM iteration; In section 5 we state the iterative lemma; In
section 6 we do the measure estimate.
2 Preliminaries and statement of the abstract KAM theo-
rem
Inthis section we introducesomenotations and state theabstract KAMtheorem. The
KAM theorem can be applied to (1.1) to prove Theorem 1.
Given two set S,S˜ Zd,d 2, S = i ,i , ,i ,S˜ = t ,t , ,t ,b,˜b 2 (for
⊆ ≥ { 1 2 ··· b} { 1 2 ··· ˜b} ≥
convenience we assume0 S S˜). Let Zd bethe complementary set of S in Z2, and Zd be
∈ 1 2
the complementary set of S˜ in Z2. Denote u = (u ) with its conjugate u¯ = (u¯ ) ,
T n n∈Zd n n∈Zd
1 1
and similarly v = (v ) with its conjugate v¯ = (v¯ ) . We introduce the weighted
n n∈Zd n n∈Zd
2 2
norm as follows:
u = u eρ|n|, v = v eρ|n| ρ > 0 (2.1)
ρ n ρ n
k k | | k k | |
nX∈Zd1 nX∈Zd2
here n = n 2+ + n 2, n = (n , ,n ) Zd for u and in Zd for v . Denote a
| | | 1| ··· | d| 1 ··· d ∈ 1 2
neighborhood of Tb+˜b ( I = 0 J = 0 ) ( u = 0 u¯ = 0 ) ( v = 0 v¯= 0 )
p
× { }×{ } × { }×{ } × { }×{ }
by
D(r,s) = (θ,ϕ,I,J,u,u¯,v,v¯) : Imθ , Imϕ < r;I,J < s2; u , u¯ , v , v¯ < s(2.2)
ρ ρ ρ ρ
{ | | | | k k k k k k k k }
3
where means the sup-norm of complex vectors.
Let|·α| = α ,β = β ,α˜ = α˜ ,β˜ = β˜ , α ,β ,α˜ ,β˜ N
{ n}n∈Zd1 { n}n∈Zd1 { n}n∈Zd2 { n}n∈Zd2 n n n n ∈
with only finitely many non-vanishing components. Denote uαu¯β = uαnu¯βn,vα˜v¯β˜ =
n n
n∈Zd
Q1
vα˜nv¯β˜n and let
n n
n∈Zd
Q2
F(θ,ϕ,I,J,u,u¯,v,v¯)= F (ξ,σ)ei(hk,θi+hk˜,ϕi)IlJ˜luαu¯βvα˜v¯β˜ (2.3)
klαβ,k˜˜lα˜β˜
klαXβ,k˜˜lα˜β˜
where (ξ,σ) Rb+˜b is the parameter set. k = (k , ,k ) Zb,k˜ = (k˜ , ,k˜ ) Z˜b
∈ O ⊆ 1 ··· b ∈ 1 ··· ˜b ∈
and l = (l , ,l ) Nb,˜l = (˜l , ,˜l ) N˜b, Il = Il1 Ilb, J˜l = J˜l1 J˜l˜b. Denote the
1 ··· b ∈ 1 ··· ˜b ∈ 1 ··· b 1 ··· ˜b
weighted norm of F by
F = sup F e(|k|+|k˜|)r
k kD(r,s),O klαβ,k˜˜lα˜β˜| klαβ,k˜˜lα˜β˜|O
(ξ,σ)∈O,kukρ,ku¯kρ,kvkρ,kv¯kρ<s
P
s2(|l|+|˜l|) uα u¯β vα˜ v¯β˜ (2.4)
× | || || || |
F = sup ∂4 F (2.5)
| klαβ,k˜˜lα˜β˜|O 0≤d≤4| (ξ,σ) klαβ,k˜˜lα˜β˜|
(ξ,σ)∈O
P
where the derivatives with respect to (ξ,σ) are in the sense of Whitney.
To a function F we define its Hamiltonian vector field by
X = (F ,F , F , F ,i F , i F ,i F , i F ) (2.6)
F I J − θ − ϕ { un}n∈Zd1 − { u¯n}n∈Zd1 { vn}n∈Zd2 − { v¯n}n∈Zd2
and the associated weighted norm is
1
X = F + F + F + F
k FkD(r,s),O k IkD(r,s),O k JkD(r,s),O s2 k θkD(r,s),O k ϕkD(r,s),O
(cid:18) (cid:19)
1
+ F + F e|n|ρ
s  k unkD(r,s),O k u¯nkD(r,s),O 
nX∈Zd1(cid:18) (cid:19) 
 
1
+ F + F e|n|ρ (2.7)
s  k vnkD(r,s),O k v¯nkD(r,s),O 
nX∈Zd2(cid:18) (cid:19) 
 
whereρ > 0is a constant andit willshrinkat each iterative step to make the smalldivisor
condition hold due to the lack of regularity of the equation.
The normal form H = N + with
0
B
N = ω(ξ,σ),I + ω˜(ξ,σ),J + Ω (ξ,σ)u u¯ + Ω˜(ξ,σ) v v¯ (2.8)
n n n n n n
h i h i
nX∈Zd1 nX∈Zd2
= (a (ξ,σ)u v¯ +b (ξ,σ)u¯ v ) (2.9)
n n n n n n
B
n∈XZd1∩Zd2
where (ξ,σ) is the parameter. Notice that apart from integrable terms, u and v¯ , v
n n n
∈O
andu¯ mayalsobecoupledandasaresultournormalformisintheformofblock-diagonal
n
with each block of degree 2.
4
For this unperturbed system, it’s easy to see that it admits a special solution
(θ,ϕ,0,0,0,0,0,0) (θ+ωt,ϕ+ω˜t,0,0,0,0,0,0)
→
corresponding to an invariant torus in the phase space. Our goal is to prove that, after
removing some parameters, the perturbedsystem H = H +P still admits invariant torus
0
provided that X is sufficiently small. To achieve this goal, we require that the
k PkDρ(r,s),O
Hamiltonian H satisfies some conditions:
(A1)Nondegeneracy: Themap(ξ,σ) (ω(ξ,σ),ω˜(ξ,σ))isaC4 diffeomorphismbetween
→ W
and its image (C4 means C4 in the sense of Whitney).
O W
(A2) Asymptotics of normal frequencies:
Ω = n 2+Ω´ , n Zd (2.10)
n | | n ∈ 1
Ω˜ = n 2+Ω´˜ , n Zd (2.11)
n | | n ∈ 2
here Ω´ ,Ω´˜ are C4 functions of (ξ,σ).
n n W
(A3) Melnikov conditions: Let
Ω a
A = n n n Zd Zd (2.12)
n bn Ω˜n ! ∈ 1 ∩ 2
and
A = Ω n Zd Zd (2.13)
n n ∈ 1 \ 2
A = Ω˜ n Zd Zd (2.14)
n n ∈ 2 \ 1
There exists γ,τ > 0, s.t. for any k + k˜ K,n Zd,m Zd, one has
| | | |≤ ∈ 1 ∈ 2
γ
k,ω + k˜,ω˜ , k + k˜ = 0 (2.15)
|h i h i| ≥ Kτ | | | | 6
and
γ
det ( k,ω + k˜,ω˜ )I +A (2.16)
| h i h i n |≥ Kτ
(cid:18) (cid:19)
and
γ
det ( k,ω + k˜,ω˜ )I A I I A T (2.17)
| h i h i ± n ⊗ 2± 2⊗ m |≥ Kτ
(cid:18) (cid:19)
Here AT denotes the transpose of matrix A and I denotes the identity matrix.
(A4) Regularity: +P is real analytic with respect to θ,ϕ,I,J,u,u¯,v,v¯ and Whitney
B
smooth with respect to (ξ,σ). And we have
X < 1, X < ε (2.18)
k BkDρ(r,s),O k PkDρ(r,s),O
(A5) Zero-momentum condition: The normal form part + P belongs to a class of
B
functions defined by:
A
f = f ei(hk,θi)+(hk˜,ϕi)IlJ˜luαu¯βvα˜v¯β˜, f
klαβ,k˜˜lα˜β˜ ∈ A
k∈Zb,k˜∈Z˜b,l∈XNb,˜l∈N˜b,α,β,α˜,β˜
5
implies
b ˜b
f = 0 = k i + k˜ t + (α β )n+ (α˜ β˜ )n =0
klαβ,k˜˜lα˜β˜ 6 ⇒ j j j j n− n n − n
jX=1 jX=1 nX∈Zd1 nX∈Zd2
(A6) To¨plitz-Lipschitz property: There exists a K > 0.
(1) For any fixed n,m Zd,c Zd 0 , the limits
∈ 1 ∈ \{ }
∂2P ∂2P ∂2P
lim , lim , lim (2.19)
t→∞ ∂un+tc∂um−tc t→∞ ∂un+tc∂u¯m+tc t→∞ ∂u¯n+tc∂u¯m−tc
exists, and moreover, when t > K, P satisfies:
| |
∂2P lim ∂2P ε e−|n+m|ρ (2.20)
k∂un+tc∂um−tc −t→∞∂un+tc∂um−tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n−m|ρ (2.21)
k∂un+tc∂u¯m+tc −t→∞∂un+tc∂u¯m+tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n+m|ρ (2.22)
k∂u¯n+tc∂u¯m−tc −t→∞∂u¯n+tc∂u¯m−tckDρ(r,s),O ≤ |t|
(2) For any fixed n Zd,m Zd,c Zd 0 , the limits
∈ 1 ∈ 2 ∈ \{ }
∂2P ∂2P ∂2P
lim , lim , lim (2.23)
t→∞ ∂un+tc∂vm−tc t→∞ ∂un+tc∂v¯m+tc t→∞ ∂u¯n+tc∂v¯m−tc
exists, and moreover, when t > K, P satisfies:
| |
∂2P lim ∂2P ε e−|n+m|ρ (2.24)
k∂un+tc∂vm−tc −t→∞∂un+tc∂vm−tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n−m|ρ (2.25)
k∂un+tc∂v¯m+tc −t→∞∂un+tc∂v¯m+tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n−m|ρ (2.26)
k∂u¯n+tc∂vm+tc −t→∞∂u¯n+tc∂vm+tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n+m|ρ (2.27)
k∂u¯n+tc∂v¯m−tc −t→∞∂u¯n+tc∂v¯m−tckDρ(r,s),O ≤ |t|
(3) For any fixed n,m Zd,c Zd 0 , the limits
∈ 2 ∈ \{ }
∂2P ∂2P ∂2P
lim , lim , lim (2.28)
t→∞ ∂vn+tc∂vm−tc t→∞ ∂vn+tc∂v¯m+tc t→∞ ∂v¯n+tc∂v¯m−tc
exists, and moreover, when t > K, P satisfies:
| |
∂2P lim ∂2P ε e−|n+m|ρ (2.29)
k∂vn+tc∂vm−tc −t→∞∂vn+tc∂vm−tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n−m|ρ (2.30)
k∂vn+tc∂v¯m+tc −t→∞∂vn+tc∂v¯m+tckDρ(r,s),O ≤ |t|
∂2P lim ∂2P ε e−|n+m|ρ (2.31)
k∂v¯n+tc∂v¯m−tc −t→∞∂v¯n+tc∂v¯m−tckDρ(r,s),O ≤ |t|
Now we state our abstract KAM theorem, and as a corollary, we get Theorem 1.
6
Theorem 2 Assume that the Hamiltonian H = N + + P satisfies condition (A1)
B −
(A6). Let γ > 0 be sufficiently small, then there exists ε > 0 and ρ > 0 such that
if X < ε, the following holds: There exists a Cantor subset with
k PkDρ(r,s),O Oγ ⊆ O
meas( ) = O(γς) (ς is a positive constant) and two maps which are analytic in θ,ϕ
γ
O\O
and C4 in (ξ,σ).
W
Φ : Tb+˜b D (r,s), ω˜ : Rb+˜b
γ ρ γ
×O → O →
where Φ is ε -close to the trivial embedding Φ : Tb+˜b Tb+˜b 0,0 0,0 0,0
γ16 0 ×O → ×{ }×{ }×{ }
and ω˜ is ε-close to the unperturbed frequency ω. Such that (ξ,σ) and (θ,ϕ) Tb+˜b,
γ
∀ ∈ O ∈
thecurvet Ψ (θ,ϕ)+ω˜t,(ξ,σ) isaquasi-periodic solutionoftheHamiltonianequation
→
(cid:18) (cid:19)
governed by H = N + +P.
B
3 Normal Form
Considertheequation set(1.1) inaview of Hamiltonian system anditcould berewrit-
ten as
i∂ u = ∂H
− t ∂u¯ (3.1)
i∂ v = ∂H
( − t ∂v¯
where the function H is a Hamiltonian:
1
H = ( ∆+M )u,u + ( ∆+M )v,v + G(u2, v 2)dx (3.2)
ξ σ
2 h − i h − i Td | | | |
(cid:18) (cid:19) Z
Expanding u,v into Fourier series
1
u= u φ , v = v φ , φ = eihn,xi (3.3)
n n n n n s(2π)d
nX∈Zd nX∈Zd
so the Hamiltonian becomes
H = λ u u¯ + λ˜ v v¯ +P(θ,ϕ,I,J,u,u¯,v,v¯;ξ,σ) (3.4)
n n n n n n
nX∈Zd nX∈Zd
whereλ is the eigenvalue of ∆+M and λ˜ is the eigenvalue of ∆+M , which means
n ξ n σ
λ = n 2+ξ if n S and λ− = n 2 if n Zd; λ˜ = n 2+σ if−n S˜ and λ˜ = n 2 if
n | | n ∈ n | | ∈ 1 n | | n ∈ n | |
n Zd.
∈ 2
Introducing action-angle variable:
u = √I eihk,θi,u¯ = √I e−ihk,θi, n S
n n n n
∈
v = √J eihk,ϕi,v¯ = √J e−ihk,ϕi, n S˜ (3.5)
n n n n
∈
The Hamiltonian (3.4) is now turned into
H = ω,I + ω˜,J + Ω u u¯ + Ω˜ v v¯ +P(θ,ϕ,I,J,u,u¯,v,v¯;ξ,σ) (3.6)
n n n n n n
h i h i
nX∈Zd1 nX∈Zd2
7
Now let’s verify condition (A1) (A6) for (3.6).
−
Verifying (A1): It’s easy to see that ∂(ω,ω˜) = I
∂(ξ,σ) b+˜b
Verifying (A2): Ω = n 2,n Zd and Ω˜ = n 2,n Zd so it’s obvious.
n | | ∈ 1 n | | ∈ 2
Verifying (A3): For convenience, we only verify the most complicated (2.18). Recall
the structure of the Hamiltonian (3.6), now a = 0,b = 0, so we only need to concentrate
n n
on each element of the diagonal of the matrix
( k,ω + k˜,ω˜ )I +A I I AT (3.7)
n 2 2 m
h i h i ⊗ − ⊗
which means that we only need to verify
γ
k,ω + k˜,ω˜ +l (3.8)
|h i h i | ≥ Kτ
After excluding a subset with measure O(γ) of the parameter set, (3.8) follows and thus
condition (A3) is verified.
Verifying (A4): It’s similar with the verification of condition (A4) in [17].
Verifying(A5): It’sverysimilartothatin[15]. Recallthenonlinearity G(u2, v 2)dx,
Td
| | | |
expand G into Taylor series in a neighbourhood of the origin and expand u,v into Fourier
R
series, it could be written as a sum of such terms:
u u u u¯ u¯ u¯ v v v v¯ v¯ v¯
a1 a2··· am b1 b2··· bm c1 c2··· cn d1 d2··· dn
with n,m 1 and
≥ m m n n
a b + c d = 0
j j j j
− −
j=1 j=1 j=1 j=1
X X X X
It belongs to a more general case :
uλu¯µvλ˜v¯µ˜ with (λ µ )n+ (λ˜ µ˜ )n = 0 (⋆)
n n n n
− −
nX∈Zd nX∈Zd
we prove that (⋆) also satisfies condition (A5). By the definition of S,S˜, we have
uλu¯µvλ˜v¯µ˜ = I λi1+µi1 I λib+µib J λ˜t1+µ˜t1 J λ˜t˜b+µ˜t˜b
i1 ··· ib t1 ··· t˜b
qei( bj=1(λij−µijq)θij+ ˜bj=1(λ˜qtj−µ˜tj)ϕtj) q
×
uλ−P bj=1λijeiju¯µ− Pbj=1µijeijvλ˜− ˜bj=1λ˜tjetjv¯µ˜− ˜bj=1µ˜tjetj (3.9)
×
P P P P
Justsetk = (λ µ , ,λ µ ),k˜ = (λ˜ µ˜ , ,λ˜ µ˜ ), α = λ b λ e ,β =
i1− i1 ··· ib− ib t1− t1 ··· t˜b− t˜b − j=1 ij ij
µ− bj=1µijeij,α˜ = λ˜ − ˜bj=1λ˜tjetj,β˜ = µ˜− ˜bj=1µ˜tjetj. Combining wiPth (⋆), it’s easy
to verify condition (A5).
P P P
Verifying (A6): Now B = 0. Because P , we have
∈ A
P = P (ξ,σ)ei(hk,ωi+hk˜,ω˜i)IlJ˜luαu¯βvα˜v¯β˜
klαβ,k˜˜lα˜β˜
klαXβ,k˜˜lα˜β˜
8
satisfying
b ˜b
k i + k˜ ˜i + (α β )n+ (α˜ β˜ )n = 0 P = 0
j j j j n − n n − n 6 ⇒ klαβ,k˜˜lα˜β˜
jX=1 jX=1 nX∈Zd1 nX∈Zd2
By the fact (n+tc) (m+tc) = n m,(n+tc)+(m tc) = n+m, we have
− − −
∂2P = ∂2P = lim ∂2P
∂un∂vm ∂un+tc∂vm−tc t→∞ ∂un+tc∂vm−tc
∂2P = ∂2P = lim ∂2P
∂un∂v¯m ∂un+tc∂v¯m+tc t→∞ ∂un+tc∂v¯m+tc
∂2P = ∂2P = lim ∂2P
∂u¯n∂vm ∂u¯n+tc∂vm+tc t→∞ ∂u¯n+tc∂vm+tc
∂2P = ∂2P = lim ∂2P
∂u¯n∂v¯m ∂u¯n+tc∂v¯m−tc t→∞ ∂u¯n+tc∂v¯m−tc
so (2.23)-(2.27) is verified, and following the same method we could verify (2.19)-(2.22)
and (2.28)-(2.31). Thus (A6) is verified.
4 KAM Iteration
We prove Theorem 2 by a KAM iteration which involves an infinite sequenceof change
ofvariables. EachstepofKAMiterationmakestheperturbationsmallerthantheprevious
step at the cost of excluding a small set of parameters. We have to prove the convergence
of the iteration and estimate the measure of the excluded set after infinite KAM steps.
At the ν-th step of the KAM iteration, we consider a Hamiltonian vector field with
H = N + +P (θ,ϕ,I,J,u,u¯,v,v¯;ξ,σ)
ν ν ν ν
B
where
N = ω(ξ,σ),I + ω˜(ξ,σ),J + Ω (ξ,σ)u u¯ + Ω˜ (ξ,σ)v v¯ (4.1)
ν n n n n n n
h i h i
nX∈Zd1 nX∈Zd2
= a (ξ,σ)u v¯ +b (ξ,σ)u¯ v (4.2)
n n n n n n
B
n∈XZd1∩Zd2(cid:18) (cid:19)
with B +P is defined in D(r ,s ) .
ν ν ν ν ν−1
×O
We construct a map
Φ :D(r ,s ) D(r ,s )
ν ν+1 ν+1 ν ν ν ν−1
×O → ×O
so that the vector field X defined on D(r ,s )satisfies
Hν◦Φν ν+1 ν+1
X = X X εκ, κ > 1
k Pν+1kD(rν+1,sν+1),Oν k Hν◦Φν − Nν+1+Bν+1kD(rν+1,sν+1),Oν ≤ ν
and the new Hamiltonian still satisfies condition (A1) (A6).
−
To simplify notations, in the following text, the quantities without subscripts refer to
quantitiesattheν-thstep,whilethequantitieswithsubscripts+denotethecorresponding
9
quantities at the (ν +1)-th step. Let’s consider the Hamiltonian defined in D(r,s) :
×O
H = N + +P
B
= ω(ξ,σ),I + ω˜(ξ,σ),J + Ω (ξ,σ)u u¯ + Ω˜ (ξ,σ)v v¯
n n n n n n
h i h i
nX∈Zd1 nX∈Zd2
+ a (ξ,σ)u v¯ +b (ξ,σ)v u¯ +P(θ,ϕ,I,J,u,u¯,v,v¯;ξ,σ,ε) (4.3)
n n n n n n
n∈XZd1∩Zd2(cid:18) (cid:19)
We assume that for (ξ,σ) , one has:
For any k + k˜ K,n∈ OZd,m Zd, the followings hold
| | | | ≤ ∈ 1 ∈ 2
γ
k,ω + k˜,ω˜ , k + k˜ = 0
|h i h i| ≥ Kτ | | | | 6
and
γ
det ( k,ω + k˜,ω˜ )I +A
| h i h i n |≥ Kτ
(cid:16) (cid:17)
and
γ
det ( k,ω + k˜,ω˜ )I +A I I AT , k + k˜ = 0
| h i h i n ⊗ 2− 2⊗ m | ≥ Kτ | | | | 6
(cid:18) (cid:19)
Expand P into Fourier-Taylor series P = P ei(hk,θi+hk˜,ϕi)IlJ˜luαu¯βvα˜v¯β˜
klαβ,k˜˜lα˜β˜
klαβ,k˜˜lα˜β˜
and by condition (A5) we get that P
P = 0 if k i + (α β )n+ k˜ t + (α˜ β˜ )n = 0
klαβ,k˜˜lα˜β˜ j j n − n j j n − n 6
1≤Xj≤b nX∈Zd1 1≤Xj≤˜b nX∈Zd2
which means that when k = 0,k˜ = 0, the terms u u¯ ,u v¯ ,v u¯ ,v v¯ are absent when
n m n m n m n m
n = m ,n = m.
| | | | 6
Now we describe how to construct a subset and a change of variables Φ :
+
O ⊆ O
D = D(r ,s ) D(r,s) such that the transformed Hamiltonian
+ + + + +
× O × O → × O
H = H Φ = N + + P satisfies conditions (A1) (A6) with new parameters
+ + + +
◦ B −
ε ,r ,s and with (ξ,σ) .
+ + + +
∈ O
4.1 Homological Equation
Expand P into Fourier-Taylor series
P = P ei(hk,θi+hk˜,ϕi)IlJ˜luαu¯βvα˜v¯β˜ (4.4)
klαβ,k˜˜lα˜β˜
klαXβ,k˜˜lα˜β˜
where k Zb,l Nb;k˜ Z˜b,˜l N˜b and the multi-indices α,β;α˜,β˜ run over the
∈ ∈ ∈ ∈
set of all infinite dimensional vectors α = ( ,α , ) ,β = ( ,β , ) ;α˜ =
··· n ··· n∈Zd1 ··· n ··· n∈Zd1
( ,α˜ , ) ,β˜= ( ,β˜ , ) with finitely many nonzero components of positive
··· n ··· n∈Zd2 ··· n ··· n∈Zd2
integers. And by (A5) we get that
P = 0 if k i + k˜ t + (α β )n+ (α˜ β˜ )n = 0(4.5)
klαβ,k˜˜lα˜β˜ j j j j n − n n − n 6
1≤Xj≤b 1≤Xj≤˜b nX∈Zd1 nX∈Zd2
10