A KAM Theorem for the Hamiltonian with Finite Zero Normal Frequencies and Its Applications (In Memory of Professor Walter Craig)

https://doi.org/10.1007/s10884-021-09972-6

A KAM Theorem for the Hamiltonian with Finite Zero Normal Frequencies and Its Applications (In Memory of Professor Walter Craig)

Yuan Wu¹·Xiaoping Yuan²

Received: 2 September 2020 / Revised: 4 February 2021 / Accepted: 13 February 2021 / Published online: 20 March 2021

© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021

Abstract

we investigate the existences of KAM tori for the infinite dimensional Hamiltonian system with finite number of zeros among normal frequencies. By constructing a constant vector we show that, for “most” tangent frequencies in the sense of Lebesgue measure, either if the vector is zero, there is a KAM torus or if the vector is not zero, there is no KAM torus in some domain. As an application, we show that the nonlinear Schrödinger equation with a zero among normal frequencies possesses many quasi-periodic solutions.

Keywords Zero normal frequency·KAM tori·Quasi-periodic solution·Schrödinger equation

1 Introduction and the Main Results Consider a Hamiltonian function

H = ω,y +

|j|≤ι

jz_j¯z_j+εR(x,y,z,z¯, ω), (1.1) where(x,y,z,z¯)∈Tⁿ×Cⁿ×H×HandHis a finite dimensional Euclidean space (ι <∞) or a Hilbert space (ι= ∞). EndowHwith symplectic structured y∧d x+id¯z∧d zwhere i²= −1.

Clearly, whenε=0,Tⁿ₀ =Tⁿ×{y=0}×{z=0}×{¯z=0}is an-dimensional invariant torus with rotational frequencyωfor the Hamiltonian system defined byH. Assume

k, ω =0, k∈Zⁿ\{0}, (1.2)

B

wuyuan@hust.edu.cn

1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, People’s Republic of China

2 School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China

k, ω ±j=0, k∈Zⁿ,|j| ≤ν, (1.3) k, ω ±i±j=0, k∈Zⁿ,|i| ≤ν,|j| ≤ν, (1.4) Melnikov [31,32] announced that the invariant torusTⁿ₀is preserved under a small analytic perturbationεRfor the finite dimensional HamiltonianH(i.e.ν <∞). Eliasson [16] gave out the detail proof. Also see Pöschel [33]. The above result is called Melnikov’s preserva- tion theorem of KAM tori. The conditions (1.3) and (1.4) are so-called the first Melnikov conditions and the second Melnikov conditions, respectively. Kuksin [23,24], Wayne [35]

initiated the study of the Melnikov’s preservation theorem for infinite dimensional Hamil- tonian systems (i.e.ν= ∞) and thus proved the existence of time-quasi-periodic solutions (KAM tori) for the nonlinear Schrödinger equations and wave equations of spatial dimension 1.

Clearly, the first and the second Melnikov conditions imply thatj=0 and multiplicity ^#_j=1, respectively. For 1-dimensional nonlinear Schrödinger equation of 0-mass

iu_t+u_{x x}+ |u|⁴u=0, x ∈T¹,

the normal frequency0 =0 is the first eigenvalue of the differential operator−∂x xwith periodic boundary conditionx ∈T¹. Considerd >1 dimensional nonlinear Schrödinger equation of massm>0

iu_t+u+m u+ |u|⁴u=0, x∈T^d,

for whichj = |j|²+mforj=(j₁, . . . ,j_d)∈Z^d. At this time, one has _j ≈ |j|^d−1→ ∞.

Naturally one proposes the following two problems:

Problem 1.Is there Melnikov preservation theorem of KAM tori for somej =0?

Problem 2.Is there Melnikov preservation theorem of KAM tori for some_j >1?

They were originally proposed by Kuksin [24]. At present time, Problem 2 has been deeply investigated when the perturbation is bounded. Bourgain [8–14] developed a new method initiated by Craig–Wayne [15] to deal with the KAM tori for the PDEs in high spatial dimension, based on the Newton iteration, Fröhlich–Spencer techniques, harmonic analysis, and semialgebraic geometry theory (see [14]). This is called the Craig–Wayne–Bourgain (C–W–B) method. We also mention the work by Eliasson–Kuksin [18] and Eliasson-Grébert–

Kuksin [17], where the classical KAM theorem is extended in the direction of [4,16,20,23, 26,34,35] to deal with high spatial dimensional nonlinear Schrödinger equations and beam equations by introducing an elegant analysis of the Töplitz–Lipschitz operator. The obtained KAM tori by [18] are linear stable. In previous KAM theorems, the normal frequencies j’s cluster at infinity for the infinite dimensional Hamiltonian system. Recently, in [38], a new KAM theorem was constructed where_j ≥1 andj’s are bounded with application to Benjianmin-Bona-Mahony equation and Pochhammer–Chree equation. Incidentally, the KAM theory is also developed to deal some 1-dimensional PDEs of unbounded perturbation.

See, for example, [1–3,5,7,19,22,25,28,29,39].

However, there are fewer results about Problem 1. In the present paper, we will investigate this problem.

In order to state our theorem, we need some notations. LetN+= {1,2, . . .}. Fix an integer b>0 and take a subsetJ ofN₊

J = {j₁< j₂<· · ·< j_b} ⊆N₊.

Denote

z^∗=(z0,z), z¯^∗=(z¯0,¯z), where

z0=(zjm ∈C,m=1, . . . ,b), z=(zj ∈C,j ∈N+\J), and

¯

z0 =(¯zjm ∈C,m=1, . . . ,b), z¯=(¯zj ∈C,j∈N+\J).

Usually the variablesz^∗andz¯^∗are regarded as being independent unless we point out that

¯

z^∗is the complex conjugate ofz^∗. Introduce the phase space (x,y,z^∗,z¯^∗)∈P^a,p=Tⁿ×Cⁿ×l^a,p×l^a,p,

wheren ∈ N+,a ≥ 0 and p ≥1 are given andl^a,p is the Hilbert space of all complex sequencesz^∗=(z₀,z)with

z^∗2a,p =

jm∈J

|z_jm|²j_m^2pe^{2a jm}+

j∈N+\J

|z_j|²j^2pe^{2a j}<∞.

EndowP^a,p with the symplectic structured y∧d x+idz¯^∗∧d z^∗. Let N(y,z^∗,z¯^∗, ξ)be an integrable Hamiltonian depending on parametersξ ∈,a parameter set of positive Lebesgue measure inRⁿ, precisely,

N(y,z^∗,z¯^∗, ξ)= ω(ξ),y + ^∗(ξ)z^∗,z¯^∗

= ω(ξ),y + 0(ξ)z0,¯z0 + (ξ)z,¯z, whereω=(ω¹, . . . , ωⁿ),^∗(ξ)=diag(0(ξ), (ξ))with

0(ξ)=diag(₀^j1(ξ), . . . , ₀^jb(ξ))∈R^b×b, (ξ)=diag(^j(ξ)∈R: j∈N+\J), and

0(ξ)z0,¯z0 =

1≤m≤b

₀^jm(ξ)zjm¯zjm,(ξ)z,z =¯

j∈N+\J

^j(ξ)zj¯zj. We always assume0(ξ)≡0, i.e.

₀^jm(ξ)≡0, m=1, . . . ,b.

Quite evidently,T₀ⁿ = Tⁿ× {y =0} × {z^∗=0} × {¯z^∗=0}is an-dimensional invariant rotational torus for the Hamiltonian system defined byN. Our aim is to give out a criterion for preservation or non-preservation of the torusT₀ⁿunder small perturbationεRfor “most”

parameters ξ ∈ in the sense of Lebesgue measure. To this end, introduce complex neighborhoods ofT₀ⁿ

D(s,r)= {(x,y,z^∗,z¯^∗)∈P^a,p : |x| ≤s,|y| ≤r²,z^∗a,p+ ¯z^∗a,p≤r}, D(s,r)= {(x,y,z^∗,z¯^∗)∈D(s,r):x,y∈Rⁿ,z¯^∗is the complex conjugate of z^∗}, wheres,r>0 are constants and| · |denotes the sup-norm for complex vectors.

Forr>0 and p¯≥ p, we define the weighted phase norm W = |X| + 1

r²|Y| +1

rUa,p¯+1

rVa,p¯,W=(X,Y,U,V)∈P^a,p¯.

Furthermore, for a mapW : D(s,r)× →P^a,p¯, ( for example,W is the Hamiltonian vector fieldX_R,) we define the norms

WD(s,r),:= sup

D(s,r)× W,

W^L_D(s,r),:= sup

D(s,r)× ∂ξW, where∂_ξis the derivative with respect toξin Whitney’s sense .

Denote byA(l^a,p,l^a,p¯)the set of all bounded linear operators froml^a,ptol^a,p¯and by|||·|||

the operator norm. For any subsetS⊂N+, let us consider a vectoru =(u_j ∈C: j ∈ S) and a matrixU =(U_{i j} ∈C:i,j ∈S). We expanduinto

˜

u=(u˜_j : j ∈N+), her eu˜=

uj, j∈S, 0, j∈N+\S and also expandU into

U˜ =(U˜i j :i,j∈N₊), her eU˜ =

U_{i j},i,j∈S, 0, i or j∈N₊\S Define||u||a,p= || ˜u||a,pand|U| = | ˜U|.

We also denote| · |2the Euclidean norm and|| · ||the operator norm induced by| · |2. Now, we state our main theorem.

Theorem 1.1 Consider a perturbation of the integrable Hamiltonian N

H(x,y,z^∗,¯z^∗, ξ)=N(y,z^∗,¯z^∗, ξ)+R(x,y,z^∗,¯z^∗, ξ) (1.5) defined on the domain D(s,r)×. Suppose the following assumptions hold.

Assumption A: (Nondegeneracy). There exist positive constants E₁ and L0 such that

|ω|,|ω|^L≤E₁and|det(∂_ξω(ξ)|≥L0.

Assumption B:(Spectral Asymptotics). There exists d≥1such that

^j(ξ)= j^d+ · · · +O(ξ^p), j∈N+\J, (1.6) where the dots stands for fixed lower order terms in j and p≥1. More precisely, there exists positive constant E₂such that||^L≤E₂.

Assumption C: (Regularity). The perturbation R is analytic in the space coordinates (x,y,z^∗,z¯^∗) ∈ D(s,r) and 1 order Whitney smooth in the parameterξ ∈ , and for eachξ ∈, its Hamiltonian vector field X_R=(R_y,−R_x,iR_z¯∗,−iR_z∗)^T (T =transpose) defines nearT₀ⁿ a real analytic map

X_R:P^a,p→P^a,p¯,

p¯≥p for d>1,

¯

p>p for d=1.

Assumption D:(Reality). For any(x,y,z^∗,z¯^∗, ξ)∈ D(s,r)×, the perturbation R is real, that is,

R(x,y,z^∗,z¯^∗, ξ)=R(x,y,z^∗,z¯^∗, ξ), (1.7) where the bar means complex conjugate.

Set E = E₁+E₂. There exists a positive constant C₀, depending on n,s,r,and E, such that, for every perturbation R described above with

ε:= X_RD(s,r),+ γ

EX_R^L_D(s,r),≤C₀γ²,

for given r>0,s>0and sufficiently small0< γ 1, there exist a sequence of constants sm ≈2^−(m+1)s, εm≈ε^(4/3)m,rm≈ε^1/3m r, γm= ^3γ₈ (1+2^−m+1),Em =E+2(ε0+ · · · + εm−1), a sequence of domains D(sm,rm)×mwith_∞

m=0m=_γ ⊂and

|\γ| =O(γ ), as well as symplectic transformations

^m−1=0◦ · · · ◦m−1: D(sm,rm)×m →D(s,r), such that the Hamiltonian H is changed by^m−1into

Hm=H◦^m−1=Nm+Rm, where the sequences of the new normal form N_m

Nm =J_m^x(ξ)+ ωm(ξ),y + m(ξ)z,z + J¯ _m^z0(ξ),z0

+J_m^¯z0(ξ),z¯₀ + J_m^z0z0(ξ)z₀,z₀ + J_m^z0z¯0(ξ)z₀,z¯₀ + J_m^¯z0z¯0(ξ)¯z₀,z¯₀, and for eachξ∈m, the perturbation Rm(x,y,z^∗,z¯^∗, ξ)is analytic on D(sm,rm). More- over, the following estimates hold:

(1)for any x∈Tⁿ, the symplectic map^m−1obeys

||^m−1−i d||D(s_m,r_m),m + γm

Em||^m−1−i d||^L_D(sm,rm),mε_m−1⁵⁶ ,

||D^m−1−I d||D(sm,rm),m+ γm

E_m||^m−1−i d||^L_D(sm,rm),mε_m⁵⁶₋₁, where D^m−1denotes the tangent map of^m−1;

(2)the frequenciesωm(ξ)andm(ξ)satisfy

|ωm(ξ)|^L_m+ |m(ξ)|^L_mE_m; (3)the perturbation Rmsatisfies

XRmD(sm,rm),m+ γm

EmXRm^L_D(sm,rm),mεm,

where uvmeans there exists a constant c>0depending on n,b such that u≤cv. Furthermore, denote

N˘^z0(ξ)=(N˘^zj1(ξ), . . . ,N˘^zjb(ξ))=

m→∞lim

Jm^zj1(ξ), . . . , lim

m→∞

Jm^zjb(ξ)

, N˘^¯z0(ξ)=(N˘^¯zj1(ξ), . . . ,N˘^z¯jb(ξ))=

m→∞lim

J_m^¯zj1(ξ), . . . , lim

m→∞

J_m^¯zjb(ξ)

, ω_∗= lim

m→∞ωm(ξ), = lim

m→∞^m−1. We have

Corollary 1.2 If N˘ := (N˘^z0(ξ),N˘^z¯0(ξ)) = 0, then there exist a Cantor setγ ⊂ , a Whitney smooth family of torus embedding

:Tⁿ× {y=0} × {z^∗=0} × {¯z^∗=0} ×γ →P^a,p,

and a Whitney smooth mapω∗:γ →Rⁿ, such that for eachξinγ the maprestricted toTⁿ× {y=0} × {z^∗=0} × {¯z^∗=0} × {ξ}is a real analytic embedding of a rotational torus with the frequenciesω∗for the Hamilton system defined by(1.5)at{ξ}.

Corollary 1.3 IfN˘ :=(N˘^z0(ξ),N˘^z¯0(ξ))=0, that is,

| ˘N^z0(ξ)|²₂+ +| ˘N^z¯0(ξ)|²₂:=δ0 >0, then there exist an integer m₀>100 log(^log_log^δ_ε⁰₀), a domain

m0 = {(x,y,z^∗,z¯^∗): |x| ≤s_m0,|y| ≤r_m²

0,z^∗a,p+ ¯z^∗a,p≤ε_m⁷⁶₀₋₁}, a Cantor setm0−1⊂, a Whitney smooth family of embedding^m0−1:m0×m0−1→ D(s,r), and a Whitney smooth mapωm₀ :m₀−1 →Rⁿ, such that for eachξinm₀, there is no torus in the domain^m0−1(m0× {ξ})⊂D(s,r)for the Hamilton system defined by (1.5).

Some remarks and a “ guide to the proof ” of Theorem1.1.

1.1 .

The basic tool for the proof of Theorem1.1is the usual Newton type iteration, as often happens in KAM theory. However, the arguments used in this paper are quite complicated since zero normal frequencies come out. Therefore, we give a “ guide to the proof ” of Theorem1.1for the readers’ convenience:

1.1.1 The Linearized Equation (Sect.2)

This is the heart of the proof. The idea consists in a quadratic-convergent iterative procedure apt to reduce at each step of the scheme, which is done in order to beat the divergence introduced by small divisors arising in the inversion of non-elliptic differential operators.

Since there are finite zero normal frequencies, the main difficulties we encounter are k, ω(ξ) ±^jm =0, 1≤m≤b, (1.8) and

k, ω(ξ) ±^jm±^jm =0, 1≤m,m≤b, (1.9) whenk=0.

To overcome these difficulties, a basic idea is that we preserve the terms related to (1.8) and (1.9).

Let

R= R^x(ξ)+ R^y(x, ξ),y + R^z0z(x, ξ)z0,z + R^z0¯z(x, ξ)z0,z¯ +R^¯z0z(x, ξ)¯z0,z + R^¯z0¯z(x, ξ)¯z0,z + R¯ ^z(x, ξ),z + R^¯z(x, ξ),z¯ +R^zz(x, ξ)z,z + R^z¯z(x, ξ)z,z¯ + R^¯z¯z(x, ξ)¯z,z¯

+R^z0(x, ξ),z₀ + R^z¯0(x, ξ),z¯₀ + R^z0z0(x, ξ)z0,z₀

+R^z0¯z0(x, ξ)z₀,z¯₀ + R^z¯0¯z0(x, ξ)¯z₀,z¯₀.

Thus, the terms we will preserve are R^z0(0, ξ),R^z¯0(0, ξ),R^z0z0(0, ξ),R^z0¯z0(0, ξ),R^z¯0z¯0 (0, ξ). Setr₀ =r, s₀ =s, γ0 = γ, ε0 = ε,E₀ = E,ω0 =ω,^∗₀ =(00, 0) = ^∗, N0 =N,R0=RandH0=H. Suppose the Hamiltonian

H₀ =N₀+R₀ = ω0(ξ),y + +^∗₀(ξ)z^∗,¯z^∗ +R₀

= ω0(ξ),y + 00(ξ)z0,¯z0 + 0(ξ)z,z +¯ R0, 00=0,(1.10) descried above satisfies assumptionsA,B,C,Dand

XR₀D(s₀,r₀),0+ γ0

E0XR₀^L_D(s0,r0),0 =ε0, there exists a symplectic transformation0, such that

H₁=H₀◦0=(N₀+R₀)◦0=N₁+R₁, (1.11) where the new normal form

N₁ = N₀^x(ξ)+ ω1(ξ),y + 1(ξ)z,z¯ + J₁^z0(ξ),z₀ + J₁^z¯0(ξ),z¯₀

+J₁^z0z0(ξ)z0,z₀ + J₁^z0z¯0(ξ)z0,z¯₀ + J₁^z¯0z¯0(ξ)¯z₀,z¯₀, (1.12) while the new perturbation is of smaller size

XR_1D(s1,r1),1+ γ1

E₁XR₁^L_D(s1,r1),1ε⁴³ :=ε1.

The parameterξ appearing in (1.12) will vary in small compact set1(of relatively large Lebesgue measure).

Obviously, after 1-th iteration, we obtain a new normal formN1, which has more terms than the usual KAM normal form. Thus, our iterative scheme fromH₁is non-standard and, from a technical point of view, represents the most novel part of the proof.

Similarly, forH₁in (1.11), there exists a symplectic transformation1, such that H₂= H₁◦1=(N₁+R₁)◦1=N₂+R₂, (1.13) where the new normal form

N₂ = 1

j=0

N^x_j(ξ)+ ω2(ξ),y + 2(ξ)z,z¯ + J₂^z0(ξ),z₀ + J₂^z¯0(ξ),z¯₀ +J₂^z0z0(ξ)z0,z0 + J₂^z0z¯0(ξ)z0,¯z0 + J₂^z¯0z¯0(ξ)¯z0,¯z0,

while the new perturbation is of smaller size XR2D(s2,r2),2+ γ2

E2XR2^L_D(s2,r2),2ε₁⁴³,

where the parameter ξ will vary in small compact set 2 (of relatively large Lebesgue measure).

Since the preserved terms are put into the normal formN1, the homological equations in this iteration are of the following forms

ω·∂xF₁+A₁F₁+F₁B₁ = R₁, (1.14) ω·∂xF₂+A₂F₂ = R₂, (1.15)

ω·∂xF₃+F₃+F₃= R₃, (1.16) where A₁,A₂,B₁ depending only onξ are not diagonal while is diagonal (Instead of 1-st iteration with normal form N₀, (1.16) is the only homological equations we have to solve.). More narrowly, Eq. (1.14) is derived from the homological equation of the coefficients F^z0z0,F^z0z¯0,F^z¯0¯z0, whose Fourier coefficient matrixes related to the preserved terms are finite dimension (less than 4b²×4b²). Thus, by introducing Kronecker product and column straightening, the coefficient equation can be solved provided that its Fourier coefficient matrixes are non-degenerate. Furthermore, observing that thek-th Fourier coefficient matrix is self-adjoint after making small changes, the essential non-resonant conditions imposed on the coefficient matrixes can be converted to its eigenvalues. For Eqs. (1.15) and (1.16), they are also solvable as long as anyk-th Fourier coefficient matrixes are non-degenerate and satisfy some non-resonant conditions. These are the places where small divisors arise. Such divisors are

(1) Rkl(1)= {ξ∈1:| k, ω1(ξ) + l, 1(ξ) |< _(|^γ_k¹_|+^l₁^d₎τ, (k,l)∈Z}, whereγ1=3/4γ0,ld =max(1,|

j^dlj|), τ >n+1 andZ= {(k,l)∈Zⁿ×Z^∞:

|l| ≤2};

(2) R1k(1)= {ξ ∈1:max|λA11(ξ)|< _|k|^γ11τ1},

whereA₁₁(ξ)= k, ω1I_3b2+B₁₁(ξ), τ1=τ−1, γ11= _3b^γ12 and the 3b²×3b²order matrixB₁₁(ξ)’s norm is small enough;

(3) _{R3j k}(1)= {ξ ∈1:max|λ_A^j

31(ξ)|< _|k|^γ31τ3},

where A₃₁^j (ξ)= (k, ω1 ±₁^j)I_4b2+B₃₁(ξ), τ3 =τ, γ31= _4b^γ12 and the 4b²×4b² order matrixB31(ξ)’s norm is small enough;

(4) R4k(1)= {ξ ∈1:max|λA₄₁(ξ)|< _|^γ_k|⁴¹τ4},

whereA41(ξ)= k, ω1I_2b2+B41(ξ), τ4=τ−1, γ41= _2b^γ12 and the 2b²×2b²order matrixB41(ξ)’s norm is small enough;

whereInis then×nidentity matrix andλAdenotes the eigenvalue of matrixA.

Therefore, the homological equation associated to (1.13) can be solved and the KAM machin- ery still works well.

1.1.2 The Iterative Lemma (Sect.4)

We want to construct, inductively, symplectic transformationsm,m≥0, such that H_m+1= H_m◦m=(N_m+R_m)◦m=N_m+1+R_m+1,

where the sequences of the new normal formN_m+1 Nm+1 =

m j=0

N^x_j(ξ)+ ωm+1(ξ),y + m+1(ξ)z,z + J¯ _m+1^z0 (ξ),z0 + J_m+1^¯z0 (ξ),z¯0 +J_m+1^z0z0(ξ)z₀,z₀ + J_m+1^z0¯z0(ξ)z₀,z¯₀ + J_m+1^z¯0z¯0(ξ)¯z₀,z¯₀,

while the sequences of perturbationsR_m+1are of smaller and smaller size X_Rm+1D(sm+1,rm+1),m+1+ γm+1

E_m+1X_Rm+1^L_{D(sm+1,rm+1),m+1}εm⁴³.

The parameterξ will vary in smaller and smaller compact setsm (of relatively large Lebesgue measure)

0⊃1⊃ · · ·m⊃m+1⊃ · · · ⊃_∞⊃ ^∞

m=0

m.

The smallness assumption onεwill allow to turn on the iteration procedure.

The symplectic map^mwill be sought of the form

^m=^m−1◦m =0◦ · · · ◦m. In order to work for the approach, one has to show that

m :D(sm+1,rm+1)×m+1→D(sm,rm), (∀m≥0), (1.17) ^m :D(sm+1,r_m+1)×m+1→D(s0,r0), (∀m≥0). (1.18) Relations (1.17) and (1.18) are checked in Sect.4.

1.1.3 Proof of Theorem1.1and Corollaries1.2,1.3(Sect.5)

Once the iterative step is set up, it has to be equipped with estimates. This technique part follows the corresponding part in [34]. Particularly, the key results of theorem1.1concerning the new HamiltonianH_mand the measure ofmfollow easily.

Moreover, when N˘ = (N˘^z0(ξ),N˘^¯z0(ξ)) = 0, from the fast convergence of H_m, there exists a family of torus embeddingsuch that(T₀ⁿ× {ξ})is an invariant torus for the HamiltonianH in (1.5).

When N˘ = (N˘^z0(ξ),N˘^z¯0(ξ)) = 0, that is,

| ˘N₀^z(ξ)|²₂+ | ˘N^¯z0(ξ)|²₂ = δ0 > 0. Since

m→∞lim J_m^z0(ξ) = ˘N^z0(ξ) and lim

m→∞J_m^¯z0(ξ) = ˘N^z¯0(ξ), there exists a sufficiently large M₀ such that for anym≥M₀,

|Jm^z0(ξ)|²₂+ |Jm^¯z0(ξ)|²₂≥ δ0

2. (1.19)

More exactly, we will choosem₀>M₀such that

δ0>28ε_m⁷⁶₀₋₁. (1.20)

Consider the Hamiltonian equation defined byHm0 =Nm0+Rm0and fix an initial value z^∗(0)_a,p+ ¯z^∗(0)_a,p≤ε_m⁷⁶₀₋₁. From (1.19), (1.20) and using some ordinary differential equation tools, we will obtain

z^∗(1)a,p+ ¯z^∗(1)a,p> ε_m⁷⁶₀₋₁.

That is, there exists no torus in the domain^m0−1(m0×{ξ})for the HamiltonianHin (1.5) when we denotem₀ = {(x,y,z^∗,¯z^∗): |x| ≤sm₀,|y| ≤r_m²₀,z^∗a,p+¯z^∗a,p ≤ε_m⁷⁶₀₋₁}.

Theorem1.1and Corollaries1.2,1.3, at this point, are completely proven.

1.2 Application to Nonlinear Schrödinger Equation (NLS) (Sect.6) Consider a specific nonlinear Schrödinger equation

iut−ux x+ |u|⁴u=0 (1.21) on the finitex-interval[0,2π]with even periodic boundary conditions

u(t,x)=u(t,x+2π),u(x,t)=u(−x,t).

From the assumptionu(x,t)=u(−x,t)which simplifies the proof, it follows_j =1. We see that0 =0. When applied the abstract Theorem1.1to (1.21), the main ingredient is to verify N˘ = (N˘^z0(ξ),N˘^¯z0(ξ)) = 0. We will arrive at this end by provingRm^z0(0, ξ) = 0,R_m^¯z0(0, ξ) = 0 in anym−th iteration. The detailed, quantitative results are collected in Sect.6.

2 The Linearized Equation

Assume that all the assumptions of Theorem1.1are satisfied. Recall that H0 =H0(x,y,z^∗,z¯^∗, ξ)=N0(y,z^∗,z¯^∗, ξ)+R0(x,y,z^∗,¯z^∗, ξ), where

N0(y,z^∗,z¯^∗, ξ)= ω0(ξ),y + 00(ξ)z0,z¯0 + 0(ξ)z,¯z

=

1≤j≤n

ω₀^jyj+

jm∈J

₀₀^jmzjmz¯jm+

j∈N+\J

₀^jzjz¯j.

In the following, we abbreviate₀₀^jm andz_jm, respectively, as^m₀₀andz_0m(m=1, . . . ,b), for convenience.

DenoteR0(x,y,z^∗,z¯^∗, ξ)= R₀^low(x,y,z^∗,¯z^∗, ξ)+R₀^high(x,y,z^∗,z¯^∗, ξ). Then we have

R^lo₀^w =

α∈Nⁿ,β,γ∈N^N,2|α|+|β|+|γ|≤2

R₀^αβγ(x, ξ)y^α{z^∗}^β{¯z^∗}^γ,

R₀^high =

α∈Nⁿ,β,γ∈N^N,2|α|+|β|+|γ|≥3

R₀^αβγ(x, ξ)y^α{z^∗}^β{¯z^∗}^γ.

We desire to eliminate the termsR₀^lowby the coordinate transformation0, which is obtained as the time-1-mapX^t_F

0|t=1of a Hamiltonian vector fieldX_F0, whereF₀(x,y,z^∗,z¯^∗, ξ)is of the form

F0(x,y,z^∗,z¯^∗, ξ)= F₀^low(x,y,z^∗,z¯^∗, ξ)

=

α∈Nⁿ,β,γ∈N^N,2|α|+|β|+|γ|≤2

F₀^αβγ(x, ξ)y^α{z^∗}^β{¯z^∗}^γ. Using Taylor formula, we have

H1= H0◦X^t_F0|t=1

= N0+ {N0,F0} + ₁

0

(1−t){{N0,F0},F0} ◦X^t_F

0dt

+R^low₀ + ₁

0

{R₀^low,F₀} ◦X^t_F

0dt+R₀^high◦X^t_F

0|t=1

= N1+R1. where

N1 = N0+ ˆN0

= N0+R₀^x(0, ξ)+ R₀^y(0, ξ),y + R^z₀⁰(0, ξ),z0 +R₀^¯z0(0, ξ),¯z₀ + R₀^z0z0(0, ξ)z₀,z₀

+R₀^z0z¯0(0, ξ)z0,z¯0 + R₀^z¯0¯z0(0, ξ)¯z0,¯z0 +

j∈N+\J

R₀^zj¯zj(0, ξ)zjz¯j,

and

R1 = ₁

0

{(1−t)Nˆ0+t R₀^low,F0} ◦X^t_F0dt+R₀^high◦X^t_F0|t=1. Then the modified homological equation writes

{N0,F₀} +R^low₀ = ˆN₀, (2.1) For anym≥0, we also denote

N_m^x(ξ)= R_m^x(0, ξ),Nm^y(ξ)=Rm^y(0, ξ),Nm^z0(ξ)=R^zm⁰(0, ξ),Nm^¯z0(ξ)=Rm^¯z0(0, ξ), Nm^z0z0(ξ)=Rm^z0z0(0, ξ),Nm^z0¯z0(ξ)=Rm^z0z¯0(0, ξ),Nm^z¯0z¯0(ξ)=Rm^¯z0z¯0(0, ξ).

2.1 The Solution of Homological Equation (2.1) To solve the equation, we need:

Lemma 2.1 Suppose H₀ = N₀+R₀descried above satisfies assumptions A,B,C and D, and

ε0:= XR0D(s₀,r₀),0+ γ0

E0XR0^L_D(s0,r0),0 ≤C0γ₀². For some fixed constantτ >n+1, let

Rkl(0)=

ξ∈0: |k, ω0(ξ) + l, 0(ξ)|< γ0ld

(1+ |k|)^τ

, whereld=max(1,|

j^dl_j|),Z= {(k,l)∈Zⁿ×Z^∞: |l| ≤2},and let

1=0\

|k|>K0,|l|≤2

Rkl(0),

where K0 will be given later. Then for eachξ ∈1, the homological equation (2.1) has a solution F₀(x,y,z^∗,z¯^∗, ξ)satisfying

F₀(x,y,z^∗,¯z^∗, ξ)=F₀(x,y,z^∗,¯z^∗, ξ), ∀(x,y,z^∗,z¯^∗, ξ)∈D(s₀−σ0,r₀)×1, XF0D(s0−σ0,r0),1+ γ0

E0XF0^L_{D(s0−σ0,r0),1}ε0

γ0,