BIRKHOFF NORMAL FORM FOR ABCD BOUSSINESQ SYSTEM ON THE CIRCLE

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BIRKHOFF NORMAL FORM FOR ABCD BOUSSINESQ SYSTEM ON THE CIRCLE

Trung Nguyen

To cite this version:

Trung Nguyen. BIRKHOFF NORMAL FORM FOR ABCD BOUSSINESQ SYSTEM ON THE CIR- CLE. 2021. �hal-03274316�

THE CIRCLE

NGUYEN TRUNG

Abstract. In this paper, we prove a Birkhoff normal form result for the abcd Boussi- nesq system on the circle withb =d > 0, a, c < 0. The result based on the fact that the nonlinear term has a so-called tame modulus and the frequencies satisfy a nonreso- nant condition. As a consequence of the Birkhoff normal form, we have that any small amplitude solution remains close to a torus for a long time.

Contents

1. Introduction 1

2. Statement of the main theorem 3

2.1. Scheme of the proof 5

3. Normal form and tame modulus 5

3.1. Nonresonant condition 6

3.2. Tame modulus 6

3.3. Homological equation 9

3.4. Remind the idea of [BG06] 9

4. Boussinesq equation’s properties 10

4.1. Nonlinearity 10

4.2. Frequencies 11

5. Proof of the Theorem 2.1 15

Appendix A. 17

Appendix B. 18

B.1. The case b6= 0 18

B.2. The case b = 0 20

References 22

1. Introduction We consider theabcd-Boussinesq system

(1.1)

(1−b∂_xx)∂_tη+∂_x(a∂_xxu+u+uη) = 0 (t, x)∈R×T (1−d∂_xx)∂_tu+∂_x(c∂_xxη+η+¹₂u²) = 0

Where η, u are real functions with zero momentum Z

T

η(t, x) dx= Z

T

u(t, x) dx= 0.

1

Here the independent variable x corresponds to distance along the channel and t is pro- portional to elapsed time. The quantity η=η(x, t) corresponds to the depth of the water at the point x and time t. The variable u(x, t) is proportional to the horizontal velocity at the height θh,where θis a fixed constant in the interval [0,1]andh is the undisturbed water depth.

We are interested in the long time behavior of small amplitude solution of (1.1). In this paper, we would like to prove a Birkhoff normal form result for (1.1). The idea comes from Theorem2.13of [BG06] (see also [Bam03], [BDGS07],[Gré07], [Bam08], [Del12]), where the authors prove a Birkhoff normal form result for Hamiltonian with nonlinearity satisfying a tame modulus condition and nonresonant frequencies. In [BG06], the authors applied that result to many partial differential equations: NLW equations, NLS equations in one dimension and higher dimension, coupled NLS in one dimension. In this paper, we state a similar result for Boussinesq systerm.

As in general, the proof of a Birkhoff normal form result is obtained by constructing a sequence of canonical transformations to eliminate the non-normalized part if the nonlin- earity step by step. This procedure relates to solving homological equations in each step.

The idea in [BG06] is to use a so-called tame inequality kuvks ≤C(kukskvk1+kvkskuk1) (1.2)

and that if u∈H^s depends only on high modes u=P

|k|≥Nu_ke^i2πkx then kuk1 ≤ kuks

N^s−1. (1.3)

This term is small whenN large enough. Then in the Birkhoff procedure, the nonlinearity, which satisfies the tame properties and has many high modes in expansion, is small and controllable. The study of frequency plays an important role in order to eliminate the remainder part of non-integrable resonant terms. However, contrary [BG06], where exter- nal potentials were used to verify the nonresonant condition, we obtain the nonresonant condition by studying frequencies as functions of parameters a, b, c, d.

Our main contribution in this work is to verify that the nonlinearity in Boussinesq system satisfies a tame property and that the frequencies satisfy a nonresonant condition(section 4) then we prove our Birkhoff normal form result(section 5). In section 2, we state our main theorem and explain the scheme of our proof. In section 3, we recall notations of tame property and nonresonant condition, which was introduced in [BG06]. We give the proof of local well-posedness in Appendix B.

The system (1.1) was originally derived by Bona, Chen and Saut [BCS02],[BCS04] in the vein of the Boussinesq original derivation [Bou72]. The equation is derived to study the two dimensional, incompressible and irrotational water wave in the shallow water regime. The abcdBoussinesq equation and its extensions have been studied extensively in the literature (see [SL08], [BCL05], [BLS08], [LPS12], [SX12], [SX17], [LY97]). Here, the parameters a, b, c, d satisfy the consistency conditions (see [BCS02], [BCS04])

a+b+c+d= 1 3. (1.4)

To consider all possible value of the parameters (a, b, c, d) present in the equation is too complicated, here we concentrate our study to the "Hamiltonian generic" case namely the

case where

b=d >0, a, c <0.

In a next paper, we would like to study the Boussinesq equation in the KdV-KdV regime (b =d = 0). It is worth saying that to prove a Birkhoff normal form result for this case is much more complicated since in this case the nonlinearity is unbounded and its tame property is failed. However, a recent paper [BG20] gives tools to deal with this problem.

Acknowledgement: I wish to thank Professor Benoît Grébert for motivating me to publish this paper with numerous suggestions and discussions. The author is partially supported by the grant BeKAM ANR-15-CE40-0001-02, and by the Centre Henri Lebesgue, ANR-11-LABX-0020-01.

2. Statement of the main theorem We expand the solution in Fourier variables

u(x) = X

k∈Z∗

u_ke^2iπkx, η(x) = X

k∈Z∗

η_ke^2iπkx (2.1)

note thatu¯_k=u_−k, η¯_k=η_−k since uand η are real, then (1.1) reads (2.2)

( ∂_tη_k=−_1+4π^i2πk2bk²((1−4π²ak²)u_k+P

j+l=ku_jη_l)

∂tu_k=−_1+4π^i2πk2_bk²((1−4π²ck²)η_k+¹₂P

j+l=kuju_l) k∈Z^∗. Simplify, we denote

D_k = 2πk

1 + 4bπ²k², ω_k=p

(1−4aπ²k²)(1−4cπ²k²) Ω_k=D_kω_k.

Before we state our result, we need to put the system (2.2) into a more convenient form by using a linear symplectic change of variables

ψ_k = 1

√2(α_ku_k+α⁻_k¹η_k), φ_k= 1

√2(α_ku_k−α⁻_k¹η_k) where α_k =

1−4aπ²k² 1−4cπ²k²

¹₄

=α_−k. Then the system can be written as (2.3)

( ∂_tψ_k =−iD_k∇ψ−kH

∂_tφ_k =iD_k∇φ−kH k∈Z^∗, where H is the Hamiltonian

H=H₀+P with

H₀ = X

k∈N∗

ω_k(ψ_kψ_−k+φ_kφ_−k)

P = 1 4√

2 X

j+l+h=0

α_hα⁻_j¹α⁻_l ¹(ψ_j+φ_j)(ψ_l+φ_l)(φ_h−ψ_h)

Through all the paper, we identify the couple function (ψ, φ) with its series of Fourier coefficients(ψ, φ) = (ψ_k, φ_k)_k∈Z∗.We also define the Sobolev space (s≥0)

H^s:={z= (ψ, φ) = (ψ_k, φ_k)_k∈Z∗ ∈ℓ²(T)×ℓ²(T)| kzk²s = X

k∈Z∗

|k|^2s(|φ_k|²+|ψ_k|²)<∞}. (2.4)

Denote byN_k=|(ψ_k, φ_k)|² =|φ_k|²+|ψ_k|² the actions of the flow(ψ, φ)(t).We also denote I_k=|φ_k|², J_k =|ψ_k|² thenN_k=I_k+J_k=α²|u_k|²+α⁻²|η_k|².Denote by B_s(R)the open ball centered at the origin of radius R inPs.

Poisson bracketIn the phase spaceH^s,we define the Poisson bracket {P, Q}= X

k∈Z∗

D_k∂_ψkP ∂_ψ

−kQ− X

k∈Z∗

D_k∂_φkP ∂_φ

−kQ

= X

k∈N∗

D_k(∂_ψkP ∂_ψ−kQ−∂_ψ−kP ∂_ψkQ)− X

k∈N∗

D_k(∂_φkP ∂_φ−kQ−∂_φ−kP ∂_φkQ).

so that the equation (2.3) reads (2.5)

( i∂_tψ_k ={ψ_k, H}

i∂_tφ_k ={φ_k, H} k∈Z^∗. Frequencies Let us write Ω_k = _1+4bπ^2πk2k²

ppk⁴+ek²+ 1 where p = 16π⁴ac, e =

−4π²(a+c). Here we assume that b = d ≥ 0, and we consider two different cases: the

"generic" case b=d >0, a, c <0 and the Kdv-Kdv caseb=d= 0, a, c >0.

In the generic case, sincea+c+ 2b= ¹₃,we haveb= ¹₆ −(a+c)/2> ¹₆ and 0< p= 16π⁴ac≤16π⁴(b−1

6)² e=−4π²(a+c) = 4π²(2b−1 3)≥0.

We denote Ib := (0, p₀ = 16π⁴(b−¹₆)²). In the Kdv-Kdv case, in order for frequencies to be real, we assumea, c > _4π¹2.Since a+c= ¹₃,we have

4

3π²−1< p≤ 4 9π⁴. Denote I0= (⁴₃π²−1,⁴₉π⁴].

For any r≥3,we define α_∗ := (7

2r³+r²+ 7r)²; N_∗(r, α, µ) := [µ^−2rα1 ]; s_∗(r, α) = 2αr²+ 2.

Then our main result is stated as follow

Theorem 2.1. Let r ≥ 1,0 < µ≪ 1. Then for any s≥s_∗, α > α_∗ there exists a subset I^µ⊂ Ib and a constant C=C(r, s, b) such that

|I^µ− Ib| ≤C|Ib|µ^α+1r+2 (2.6)

and for any p∈ I^µ,for |t| ≤µ^−r+3/2,there exists a transformation T :B_s(µ/3)→B_s(µ) satisfying

H◦ T =H₀+Z+R. (2.7)

here Z is a polynomial of degree at most r+ 2 that commutes with the actions N_k, i.e., {Z, N_k}= 0 ∀k∈Z^∗

(2.8)

and R ∈C^∞(B_s(µ)) fulfills the estimate sup

k(ψ,φ)k^s≤µ/3kX_Rk^s≤Cµ^r+32. (2.9)

The canonical transformations and its inverse are close to identity

sup

k(φ,ψ)k^s≤µ/3k(φ, ψ)− T(φ, ψ)k^s ≤Csµ² sup

k(φ,ψ)k^s≤µ/3k(φ, ψ)− T⁻¹(φ, ψ)k^s ≤Csµ². (2.10)

Corollary 2.2. Fix r ≥ 1, assume Ω is nonresonant, then for any s sufficiently large, there exist µ₀ and csuch that if the initial data is in B_s(µ), µ≤µ₀ then

• k(u, η)(t)ks ≤2µ fort≤cµ^−r+3/2

• α²_k(|u_k|²(t)− |u_k|²(0)) +α⁻_k²(|η_k(t)|²− |η_k(0)|²)≤ _k^µ2s3 fort≤cµ^−r+3/2; k∈Z∗. The difference between the "generic" system and the Kdv-Kdv system comes fromD_k. In the generic caseb > ¹₆,we have |D_k| ≤1, which makes it easy to prove tame property, while in the Kdv-Kdv case D_k= 2πk is unbounded so that the nonlinear termP does not satisfies tame property. In the fullness of time, we would like to study this case, which is much more complicated.

2.1. Scheme of the proof. In order to prove the Theorem2.1, we search iterative changes of variables Tⁿ,1≤n≤r, such that

H◦ Tn=H₀+Z₄(I, J) +Z₆(I, J, Y) +· · ·+Z_n(I, J, Y) +R_n+1. (2.11)

Where R_n+1 is a perturbation term which is of the sizeµⁿ⁺³² and I = (I_k)_k∈Z∗, J = (J_k)_k∈Z∗, Y = ( Y

P

kj=0

Y_kj), Y_k=ψ_kφ_k.

Here we callY_k the pseudo actions, which satisfy{Y_k, H₀}= 0, k∈Z^∗.Moreover {I_k+J_k, Y

P

kj=0

Y_kj}= 0 ∀k, k_j ∈Z^∗. (2.12)

The condition P

k_j = 0 appear naturally since H comes from an integration of a real function on the circle. As a consequence of the appearance of Y, the exchange of energy mainly occurs between I_k and J_k for a long time |t| ≤µ^−r+32,∀k∈Z^∗.

3. Normal form and tame modulus In this section, we recall some notations and lemmas used in [BG06].

3.1. Nonresonant condition.

Definition 3.1. Fix two positive parametersκandα,and a positive integerN.A function Z is said to be in (κ, α, N)−normal form with respect to Ωif Z_kl6= 0 implies

|Ω(k−l)|< κ

N^α and X

|j|≥N+1

|kj|+|lj| ≤2.

Definition 3.2 (Nonresonant condition). Let r be a positive integer, we say that the frequency Ω is nonresonant if there exist κ > 0, and α ∈ R such that for any N large enough one has

(3.1) |X

j∈N∗

Ω_jk_j| ≥ κ N^α for any k∈Z∞,fulfilling 06=|k|:=P

j|k_j| ≤r+ 2,P

|j|>N|k_j| ≤2.

Remark 3.3. If the frequencyΩis nonresonant then any polynomial which is in(κ, α, N)−normal form depends only on I, J and Y.

Example 3.4. If the frequency Ω is nonresonant then a polynomial of order 6, which is in(κ, α, N)−normal form with respect toΩwould be

Z₆= X

k,l∈Z∗

a_k,lY_kY_lY_−k−l+ X

k,l,h∈Z∗

b_k,l,hI_kI_lI_h+c_k,l,hI_kI_lJ_h+d_k,l,hI_kJ_lJ_h+e_k,l,hJ_kJ_lJ_h where a_k,l, b_k,l,h, c_k,l,h, d_k,l,h, e_k,l,h are real constants.

3.2. Tame modulus. Let us consider a homogeneous polynomial of degreer+ 1f :Ps→ R,which can be written as

f(z) := X

|j|=r+1

f_jz^j, j= (. . . , j_−l1, j_−l2, . . . , j₋₁₁, j₋₁₂, j₁₁, j₁₂, . . . , j_l1, j_l2, . . .)

z^j := (ψ, φ)^j =. . . ψ₋^j−l1_l φ^j₋^−l2_l . . . ψ^j_l^l1φ^j_l^l2. . . , |j|:=X

l

(|j_l1|+|j_l2|), We can associatef to a symmetricr+ 1−linear form f˜as

f(z) = ˜f(z⁽¹⁾, . . . , z^(r+1)) = X

|j|=r+1

fj1,...,jr+1z⁽¹⁾_j1 . . . z_j^(r+1)_r+1 , j_l= (j_l1, j_l2).

We say thatf is bounded if there exists a constantC such that

|f(z)| ≤Ckzk^r+1s , ∀z∈ Ps, or equivalently

|f˜(z⁽¹⁾, . . . , z^(r+1))| ≤Ckz⁽¹⁾ks. . .kz^(r+1)ks.

Now consider X_f := (_1+4π^2πk2bk²∂_ψkf,−_1+4π^2πk2_bk²∂_φkf)the vector field of f.We write it as X_f(z) =X

l

X_l(z)e_l

where e_l is l-th standard basis of Ps and X_l(z) is a real valued homogeneous polynomial of degreer. We also consider ther−linear formX˜_f so that X_f(z) = ˜X_f(z, z, . . . , z).

Definition 3.5. Let f is a homogeneous polynomial of degree r, we define its modulus

⌊f⌉ by

⌊f⌉:= X

|j|=r

|f_j|z^j. We also define the modulus of its vector field X_f as

⌊X_f⌉:= X

l∈Z\{0}

⌊X_l⌉e_l.

Definition 3.6. Lets≥1, a homogeneous polynomial vector field X of degree r is said to be an s−tame map if there exists a constantCs such that

kX(z˜ ⁽¹⁾, . . . , z^(r))ks≤C_s1 r

r

X

l=1

kz⁽¹⁾k1. . .kz^(l−1)k1kz^(l)kskz^(l+1)k1. . .kz^(r)k1. (3.2)

If a map is s-tame for any s ≥1 then it is said to be tame. Moreover if X is the vector field of a homogeneous polynomial f and ⌊X_f⌉ is ans−tame map also, then we define

|f|s:=sup kX˜_⌊f⌉(z⁽¹⁾, . . . , z^(r))ks 1

r

P_r

l=1kz⁽¹⁾k1. . .kz^(l−1)k1kz^(l)kskz^(l+1)k1. . .kz^(r)k1

thestame norm of f. We denote byT_M^s the set of function has finitestame norm.

Remark 3.7. Iff ∈T_M^s is a homogeneous polynomial of degreer+ 1then one has kX_f(z)ks≤ kX_⌊f⌉ks ≤ |f|skzk^r1⁻¹kzks.

(3.3)

The important property ofstame norm is that the Poisson bracket of two homogeneous polynomials in T_M^s is also in T_M^s (see lemma 4.12 in [BG06]).

Lemma 3.8. Assume that f, g ∈ T_M^s are two homogeneous polynomials of degrees n+ 1 andm+ 1respectively, then {f, g} is a homogeneous polynomial of ordern+min T_M^s with

|{f, g}|s≤2(n+m)|f|s|g|s. (3.4)

For a non homogeneous polynomial, we consider its Taylor expansion f =X

f_r

where f_r is homogeneous function of degree r. Then we define (3.5) Ts,R:={f =X

fr|fr ∈T_M^s ; h|f|i^s,R :=X

r≥2

|fr|^sR^r−1<∞}. Following lemma4.13 in [BG06] we have

Lemma 3.9. Let f, g∈T_s,R,then for any d < R we have{f;g} ∈T_s,R−d and h|{f;g}|is,R−d≤ 2

dh|f|is,Rh|g|is,R.

Proof. Writef =P

jf_j and g=P

kg_k withf_j homogeneous of degreej and similarly for g.One has

{f, g}=X

j,k

{f_j, g_k}. We have estimate for each term of the series

h|{f_j;g_k}|is,R−d=|{f_j, g_k}|s(R−d)^j+k−3

≤2|f_j|s|g_k|s(j+k−2)(R−d)^j+k−3

≤2|fj|^s|g_k|^s1

dR^j+k−2

= 2

dh|f_j|is,Rh|g_k|is,R

here we use the estimate

k(R−d)^k−1 < R^k

d , ∀0< d < R.

Then as in lemma 4.14 in [BG06], we deduce

Lemma 3.10. Let g, χ ∈ T_s,R be two analytic functions. Denote g_l = {g_l−1, χ} for all l >0 withg₀ =g. Then for any0< d < R,one has g_n∈T_s,R−d and

h|g_n|is,R−d≤ h|g|is,R(2e

dh|χ|is,R)ⁿ. (3.6)

Proof. Denote δ = ^d_n.Apply iteratively Lemma3.9, one has h|g_l|is,R−δl ≤ 2

δh|g_l−1|is,R−δl−1h|χ|i^s,R≤ · · · ≤ 2ⁿ n!(n

dh|χ|i^s,R)ⁿh|g|i^s,R. (3.7)

Using the inequality nⁿ< N!eⁿ,one has the thesis.

Lemma 3.11. Let χ, g be two analytic functions with Hamiltonian vector fields analytic in B_s(R). Fix0< d < R assume kX_χks,R < ^d₃, then for |t| ≤1,one has

sup

kzk^s≤R−dkϕ^t_χ(z)−zks≤ kX_χks,R

(3.8)

kX_g◦ϕ^t_χks,R−d≤(1 + 3

dkX_χks,R)kX_gks,R. (3.9)

For the proof see [Bam99] proof of lemma 8.2

Remark 3.12. For any positive integer numberN,we writez= ¯z+ˆzwithz¯= (φj, ψj)_|j|≤N and zˆ= (φj, ψj)_|j|≥N.Then if homogeneous polynomial f ∈T_M^s of orderr+ 1has a zero of order three in z,ˆ one has

(3.10)

kX_fks,R≤ |f|s(1 r

r

X

l=1

kz⁽¹⁾k1. . .kz^(l−1)k1kz^(l)kskz^(l+1)k1. . .kz^(r)k1)

≤ |f|skzˆk1kzk^r1⁻³kzks≤ |f|sR^r

N^s−1 = h|f|is,R

N^s−1 .

Here we use estimates

kzˆk1 ≤ kzks

N^s−1

kz¯ks ≤ kzks, kzˆks≤ kzks

kzk1 ≤ kzks.

For a non homogeneous polynomial f ∈T_s,R of order less or equal r+ 2

f = X

j≤r+2

fj

then

(3.11) kX_fks,R≤ X

j≤r+2

kX_fjks,R≤ P

j≤r+2|f_j|sR^j−1

N^s−1 =rh|f|is,R

N^s−1 .

3.3. Homological equation. In each step of Birkhoff normal form procedure, we need to solve a homological equation

−i{H₀, χ}+Z =f (3.12)

with Z is in (κ, α, N)−normal form. Then after each step, we have the new nonlinearity term of size{Z4, χ}+{f, χ}. Then we need to estimate the nonlinearity χand the normal formZ.

Lemma 3.13. Let f be a polynomial in T_M^s which is at most quadratic in the variables z.ˆ Assume nonresonant condition, then there exist χ, Z ∈ T_s,R with Z in (κ, α, N)−normal form solving

{H0, χ}+Z =f.

(3.13)

Moreover Z andχ satisfy the estimates

h|χ|i^s,R≤ N^α

κ h|f|i^s,R, h|Z|i^s,R ≤ h|f|i^s,R. (3.14)

For the proof see [BG06] proof of lemma 4.7.

3.4. Remind the idea of [BG06]. Here we recall the idea of the proof of the Theorem 2.13 in [BG06]. The proof consists in using iterative Lie transformations. Consider an Hamiltonian functionχ with the corresponding Hamiltonian equation

( ˙ψ,φ) =˙ X_χ((ψ, φ)).

Denote byϕ^t_χ the corresponding flow and call ϕ_χ=ϕ¹_χ the Lie transform generated by χ.

Then for any polynomialg, we have an important property d

dt(g◦ϕ^t_χ) ={g, χ} ◦ϕ^t_χ.

To prove the theorem we find Hamiltonian functionsχ_nand Hamiltonians H_n such that

(3.15)























H_n=H₀+Z_n+f_n+Rn

H_n+1 =H_n◦ϕ_χn τ =ϕ_χ0 ◦ · · · ◦ϕ_χr H_n+1 =H_n− {H_n, χ_n}+P

n≥2 (−1)ⁿ

n! ad_χn(H_n) Z_n+1=Z_n+f_n+−{H₀, χ_n}

f_n+1=−{f_n, χ_n}+P_r

l≥2−1^l

l! ad^l_χn(H₀+Z_n+f_n) Rn+1 =Rn◦ϕ_χn+ ⁽⁻_(r+1)!^1)r+1R1

0 ad^r+1_χn (H₀+Z_n+f_n)◦ϕ^τ_χndτ.

Heref_nis a polynomial of orderr,Rnis a small term. This leads us to solve the homological equation

{H₀, χ}+Z =f (3.16)

withZ is in(κ, α, N)−normal form with respect to Ω. Then by [BG06] iff₀ =P has the tame modulus property and frequenciesΩ_k satisfy a nonresonant condition, there existsχ solving (3.16) and

h|X_χ|is,R ≤CN^α

κ h|X_f|is,R, h|X_Z|is,R≤Ch|X_f|is,R. (3.17)

In our case, we formally expand f in Taylor series f(φ, ψ) =X

j,l

f_j,lΠ_k,hψ_k^jkφ^l_h^h

and similarly forχ, Z. The equation (3.16) becomes (Ω(j−l))χ_j,l+Z_j,l= X

k∈N∗

(Ω_k(j_k−j_−k−l_k+l_−k))χ_j,l+Z_j,l=f_j,l. (3.18)

Here we use

{H₀, ψ_k^jkφ^l_h^h}= (Ω_kj_k−Ω_hl_h)ψ_k^jkφ^l_h^h, Ω_k=−Ω_−k.

Then whenΩsatisfies the nonresonant condition, we define the solution for the homological equation as following

χ_jl:= f_jl

(Ω(j−l)) with |Ω(j−l)| ≥ κ N^α Z_jl:=f_jl with |Ω(j−l)|< κ

N^α.

4. Boussinesq equation’s properties 4.1. Nonlinearity. Now consider

P = X

k1+k2+k3=0

α⁻_k1¹α⁻_k2¹α_k3(ψ_k1 +φ_k1)(ψ_k2 +φ_k2)(ψ_k3−φ_k3).

We prove thatP has tame modulus.

Lemma 4.1. The nonlinearity P has tame modulus.

Proof. One has that its modulus is

⌊P⌉= X

k1+k2+k3=0

|α⁻_k¹

1 α⁻_k¹

2α_k3|(ψ_k1 +φ_k1)(ψ_k2+φ_k2)(ψ_k3 +φ_k3) and

⌊X˜_P⌉= (|D_k|∂_φ

−k⌊P⌉,|D_k|∂_φ

−k⌊P⌉)_k∈Z∗

with

|D_k|∂_φk⌊P⌉=|D_k|∂_ψk⌊P⌉= 1 2√

2|D_k|



 X

j+l+k=0

1

2|α_kα⁻_j¹α⁻_l ¹|(ψj+φj)(ψ_l+φ_l)





+ 1 2√

2|D_k|



 X

j+l+k=0

|α⁻_k¹α⁻_j¹α_l|(ψ_j+φ_j)(ψ_l+φ_l)





= 1 2√

2|D_k| X

j+l+k=0

(1

2|α_kα⁻_j¹α⁻_l ¹|+|α⁻_k¹α⁻_j¹α_l|)(ψ_j +φ_j)(ψ_l+φ_l).

In the case b6= 0,we have that |D_k|= _1+4π^|2πk_2bk^| ₂ ≤ _2π¹_|b|. Since|α_k| ≤1, we verifys−tame map property of⌊X_P⌉as

k⌊X_P⌉k²s = 1 4

X

k∈Z∗

hki^2s|D_k|²( X

j+l+k=0

(1

2|α_kα⁻_j¹α⁻_l ¹|+|α⁻_k¹α⁻_j¹α_l|)(ψ_j +φ_j)(ψ_l+φ_l))²

≤ 9

64π²b² X

k∈Z∗

hki^2s( X

j+l+k=0

(ψ_j +φ_j)(ψ_l+φ_l))²

≤ 9

16π²b²

kz⁽¹⁾k²skz⁽²⁾k²1+kz⁽¹⁾k²1kz⁽²⁾k²s

i.e.

k⌊X_P⌉ks≤ 3 4π|b|

kz⁽¹⁾kskz⁽²⁾k1+kz⁽¹⁾k1kz⁽²⁾ks

(4.1)

So that P has tame modulus.

4.2. Frequencies. We now consider the frequencies Ω_k = 2π k

2 + 4π²bk²

p(1−4aπ²k²)(1−4cπ²k²) = 2π k 1 + 4π²bk²

ppk⁴+ek²+ 1 where

e=−4π²(a+c)>0, p= 16π⁴ac∈(0, p₀ = 16π⁴(1

6 −b)²) =Ib.

The parameter s and p are well-defined, positive and independent. In this subsection, we study frequencies Ω_k as functions of p. More precisely, the goal of this part is to prove the following theorem, which has been demonstrated in slightly different contexts in [Bam03],[BG06] and [EGK16].

Theorem 4.2. There exists a set J ∈ I of full measure such that for any p ∈ J, fixed r≥1,there exist κ=κ(r, p)>0, α=α(r, p) >0 such that for any N large enough then

|X

j≥1

Ω_jk_j| ≥ κ N^α, (4.2)

for any k∈Z^∞, fulfilling06=|k|=P

j|k_j| ≤r+ 2,P

|j|>N|k_j| ≤2.

In order to prove Theorem4.2, we need to construct some lemmas.

Lemma 4.3. For any r ≤ N, consider r indexes j₁ < j₂ < j₃ < ...j_r ≤N; consider the determinant

D=:

Ω_j1 Ω_j2 . . . Ω_jr

dΩj1

dp

dΩj2

dp . . . ^dΩ_dp^jr

. . . .

. . . .

d^r−1Ωj1

dp^r−1

d^r−1Ωj2

dp^r−1 . . . ^dr−1_dpr−1^Ωjr

Then

D=±





r−1

Y

j=1

(2j−3)!

2^j−2(j−2)!2^j





r

Y

ℓ=1

Ω_jℓ

!

 Y

1≤ℓ<k≤r

(x_jℓ−x_jk)





≥C(r, b) 1 N^12(5r2−3r).

Here x_j = _pj4+ej^j42+1.

Proof. We compute the iterative derivatives ofΩ_k with respect topin the spirit of section 3 in [EGK16]

d^jΩ_k

dp^j = 2π k 1 + 4π²bk²

(2j−1)!

2^j−1(j−1)!2^j

(−1)^jk^4j (pk⁴+ek²+ 1)^j−12

= (−1)^j(2j−1)!

2^j−1(j−1)!2^jΩ_kx^j_k.

Substituting this into the determinant then D=±





r

Y

j=1

(2j−3)!

2^j−2(j−2)!2^j





r

Y

ℓ=1

Ω_jℓ

!

×

1 1 1 . . . 1

x_j1 x_j2 x_j3 . . . x_jr x²_j1 x²_j2 x²_j3 . . . x²_jr

. . . .

. . . .

x^r_j1⁻¹ x^r_j2⁻¹ x^r_j3⁻¹ . . . x^r_jr⁻¹

=±





r

Y

j=1

(2j−3)!

2^j−2(j−2)!2^j





r

Y

ℓ=1

Ω_jℓ

!

 Y

1≤ℓ<k≤r

(x_jℓ−x_jk)



.

Since b≥ ¹₆,we have estimate

|

r

Y

ℓ=1

Ωjℓ|=

r

Y

ℓ=1

2π |j_ℓ| 1 + 4π²bj_ℓ²

q

pj_ℓ⁴+ej_ℓ²+ 1≥ 1 (4πb)^r

r

Y

ℓ=1

1

|j_ℓ| and

Y

1≤ℓ<k≤r

|(x_jℓ−x_jk)|= Y

1≤ℓ<k≤r

|j_ℓ²+j_k²+ej_ℓ²j_k²||j_ℓ²−j²_k| (pj_ℓ⁴+ej_ℓ²+ 1)(pj_k⁴+ej²_k+ 1)

≥ Y

1≤ℓ<k≤r

2(|j_ℓ|+|j_k|) (p+e+ 1)²j_ℓ³j_k³

≥ Y

1≤ℓ<k≤r

2(|j_ℓ|+|j_k|) (4π²(b−¹₆) + 1)⁴j_ℓ³j_k³. Then

D≥ |

r

Y

j=1

(2j−3)!

2^j−2(j−2)!2^j 1 (4πb)^r

r

Y

ℓ=1

1

|j_ℓ| Y

1≤ℓ<k≤r

2(|j_ℓ|+|j_k|)

(4π²(b−¹₆) + 1)⁴j³_ℓj_k³ ≥C(r, b) 1 N^12(5r2−3r) where

C(r, b, p) =|

r

Y

j=1

(2j−3)!

2^j−2(j−2)!2^j| 1 (4πb)^r

2^r(r−1)/2

(4π²(b− ¹₆) + 1)^2r(r−1).

Now we recall a result from [BGG85] appendix B

Lemma 4.4. Let u⁽¹⁾,· · · , u^(r) be r independent vectors in R^r of norm at most one, and let w∈R^r be any non-zero vector. Then there exists i∈[1,· · · , r]such that

|hu⁽ⁱ⁾, wi| ≥ |w||det(u⁽¹⁾,· · ·, u^(r))|

r³² .

By [XYQ97], we have the lemma

Lemma 4.5. Suppose that g(p) be a C^r-smooth function on an interval J ⊂ R. Let J_h :={p∈ J : |g(p)|< h}, h >0. If |g^(r)(p)| ≥dthen |J_h| ≤M h^1/r,where

M := 2(2 +· · ·+r+1 d).

Now let us consider the function g(p) =|k|⁻¹(P_N

|a|=1k_aΩ_a+c), with|k| ≤r < N,c is a fixed constant. Define

R_kc(κ, α) :={p∈ Ib| |g(p)| ≤ κ

|k|N^α}

By lemma 4.4one has max_1≤k≤rmin_p|∂^kg(p)| ≥C(r, b)N^{−12(5r2−3r)−2} so that

|R_kc(κ, α)| ≤ 2

C(r, b)κ^1/rN^τ withτ = ¹₂(5r−3r) + 4−^α_r.

Lemma 4.6. Fix α > ⁵₂r³+r²+ 7r, and let I_κ =I \S

kcR_c,k(κ, α) then for any p∈ I_κ and for any c∈βZ, k∈Z^N with 06=|k| ≤r,β >0, one has

|

N

X

|a|=1

k_aΩ_a+c| ≥ κ N^α

Moreover

|I −I_κ| ≤Cκ^1r where C= ²_C(r,b)πb^√p+e+1.

Proof. Since

|Ω_k|=| 2πk 1 + 4π²bk²

ppk⁴+ek²+ 1| ≤

√p+e+ 1 2πb |k| We have that |P_N

|a|=1k_aΩ_a+c| ≥1for all |c| ≥ ^√p+e+1_2πb rN. hence

|[

kc

R_c,k(κ, α)| ≤ X

|k|≤r,|c|≤CrN

|R_c,k(κ, α)|

≤2

√p+e+ 1

2πb rN 2

C(r, b)r^2rN^1+τκ^1/r

≤ 2√

p+e+ 1 C(r, b)πb κ^1/r.

Lemma 4.7. For any0< κ≪1, there exist α^′ >0 and a set Jκ satisfying

|I − Jκ| ≤Cκ^α+1r+2 Such that ∀p∈ Jκ one has

|Ω^N·k+e1Ωj+e2Ω_l| ≥ κ N^α′ (4.3)

for any k∈Z^N, |ei| ≤1,|j| ≥ |l|> N, and |k|+|e1|+|e2| 6= 0, |k| ≤r+ 2.

Proof. One has an estimate for Ω_j (and Ω_l also) Ω_j = 2π

√ac

b j+a_j with|a_j| ≤ C j. Hence

e₁Ω_j+e₂Ω_l= 2π

√ac

b (e₁j+e₂l) +e₁a_j+e₂a_l.

If j, l > CN^α/κ then e₁a_j +e₂a_l is just an irrelevant terms, so that the estimate (4.3) follows from Lemma 4.6 with β = 2π^√_b^ac. For the case j, l ≤ CN^α/κ, we reapplies the lemma withN, rreplaced byN^′ =CN^α/κandr^′=r+2.Then the nonresonance condition (4.3) holds providedα^′ =α²≈r⁶and the complement set is measured by a constant times κ^α+1r+2.

Proof for Theorem 4.2. DefineJ :=S

κ>0Jκ then it satisfies the Theorem.

5. Proof of the Theorem 2.1

SinceP has tame modulus, there exists a positive numberµ_# such that h|P|is,µ≤Aµ², ∀0< µ≤µ_#

(5.1)

hereA= _2π³

|b|. Fixr_∗ >1, N >1,for any r≤r_∗,we define µ_r andµ_∗ by µ_∗ := κ

48er_∗N^αA (5.2)

µ_r :=µ(1− r

2r_∗) ∀µ_r≤µ_#, u_r µ_∗ ≤ 1

2. (5.3)

Then by induction, we can construct sequences of Hamiltonianχ_r, H_r, Z_r and continuous functionsfr for 0< r≤r^∗ as in (3.15). Precisely

Proposition 5.1. Fixr_∗ >1, N >1,for anyr ≤r_∗,there exists a canonical transforma- tion ϕ_r =ϕ_χr which puts H in the form

H_r:=H◦ϕ_r=H₀+Z_r+f_r+R^Nr +R^Tr

(5.4)

here Zr is a polynomial of degree at mostr+ 2 having a zero of order 3 at the origin and is in (κ, α, N)−normal form; f_r is a polynomial having a zero of order r+ 3at the origin.

Moreover

h|Zr|i^s,µr ≤

(0 r= 0 Aµ²Pr−1

l=0(_µ^µ

∗)^l r≥1 (5.5)

h|f_r|is,µr ≤Aµ²( µ µ_∗)^r (5.6)

the remainder terms R^Nr ,R^Tr ∈C^∞(B_s(µ_r))satisfy kX_R^T

rks,µr ≤2^rµ²(µ µ_∗)^r (5.7)

kX_RN

r ks,µr ≤ µ² N^s−1 (5.8)