Existence of quasiperiodic solutions for the second-order approximation Boussinesq system

Nonlinear Analysis 69 (2008) 3734–3748www.elsevier.com/locate/na

Existence of quasiperiodic solutions for the second-orderapproximation Boussinesq system

Claudia Valls

Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal

Received 19 July 2007; accepted 4 October 2007

Abstract

In this paper we study analytically a class of waves in the variant of the classical second-order approximation Boussinesq systemgiven by

∂t u − b∂xxt u + c∂xxxxt u = −∂xv − ∂x (uv)−

(13− 2b

)∂xxxv + b∂xxx (uv)+ b∂x (u∂xxv)− a∂xxxxxv,

∂tv − b∂xxtv + c∂xxxxtv = −∂x u +12∂xxx u − v∂xv − ∂x (u∂xx u)+ b∂x (v∂xxv)− d∂xxxxx u,

where a, b, c, d are some real constants. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless,we show that, for almost every a, b, c and d it admits solutions that are quasiperiodic in time.c© 2007 Elsevier Ltd. All rights reserved.

MSC: primary 35Qxx; 37Kxx

Keywords: Boussinesq system; Normal forms; Hamiltonian formalism

1. Introduction

We study two-directional waves on the surface of an inviscid fluid in a flat channel without excluding the effects ofwave interactions and/or wave reflections. In this case, a restricted four-parameter family of systems (see [2]),

∂t u − B∂xxt u + B1∂xxxxt u = −∂xv − ∂x (uv)− A∂xxxv + B∂xxx (uv)

−

(A + B −

13

)∂x (u∂xxv)− A1∂xxxxxv

∂tv − D∂xxtv + D1∂xxxxtv = −∂x u − C∂xxx u − v∂xv − C∂xx (v∂xv)− ∂x (u∂xx u)

+ (C + D − 1)∂xv∂xxv + (C + D)v∂xxxv − C1∂xxxxx u

(1)

may be used. This system represents the second-order approximations to the Euler equations that govern the waves onthe surface on an ideal fluid under the force of gravity. The dimensionless variables u(x, t), v(x, t), x and t are scaled

E-mail address: [email protected].

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.10.009

http://www.elsevier.com/locate/na

mailto:[email protected]

http://dx.doi.org/10.1016/j.na.2007.10.009

C. Valls / Nonlinear Analysis 69 (2008) 3734–3748 3735

by the length scale h0 and time scale (h0/g)1/2 where h0 denotes the still water depth and g denotes the accelerationof gravity. The variable u(x, t) is the non-dimensional deviation of the water surface from its undisturbed position andv(x, t) is the non-dimensional horizontal velocity at a height above the bottom of the channel corresponding to θh0with 0 ≤ θ ≤ 1. The constants A, B,C, D are called dispersive constants which satisfy the physical constraints

A + B + C + D =13

and C + D =1− θ2

2≥ 0.

It is shown in [1,3] that this system has the capacity to capture the main characteristics of the flow in an ideal fluid.In this paper, we will study system (1) when A = 1

3 − 2D, C = − 12 , D = B := b, D1 = B1 := c, A1 := a,

C1 = d. Note that in this case Eq. (1) becomes

∂t u − b∂xxt u + c∂xxxxt u = −∂xv − ∂x (uv)−

(13− 2b

)∂xxxv + b∂xxx (uv)+ b∂x (u∂xxv)− a∂xxxxxv,

∂tv − b∂xxtv + c∂xxxxtv = −∂x u +12∂xxx u − v∂xv − ∂x (u∂xx u)+ b∂x (v∂xxv)− d∂xxxxx u,

(2)

where a, b, c, d are real constants and where we have used that

12∂xx (v∂xv)+

(b −

32

)∂xv∂xxv +

(b −

12

)v∂xxxv = b∂x (v∂xxv).

We set

w1,k(a, b) = 1−(

13− 2b

)k2+ ak4, w2,k(d) = 1+

12

k2+ dk4 (3)

and we restrict the values of a, b, c and d to the set S1 defined by

S1 =

(a, b, c, d) ∈ R4

: w1,k(a, b)w2,k(d) < 0 for k ∈ 1, 2.

We note that we could have also worked with the values of the parameters A, B,C, D, A1, B1,C1, D1 satisfying

(1− Ak2+ A1k4)(1− Ck2

+ C1k4)

(1+ Dk2 + D1k4)2< 0 (4)

for some k ∈ 1, . . . , N and some finite integer N ≥ 2. For any other value of the parameters in (1) notsatisfying Eq. (4), additional techniques may be used in order to prove the existence of quasiperiodic solutions forsystem equation (2). However, due to tedious calculations in the computation of the Hamiltonian, of the normalform and in the verification of the nondegeneracy conditions for the KAM theory, we restrict to the values ofA, B,C, D, A1, B1,C1, D1 explained above with a, b, c, d ∈ S1.

The present paper deals with the existence of periodic and quasiperiodic solutions (with two frequencies) for systemEq. (2) that generalize the linear oscillations of the normal flow to the complete system. These solutions are obtainedinside a certain invariant center manifold for the dynamics, which turns out to be finite-dimensional. The approach ofreducing to a center manifold can be traced back to the work of Kirchgassner [7], and the method is sometimes calledKirchgassner reduction. The application of this approach strongly depends on the desired results and appropriateadditional techniques may need to be developed in each particular case, mainly due to the infinite-dimensional natureof the problems. For later developments and applications of this method we refer the reader to the works [4–6,8–10,12]and to the references therein.

Once we have proven the existence of a center manifold, we will study the dynamics of the restriction to this centermanifold. To do it, first we observe that system (2) can be written as a Hamiltonian in infinitely many coordinates,which turns out to be real analytic near the origin. Then we will compute the Birkhoff normal form up to fourth orderof the Hamiltonian restricted to the center manifold. This will allow us to check that the nondegeneracy conditionsfor the KAM theorem are satisfied and thus by means of KAM techniques to prove the existence of quasiperiodicsolutions for this transformed Hamiltonian on the center manifold.

The paper has been organized in the following way: in Section 2 we provide the main result of this paper and itsproof by stating some auxiliary results like the existence of finite center manifolds, the computation of the Hamiltonian

3736 C. Valls / Nonlinear Analysis 69 (2008) 3734–3748

formalism and the Birkhoff normal form. For clearness and an straightforward reading of the paper, we have postponedthe proof of those auxiliary results to two Appendices. More concretely, Appendix A deals with the proof of theexistence of the center manifold, while Appendix C deals with the complete description and computation of theBirkhoff normal form up to the fourth order. Finally, we have included an Appendix B which contains some technicallemmas used in the computation of the normal form in Appendix C.

2. Setup and main result

2.1. Notations and existence of center manifolds

We consider the system (2) on the finite x-interval [0, 2π ], with the periodic boundary condition u(t, 0) = u(t, 2π),v(t, 0) = v(t, 2π), ux (t, 0) = ux (t, 2π), vx (t, 0) = vx (t, 2π) for 0 ≤ t < ∞ and zero mean,

∫ 2π0 u(t, x)dx = 0,∫ 2π

0 v(t, x)dx = 0. Furthermore, we rewrite it as a Hamiltonian system. Namely, introducing the variable z = (u, v)and the Hamiltonian

H(z) = −12

∫ 2π

0

[u2+ v2−

(13− 2b

)(∂xv)

2+ a(∂xxv)

2+

12(∂x u)2 + d(∂xx u)2 + uv2

− 2duv∂xxv + u2∂xx u

]dx (5)

the system (2) can be written in the form

∂t z = J∇z H(z) with J =

(0 J1J1 0

),

where

J1 = (1− b∂2x + c∂4

x )−1∂x .

We will study system (2) on the space Hm−10 ([0, 2π ]) × Hm−1

0 ([0, 2π ]), where Hm−10 ([0, 2π ]) is the Sobolev

space of functions on [0, 2π ] with periodic boundary conditions and zero mean. Moreover, m ≥ 1.We will work with the variables u(t, x) and v(t, x) decomposed into Fourier series with respect to the x variables.

We will focus our attention on the variables of the form

u(t, x) =∑k≥1

λk(t) cos(kx), v(t, x) =∑k≥1

bk(t) sin(kx). (6)

The variables u(t, x) and v(t, x) correspond to an invariant subspace of (formal) solutions of system (2). We note thatthe fact that we restrict to this subset guarantees that the Birkhoff normal form computed below is not resonant.

Note that the linear variational equation of (2) is

∂t u − b∂xxt u + c∂xxxxt u = −∂xv −

(13− 2b

)∂xxxv − a∂xxxxxv,

∂tv − b∂xxtv + c∂xxxxtv = −∂x u +12∂xxx u − d∂xxxxx u

(7)

with frequencies, for k ≥ 1νk(a, b, c, d) = ±iνk(a, b, c, d) where taking the notation

w5,k(b, c) = 1+ bk2+ ck4 (8)

we get

νk(a, b, c, d) = k

√w1,k(a, b)w2,k(d)

w5,k(b, c)2. (9)

We remark that by the choice of the subset S1, the frequency νk(α) is purely imaginary only for k ∈ 1, 2. Then, wehave elliptic behavior on two-dimensional tori. Restricting the values of a, b, c, d to the full Lebesgue measure set


S = a, b, c, d ∈ S1, νk(a, b, c, d) 6= 0,∀k ≥ 1,

applying the work of Mielke in [9, Chapter 2], we can prove for each r ≥ 3 the existence of a Cr center manifoldaround the origin, Wc, for the Eq. (2). For a precise statement of our center manifold theorem and a verification of theconditions which allow us to apply Mielke’s results see Appendix A. We list here some important properties of thecenter manifold:

(1) The space T0Wc is spanned by the eigenvectors of the linear part of (2) with purely imaginary eigenvalues. Thus,the restriction of the dynamics to the center manifold is finite-dimensional. We note that the center manifold isprecisely where one can expect to have invariant tori persisting, and thus, some kind of stability. Since in our casethe dynamics restricted to the center manifold is finite-dimensional this will allow us to use (finite-dimensional)perturbative techniques to establish this stability.

(2) The dynamics on the center manifold is Hamiltonian with a Hamiltonian that coincides with the restriction of theoriginal one in (5) to the center manifold (see [9] for details).

2.2. Main results

First, we note that by a well-known theorem in the book by Siegel and Moser [11, Section 16], since νk/νk0 6∈ Zfor 1 ≤ k 6= k0 ≤ 2 (see Eq. (9)) for every a, b, c, d ∈ S with an exception of a countable set, there exists aone-parameter analytic family of periodic orbits in (2). In view of this result we will now, in the rest of the paper,prove the existence of quasiperiodic solutions with two frequencies. Same results may also be true for the existenceof quasiperiodic solutions with more than two frequencies. However, the process imply very tedious calculations andtherefore, we do not compute it here.

We consider the action-angle variables (I j , θ j ) for Eq. (7) (see Eq. (13) for the definition) and let τ > 1. For eachreal number m > 0 we define the set

Dm = I j > 0: m ≤ I j ≤ 2m, j ∈ 1, 2 × T2.

We also consider the Diophantine condition

|〈w(a, b, c, d), k〉| ≥ γ ‖k‖−τ for all k ∈ Z2\ 0, (10)

where

w(a, b, c, d) =

(−π

√wk(a, b, c, d)

k

)k=1,2

,

where

wk(a, b, c, d) =

(1−

(13− 2b

)k2+ ak4

)(1+

12

k2+ dk4

)(1+ bk2

+ ck4)2.

We now state our main theorem.

Theorem 2.1. Given τ > 1 and γ > 0, for almost every a, b, c, d ∈ S such that w(a, b, c, d) satisfies Eq. (10), andevery m = m(a, b, c, d) sufficiently small, all invariant tori with frequencies w(a, b, c, d) persist in the perturbedHamiltonian equation (5) restricted to the center manifold, in the sense that there is a map φ: T2

→ Dm of class Cκ

with κ < 2τ + 9 such that the flow on Dm is the image by φ of the linear flow defined by w(a, b, c, d) on T2.

2.3. Sketch of the proof of Theorem 2.1

To prove Theorem 2.1, we will proceed as follows: first, we obtain a Hamiltonian in infinitely many coordinatesuk, vkk≥1, which is real analytic near the origin. The linear equation gives rise to the quadratic Hamiltonian H2. Thenonlinearity, gives rise to terms of degree three. Thus

H = H2 + H3


corresponding to an elliptic fixed point in infinitely many degrees of freedom. This computation of the Hamiltonianwill be done for k ≥ 1. Once we have the Hamiltonian expanded around u = v = 0, we perform the reduction to thecenter manifold. This process is based on performing a partial normal form scheme, uncoupling (up to a higher order)the hyperbolic directions for the elliptic ones). The restriction to the invariant manifold is a finite-dimensional degreesof freedom Hamiltonian system with an elliptic equilibrium at the origin. To study the dynamics inside this restrictedHamiltonian, i.e., to look for the quasiperiodic solutions inside this center manifold, we use KAM techniques. Thosetechniques always require some nondegeneracy conditions, and thus, we need to compute the Birkhoff normal formup to fourth order for Eq. (5).

The upshot is that, for almost every a, b, c, d ∈ S, there exists a change of coordinates so that

H = H2 + H4 + O(‖(u, v)‖5), with H4 =∑

(k,l)∈1,22H4,k,lλkλkblbl .

Thus, the Hamiltonian is integrable up to fourth order. Now, KAM theory comes into play. It is straightforwardto verify the nondegeneracy conditions for the KAM theory for almost every a, b, c, d ∈ S. Once we prove thenondegeneracy conditions for the KAM theory, we will show that, given the nondegeneracy conditions, we can reducethe Hamiltonian in the center manifold to a Birkhoff normal form up to terms of sufficiently high order. By takingm sufficiently small in the definition of the domain Dm , the Hamiltonian will be a small perturbation of the Birkhoffnormal form. This will allow us to reduce Theorem 2.1 to a suitable quantitative KAM theorem of Zehnder in [13].

It is important to note that the size of the perturbations to which the KAM theorem is applied corresponds tothe size of the difference between the Hamiltonian and its Birkhoff normal form. Hence, it is possible to choose theneighborhood in the center manifold, or in Dm , to be so close to the origin so that all necessary smallness conditionsare verified. On the other hand, since the reduction to the Birkhoff normal form involves a change to polar coordinates,which are degenerate near the origin, it is necessary to obtain a quantitative KAM theorem (see Theorem 3.2 below),which will be used to overcome some problems arising from the polar coordinates.

2.4. The Hamiltonian formalism

We consider the Hamiltonian for Eq. (2). We will work with the variables u(t, x) and v(t, x) as in (6). From (5)and by Parseval’s identity, we obtain that the Hamiltonian H(q, p) can be written as

H(λ, b) = −12

∫ 2π

0

[u2+ v2−

(13− 2b

)(∂xv)

2+ a(∂xxv)

2+

12(∂x u)2 + d(∂xx u)2 + uv2

− 2duv∂xxv + u2∂xx u

]dx = H2(λ, b)+ H3(λ, b),

with λ = (λk)k≥1, b = (bk)k≥1 and H2, H3 given by

H2 = −π∑k≥1

λ2kw1,k(a, b)+ b2

kw2,k(d),

with w1,k(a, b) and w2,k(d) introduced in (3) and

H3 = −π

4

∑k,l≥1

δσ (l − k)[1+ 2d(klσ )

2]λkblbkl

σ+π

4

∑k,l≥1

(klσ )

2λkλlλklσ,

where we have introduced the notation klσ = |k + (−1)σ l|, with σ ∈ 0, 1 and

δσ (l − k) =

1, if σ = 0 or σ = 1 and l > k−1, if σ = 1 and l < k.

Now we compute the expression of J in the Fourier series basis. To this end, we take again the expression of u(t, x)and v(t, x) given in Eq. (6) and the fact that

J(u

v

)=

((1− b∂2

x + c∂4x )−1∂xv

(1− b∂2x + c∂4

x )−1∂x u

).


The real form of J , in the coordinates λk, bkk≥1 is

J =

J1

J2. . .

Jk. . .

, Jk =

0k

1+ bk2 + dk4

−k

1+ bk2 + dk4 0

,

and the symplectic two-form Ω , is

Ω =∑k≥1

k

1+ bk2 + dk4 [dλk ∧ dbk].

It is very convenient to simplify H2, such that it has a diagonal form. To do it, we recall the definition of w5,k(b, c) in(8) and introduce the variables

qk =

√k

2w5,k(b, c)

(bkw

1/42,k (d)

w1/41,k (a, b)

− iλkw

1/41,k (a, b)

w1/42,k (d)

),

and

pk =

√k

2w5,k(b, c)

(λkw

1/41,k (a, b)

w1/42,k (d)

− ibkw

1/42,k (d)

w1/41,k (a, b)

).

With respect to those variables, we can define the Poisson bracket of two functionals S(p, q) and R(p, q) as

S, R =∑n≥1

∂S

∂pn

∂R

∂qn−∂S

∂qn

∂R

∂pn

.

Furthermore, in the variables p, q

H2 = −2π i∑k≥1

µk(a, b, c, d)pkqk, µk(a, b, c, d) =

√w1,k(a, b)w2,k(d)w5,k(b, c)

k(11)

and setting N1 = 0, 1 × N2,

F0k,l,kl

σ=

w1/25,k (b, c)w1/2

5,l (b, c)w1/25,kl

σ(b, c)√

klklσ

,

F1k,l,kl

σ=

w1/42,k (d)w

1/41,l (a, b)w1/4

1,klσ(a, b)

w1/41,k (a, b)w1/4

2,l (d)w1/42,kl

σ(d)

,

F2k,l,kl

σ=

w1/42,k (d)w

1/42,l (d)w

1/42,kl

σ(d)

w1/41,k (a, b)w1/4

1,l (a, b)w1/41,kl

σ(a, b)

,

we obtain

H3 = −π

8√

2

∑(σ,k,l)∈N1

δσ (l − k)[1+ 2d(klσ )

2]F0

k,l,klσ

F1k,l,kl

σ−pk pl pkl

σ− iqk pl pkl

σ+ pkql pkl

σ

− qkql pklσ+ i pk plqkl

σ− qk plqkl

σ+ pkqlqkl

σ+ iqkqlqkl

σ

+π

8√

2

∑(σ,k,l)∈N1

δσ (l − k)(klσ )

2 F0k,l,kl

σF2

k,l,klσpk pl pkl

σ+ iqk pl pkl

σ+ ipkql pkl

σ− qkql pkl

σ

+ ipk plqklσ− qk plqkl

σ− pkqlqkl

σ− iqkqlqkl

σ. (12)


2.5. The process of normal form

In this subsection we compute the normal form up to fourth order of the reduction to the center manifold for theHamiltonian H = H2 + H3 (introduced in (11) and (12)). This means that we want to compute a normal form suchthat on the center manifold H = H2 + H4 + R where R contain the terms with degree greater than or equal to five.

The process of reducing to the center manifold is based on removing some monomials in the expansion of theHamiltonian in order to produce an invariant manifold tangent to the elliptic directions of H2, that is, the modeswith indices k ∈ 1, 2. To do it, and since we already have proven the existence of a center manifold, we just needto cancel the monomials in H3 with one hyperbolic direction (i.e., the monomials with k > 2) and which, in thisdirection, has degree one (this ensures that when restricted to the elliptic directions (i.e., the monomials with k ≤ 2),the Hamiltonian H = H2 + H3 exhibits a decoupling in these two directions and approximates the dynamics in thecenter manifold up to third degree). Indeed, considering that H = H2 + H3 with H2 given in (11) and the monomialspk, qk with k > 2 in H3 have degree greater than or equal to two, then setting in H , pk = qk = 0 for k > 2, we getthat the equations of motion coming from H satisfy pk = qk = 0 for k > 2. Thus, the set

C = pk = qk = 0 for k > 2

is invariant under the Hamiltonian flow and H restricted to C only contain monomials p j , q j with j ≤ 2 and thus,represents the dynamics inside the center manifold up to third degree. Then, we do the same process for the fourth-order terms in the new Hamiltonian. Since we already have proved the existence of the center manifold, and in thissection we just want to compute it approximately up to fourth degree, together with the fact that to compute a termin the expansion of the center manifold we only need to work with a finite number of Fourier modes (which is analgebraic expression of the coefficients of the Hamiltonian), we can proceed formally ignoring questions of domainsof the operators and convergence of series.

The restriction of H to the center manifold, that we denote by ΠWc (H), will be obtained in the following manner.We define the action-angle variables Ik and θk by

pk =√

Ikeikθk , qk = −i√

Ike−ikθk , k = 1, 2. (13)

We will show that

ΠWc (H) = H2 + H4 + R

where H2 contains the second-order terms, H4 contains the fourth-order terms and R contains the term of order atleast five.

Theorem 2.2. For almost every a, b, c, d ∈ S, there exists a canonical change of coordinates that takes the reductionof H to the center manifold into its normal form up to fourth order, i.e.,

ΠWc (H) = H2 + H4 + R,

where R contains the terms of degree at least five,

H2 = −π(µ1(a, b, c, d)I1 + µ2(a, b, c, d)I2)

with µ1(a, b, c, d) and µ2(a, b, c, d) were introduced in (11) and (I1, I2) as in (13). Moreover,

H4 =−π

256(A1,1 I 2

1 + A1,2 I1 I2 + A2,2 I 22 ),

with A1,1, A1,2 and A2,2 given respectively by Eqs. (25), (27), (28) and (26). Furthermore, we have H4 = 〈AI, I 〉with I = (I1, I2) and

A =

(A1,1 A1,2/2

A1,2/2 A2,2

)with det(A) 6= 0.

The proof of this theorem is rather technical and relies on a large number of calculations. In order to make thepaper more readable, the proof is deferred to Appendix B.


3. Proof of Theorem 2.1

In this section we establish the existence of quasiperiodic solutions for the restriction to the center manifold of theHamiltonian H = H2 + H3 given in (11) and (12). Set I = (I1, I2), θ = (θ1, θ2) where (I j , θ j ) were introducedin (13).

Proposition 3.1. For almost every a, b, c, d ∈ S, there exists a canonical change of variables C defined in aneighborhood of the origin such that

ΠWc (H C) = H2(I )+ H4(I )+ · · · + H2N (I )+ R(I, θ),

where H2N (I ) are polynomials in the I -variables of degree N vanishing at the origin and R(I, θ) are divisible by√

I 2N+1.

The proof of the Proposition 3.1 is straightforward if one restricts the values of the parameter α to a full Lebesguemeasure set.

By Theorem 2.2,

H2(I ) = 〈ω, I 〉, H4(I ) = 〈AI, I 〉,

with ω = −π(µ1(a, b, c, d), µ2(a, b, c, d)) (see Eq. (11) for the definition of µ1 and µ2) and A was introduced inTheorem 2.2. To establish the nondegeneracy conditions of KAM theory, we need to prove that det(A) 6= 0, which inthis case is immediate in view of Theorem 2.2 if one restricts to a full Lebesgue measure set of values of α.

Once the Hamiltonian is transformed into the Birkhoff normal form up to order 2N with the nondegeneracyproperty, we now need to prove the existence of invariant Cantor manifolds for Hamiltonians in such normal forms.Since we are dealing with nonlinearities which are not analytic (only Cr ) we will deduce the result from the technicalKAM theorem of Zehnder [13]. To state the quantitative version of the KAM theorem by Zehnder (see below), werecall that we say that the functions F1, F2, . . . , Fn are in involution if Fi , F j = 0 for i 6= j . Moreover, theyare independent if the one-forms d F1, . . . , d Fn are linearly independent over a full Lebesgue measure subset of thecommon definition domain of F j for j = 1, . . . , n. A Hamiltonian system with n degrees of freedom having nindependent functions which are constant over the trajectories of the system and are in involution is called integrable.

Theorem 3.2. Let F0( I ) be a real analytic, integrable Hamiltonian which is nondegenerate (that is, det ∂2 F0

∂ I 2 ( I ) 6= 0).

Moreover, assume that the perturbed Hamiltonian F( I , θ ) = F0( I )+ F1( I , θ ), on

I ∈ Am = I j : m ≤ I j ≤ 2m, 1 ≤ j ≤ N , θ ∈ TN

is of class C l with l ≥ 4τ + 10. Then, there exists K depending on τ > 1, γ such that if |F1|C l (Am×TN ) ≤ K , then all

the invariant tori for the frequencies w = ∂ I F0( I ) satisfying

|〈w, k〉| ≥ γ ‖k‖−τ for all k ∈ ZN\ 0, (14)

persist in the perturbed Hamiltonian F( I , θ ) in the sense that there is a map φ : TN→ Am × TN , where

φ ∈ Cκ(Am × TN ) with κ = l − 2τ − 1, and the flow on Am × TN is the image by φ of the linear flow givenby ω on TN .

We can finally show how Theorem 3.2 implies Theorem 2.1.

Proof of Theorem 2.1. First of all, notice that from Proposition 3.1 the Hamiltonian

H2(I )+ H4(I )+ · · · + H2N (I ) (15)

is a real polynomial of degree N in the I -variables and, thus, it is real analytic as a function of I . Furthermore, it isintegrable. From the hypothesis of Theorem 2.1, w(a, b, c, d) = ∂I H2(I ) satisfies Eq. (10), i.e., for τ > 1,

|〈w(a, b, c, d), k〉| ≥ γ ‖k‖−τ for all k ∈ Z2\ 0.

Thus, taking m = m(a, b, c, d) sufficiently small,

w = ∂I(H2(I )+ · · · + H2N (I )

)


also satisfies Eq. (14). Moreover, as pointed out before, H2(I ) + H4(I ) is nondegenerate. Therefore, takingm = m(a, b, c, d) sufficiently small, the Hamiltonian given in Eq. (15) is also nondegenerate.

By Proposition 3.1, we have that R(I, θ) is divisible by√

I 2N+1, and, thus,

‖R‖C l (Am×T2) ≤ C(‖R‖C N (Am×T2)

)m

2N+12 −l .

Therefore, for a given l ≥ 4τ +10, taking N > 2l in Proposition 3.1, and also taking m small enough, we can achieve

‖R‖C l (Am×T2) ≤ K

for any positive K > 0. Therefore, taking K sufficiently small, Theorem 3.2 implies that the Hamiltonian H hasC l−2τ−1-invariant tori on which the flow is linear.

Acknowledgements

The author wishes to thank Luis Barreira for his useful comments and remarks. The work was supported by theCenter for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundacao para a Ciencia e aTecnologia by the Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/26465/2006.

Appendix A

For the convenience of the reader, we have included this appendix where we make precise the statement of thecenter manifold theorem. We also present a sketch of the proof.

Theorem A.1. For each r ≥ 1, there is a Cr finite-dimensional manifold, Wc ⊂ Hm−10 ([0, 2π ])×Hm−1

0 ([0, 2π ]), form ≥ 2, containing 0, which is locally invariant under Eq. (2) in some neighborhood of the origin, and such that T0Wcis obtained from the spanning of the linear part of Eq. (2) with purely imaginary eigenvalues, i.e., the eigenvaluesνk(a, b, c, d)k∈1,2 (see Eq. (9)).

Proof. This theorem is a straightforward application of Theorem 2.1 in Mielke [9].We just need to verify the conditions in this result. The eigenvalues νk(a, b, c, d)k≥1 of the linearized part

K : Hm0 ([0, 2π ])× Hm

0 ([0, 2π ])→ Hm−10 ([0, 2π ])× Hm−1

0 ([0, 2π ]),

with

K (u, ut ) =

(−(1− b∂2

x + c∂4x )−1∂x (1+ (1/3− 2b)∂2

x + a∂4x )v

−(1− b∂2x + c∂4

x )−1∂x (1− 1/2∂2

x + d∂4x )u

)are given in (9). Therefore, only finitely many lie on the imaginary axis, and there are no zero eigenvalues (sincewe have the zero mean condition). Furthermore, K can be written as K = K1 + K2 in which K1 corresponds to theprojection in the imaginary space and so et K1 generates a strongly continuous group that satisfies ‖eK1t

‖ ≤ C(1+|t |)m

for some C,m > 0 and K2 satisfies the resolvent estimate

‖(K2 − iξ)−1‖ ≤

C

1+ ‖ξ‖, ξ ∈ R,

for some C > 0. Furthermore, the nonlinearity is clearly Cr+1 from

Hm0 ([0, 2π ])× Hm

0 ([0, 2π ])→ Hm−10 ([0, 2π ])× Hm−1

0 ([0, 2π ]).

We have thus verified all the conditions which allow us to apply Theorem 2.1. Our statement is simply a reformulationof Theorem 2.1 in Mielke [9].

Appendix B

In this appendix we will establish two technical lemmas. We consider that n is a fixed integer.


Lemma B.1. The following relations hold:∑k≥1

∑σ=0,1

δσ (n − k)Ak,n,knσ

pkqknσ=

∑k≥1

∑σ=0,1

(−1)σ+1 Aknσ ,n,k pkn

σqk,∑

k≥1

∑σ=0,1

δσ (k − n)Ak,n,knσ

pkqknσ=

∑k≥1

∑σ=0,1

δσ (k − n)Aknσ ,n,k pkn

σqk .

Proof. We will only prove the first identity since the other is done in the same way. From the definition of knσ and

δσ (n − k) in Section 2.4 we have∑k≥1

∑σ=0,1

δσ (n − k)Ak,n,knσ

pkqknσ=

∑k≥1

Ak,n,k+n pkqk+n +

n−1∑k=1

Ak,n,n−k pkqn−k −∑

k≥n+1

Ak,n,k−n pkqk−n

=

∑r≥n+1

Ar−n,n,r pr−nqr +

n−1∑r=1

An−r,n,r pn−r qr −∑r≥1

Ar+n,n,r pr+nqr

=

∑k≥1

∑σ=0,1

(−1)σ+1 Aknσ ,n,k pkn

σqk .

Lemma B.2. The following relations hold:

∂

∂pn

∑k,l≥1

∑σ=0,1

δσ (k − l)Ak,l,klσ

pklσ=

∑k,l≥1

∑σ=0,1

δσ (k − l)∂Ak,l,kl

σ

∂pnpkl

σ+

∑k≥1

∑σ=0,1

(−1)σ+1 Ak,knσ ,n,

∂

∂pn

∑k,l≥1

∑σ=0,1

δσ (l − kl)Ak,l,klσ

pklσ=

∑k,l≥1

∑σ=0,1

δσ (l − k)∂Ak,l,kl

σ

∂pnpkl

σ+

∑k≥1

∑σ=0,1

δσ (l − k)Ak,knσ ,n .

Proof. We will only prove the first relation since the second one is done in the same way.

∂

∂pn

∑k,l≥1

∑σ=0,1

δσ (l − k)Ak,l,klσ

pklσ

=

∑k,l≥1

∑σ=0,1


σ

∂pnpkl

σ+

n∑k=1

Ak,n−k,n +∑

k≥n+1

Ak,k−n,n −∑k≥1

Ak,k+n,n

=

∑k,l≥1

∑σ=0,1


σ

∂pnpkl

σ+

∑k≥1

∑σ=0,1

(−1)σ+1 Ak,knσ ,n .

Appendix C. Proof of Theorem 2.2

C.1. Removal of some third-order terms in the Hamiltonian

In view of the explanation given above, to produce a Hamiltonian H3 of order three with at most one hyperbolicdirection and which, in this direction, has degree one, we need to cancel, if possible, in H3 all the monomials suchthat for any σ ∈ 0, 12, have indices (k, l, kl

σ ) ∈ Γ3(α) where

Γ3 := Γ3(a, b, c, d) =4⋃

j=1

Γ3, j (a, b, c, d)

with Γ3,1(a, b, c, d) = (k, l, klσ ) ∈ N3

: (k, l, klσ ) ∈ 1, 23 and

Γ3,2(a, b, c, d) = (k, l, klσ ) ∈ N3

: (k, l) ∈ 1, 22 \ Γ3,1(a, b, c, d),

Γ3,3(a, b, c, d) = (k, l, klσ ) ∈ N3

: (k, klσ ) ∈ 1, 22 \ Γ3,1(a, b, c, d),

Γ3,4(a, b, c, d) = (k, l, klσ ) ∈ N3

: (l, klσ ) ∈ 1, 22 \ Γ3,1(a, b, c, d).


We take Γ1 = X tG3|t=1 the time 1-map of the flow of the Hamiltonian vector field X G3

given by the Hamiltonian G3

defined as:

G3 =i

16√

2

∑(σ,k,l,kl

σ )∈Γ3

δσ (l − k)[1+ 2d(klσ )

2]F0

k,l,klσ

F1k,l,kl

σ−C−1,−1,−1

k,l,klσ

pk pl pklσ

− iC1,−1,−1k,l,kl

σqk pl pkl

σ+ C−1,1,−1

k,l,klσ

pkql pklσ− C1,1,−1

k,l,klσ

qkql pklσ+ iC−1,−1,1

k,l,klσ

pk plqklσ

−C1,−1,1k,l,kl

σqk plqkl

σ+ C−1,1,1

k,l,klσ

pkqlqklσ+ iC1,1,1

k,l,klσqkqlqkl

σ

−i

8√

2

∑(σ,k,l,kl

σ )∈Λ3

δσ (l − k)(klσ )

2 F0k,l,kl

σF2

k,l,klσC−1,−1,−1

k,l,klσ

pk pl pklσ

+ iC1,−1,−1k,l,kl

σqk pl pkl

σ+ iC−1,1,−1

k,l,klσ

pkql pklσ− C1,1,−1

k,l,klσ

qkql pklσ+ iC−1,−1,1

k,l,klσ

pk plqklσ

−C1,−1,1k,l,kl

σqk plqkl

σ− C−1,1,1

k,l,klσ

pkqlqklσ− iC1,1,1

k,l,klσqkqlqkl

σ, (16)

where, for a fixed (k1, k2, k3) ∈ N3, in the notation given in (11),

Cn,m,lk1,k2,k3

=1

nµk1(a, b, c, d)+ mµk2(a, b, c, d)+ lµk3(a, b, c, d).

It is easy to see by this definition that for almost every a, b, c, d ∈ S, G3 can be chosen so that it is well-defined(the denominators are different from zero) and, moreover, is a polynomial.

It is well-known that the transformed Hamiltonian, H , by the time 1-map of the flow of the Hamiltonian vectorfield XG3 is given by

H = H X tG3|t=1 = H2 + H3 + H2,G3 + H3,G3 +

12H2,G3,G3 +

12H3,G3,G3

+

∫ 2π

0

(1− t +

t2

2

)H,G3,G3,G3 X t

G3dt, (17)

in which H,G3 denotes the Poisson bracket of H = H2 + H3 and G3. It is clear that the terms of degree greaterthan or equal to five are

R =12H3,G3,G3 +

∫ 2π

0

(1− t +

t2

2

)H,G3,G3,G3 X t

G3dt. (18)

Furthermore, H3 + H2,G3 = H3 and due to the form of Γ3, this term does not contribute to the center manifold.

C.2. Computation of higher-order terms

With this choice of G3, and from Eqs. (17) and (18) we obtain H = H2 + H3 + H4 + R, where

H4 = H3,G3 +12H2,G3,G3 =

12

∑n≥1

∂H3

∂pn

∂G3

∂qn−∂H3

∂qn

∂G3

∂pn

. (19)

To compute H4, we will first compute the derivatives ∂H3/∂pn , ∂H3/∂qn , ∂G3/∂pn and ∂G3/∂qn . At this point, werecall that from the expressions of H3 and G3 given in (12) and (16) and Lemmas B.1 and B.2 we conclude that forn = 1, 2, 3, 4 we have

∂H3

∂pλ1n ∂qλ2

n

= −π

8√

2

∑σ=0,1

∑(k,kn

σ )∈N2

F0k,n,kn

σiλ2 Aλ1,λ2

k,n (σ )pk pknσ+ iλ2 Bλ1,λ2

k,n (σ )qkqknσ

+ 2iλ1Cλ1,λ2k,n (σ )qk pkn

σ, (20)


and

∂G3

∂pλ1n qλ2

n

=i

16√

2

∑σ=0,1

∑(k,kn

σ )∈N2

F0k,n,kn

σ(α)iλ2 Aλ1,λ2

k,n (σ )C−1,(−1)λ1 ,−1k,n,kn

σpk pkn

σ

+ iλ2 Bλ1,λ2k,n (σ )C1,(−1)λ1 ,1

k,n,knσ

qkqknσ+ 2iλ1Cλ1,λ2

k,n (σ )C1,(−1)λ1 ,−1k,n,kn

σqk pkn

σ, (21)

where

Aλ1,λ2k,n (σ ) = −δσ (k − n)F1

n,k,knσ(1+ 2d(kn

σ )2)− F2

k,n,knσ(2(kn

σ )2+ n2)

+ 2(−1)λ1δσ (n − k)F1k,n,kn

σ(1+ d(kn

σ )2+ dn2),

Bλ1,λ2k,n (σ ) = δσ (k − n)F1

n,k,knσ(1+ 2d(kn

σ )2)− F2

k,n,knσ(2(kn

σ )2+ n2)

+ 2(−1)λ1δσ (n − k)F1k,n,kn

σ(1+ d(kn

σ )2+ dn2),

Cλ1,λ2k,n (σ ) = (−1)λ1+1δσ (k − n)F1

n,k,knσ(1+ d(kn

σ )2+ dk2)+ (−1)λ1 F2

k,n,knσ((kn

σ )2+ n2

+ k2)

− δσ (n − k)F1k,n,kn

σ(1+ d(kn

σ )2+ dn2)+ (−1)σ+1 F1

knσ ,n,k

(1+ dk2+ dn2).

Furthermore, if we define the set

A4 =

m ∈ 0, 14, n ∈ 0, 14

∣∣∣∣∣ 4∑i=1

mi + ni = 4

,

then, H4 can be written as

H4 =∑

(σ,ρ)∈0,12

(r,k,l, j1, j2)∈N5, j1=krσ , j2=lrρ

m,n∈A4

h4,m,n,σ,ρk,l, j1, j2

pmkk pml

l pm j1j1

pm j2j2

qnkk qnl

l qn j1j1

qn j2j2, h4,m,n,σ,ρ

k,l, j1, j2∈ C. (22)

C.3. Restriction to the center manifold and normal form for the elliptic part of H4

Now we want to compute the restriction to the center manifold of H4 and approximate the dynamics to the centermanifold up to degree four. Thus, we need to make a normal form which only cancels the monomials in H4 withone hyperbolic direction and which, in this direction, has degree one. Furthermore, since we also want to put innormal form the monomials in H4 with elliptic directions to introduce the action-angle variables, we need to cancelthe monomials in H4 such that for any (σ, ρ) ∈ 0, 12, r ∈ N have indices (k, l, kr

σ , lrρ) ∈ Γ4 := Γ4(a, b, c, d) being

Γ4(a, b, c, d) =4⋃

j=1

Γ4 j (a, b, c, d)

with Γ4,1(a, b, c, d) = (k, l, krσ , l

rρ) ∈ N4

: (k, l, krσ , l

rρ) ∈ 1, 24 and

Γ4,2(a, b, c, d) = (k, l, krσ , l

rρ) ∈ N4

: (k, l, krσ ) ∈ 1, 23 \ Γ4,1(a, b, c, d),

Γ4,3(a, b, c, d) = (k, l, krσ , l

rρ) ∈ N4

: (k, l, lrρ) ∈ 1, 23 \ Γ4,1(a, b, c, d),

Γ4,4(a, b, c, d) = (k, l, krσ , l

rρ) ∈ N4

: (l, krσ , l

rρ) ∈ 1, 23 \ Γ4,1(a, b, c, d).

Therefore, we take Γ2 = X tG4|t=1 the time 1-map of the flow of the Hamiltonian vector field XG4 given by the

Hamiltonian G4 defined as:

G4 =∑

(σ,ρ)∈0,12

(r,k,l, j1, j2)∈N×Γ4, j1=krσ , j2=lrρ

m,n∈A4

g4,m,n,σ,ρk,l, j1, j2

pmkk pml

l pm j1j1

pm j2j2

qnkk qnl

l qn j1j1

qn j2j2, (23)


in which, with the notation, Dσ,ρk,l, j1, j2(a, b, c, d) =

∑4j=1 µξ j (a, b, c, d)(nξ j − mξ j ), with (ξ1, ξ2, ξ3, ξ4) = (k, l,

j1, j2), we have

g4,m,n,σ,ρk,l, j1, j2

=

0, if Dσ,ρk,l, j1, j2

(a, b, c, d) = 0

ih4,m,n,σ,ρk,l, j1, j2

πDσ,ρk,l, j1, j2(a, b, c, d)

, otherwise.

At this point it is easy to see that G4 is a polynomial and that for almost every a, b, c, d ∈ S , Dσ,ρk,l, j1, j2(a, b, c, d)

= 0 if and only if (σ, ρ, r, k, l, j1, j2) ∈ R where R denotes the set of values of

(σ, ρ, r, k, l, j1, j2) ∈ 0, 12 × 1, 2, 3, 4 × Γ4

such that, j1 = krσ , j2 = lr

ρ and mk = nk , ml = nl , m j1 = n j1 , m j2 = n j2 in Eq. (23).

In this case, the transformed Hamiltonian, H , by the time 1-map of the flow of the Hamiltonian vector field XG4 isgiven in a neighborhood of the origin by

H = H2 + H3 + H4 + R + H2,G4 + H3,G4 + H4,G4 + R,G4

+

∫ 2π

0(1− t)H ,G4,G4 X t

G4dt.

It is clear from the explanations given during this section, that H3 and H3,G4 do not contribute to the center man-ifold. Furthermore, from (11), we have that the restriction of H2 to the center manifold, H2 = ΠWc (H2), is given by

H2 = −π

2∑k=1

µk(α)Ik .

Moreover, the higher-order terms taking part in the center manifold are

R = ΠWc

(R + H4,G4 + R,G4 +

∫ 2π

0(1− t)H ,G4,G4 X t

G4dt

).

Furthermore, by the definition of G4 we obtain that H4 + H2,G4 = H4 + H4 where

H4

∑(σ,ρ,r,k,l, j1, j2)∈R

h4,m,n,σ,ρk,l, j1, j2

pmkk pml

l pm j1j1

pm j2j2

qnkk qnl

l qn j1j1

qn j2j2

and, from the choice of G4 (see the explanation given after equation Eq. (22)), the term H4 does not contribute to therestriction to the center manifold of H4. In order to be able to compute H4 (the fourth-order terms in the Hamiltonianrestricted to the center manifold) we need to obtain the setR. We proceed as follows: the only monomials taking partin the Hamiltonian H4 are, for k, l = 1, 2, pkqk plql . Thus, setting

Gk,l,krσ ,lr

ρ ,r = F0k,r,kr

σF0

l,r,lrρ

from (19)–(21) we get

H4 =π

256

∑(σ,ρ,r,k,l,kr

σ ,lrρ )∈R

Gk,l,krσ ,lr

ρ ,r [A1,0l,r (ρ)B

0,1k,r (σ )[C

1,1,1k,r,kr

σ+ C1,1,1

l,r,lrρ]qkqkr

σpl plr

ρ

+ A0,1l,r (ρ)B

1,0k,r (σ )[C

−1,1,−1k,r,kr

σ+ C1,−1,1

l,r,lrρ]qkqkr

σpl plr

ρ+ [C1,0

k,r (σ )C0,1l,r (ρ)C

1,1,−1l,r,lr

ρ

−C0,1k,r (σ )C

1,0l,r (ρ)C

1,−1,−1l,r,lr

ρ]qk pkr

σql plr

ρ]. (24)

Furthermore, (σ, ρ, k, l, r, krσ , l

rρ) ∈ R if and only if k = l and σ = ρ with (k, kr

σ , l, lrρ) ∈ 1, 24 or k = lr

ρ andl = kr

σ with (k, krσ , l, l

rρ) ∈ 1, 24. Then, a straightforward computation shows that (σ, ρ, k, l, r, kr

σ , lrρ) ∈ R if and

only if k = l = 1, r = 2, σ = ρ = 1; k = l = 2, r = 4, σ = ρ = 1; k = l = r = 1, σ = ρ = 0; k = l = 1, r = 3,


σ = ρ = 1; k = l = 2, r = 1, σ = ρ = 1; k = l = 2, r = 3, σ = ρ = 1; k = 1, l = 2, r = 1, σ = 0, ρ = 1;k = 1, l = 2, r = 3, σ = ρ = 1; k = 2, l = 1, r = 3, σ = ρ = 1 and k = 2, l = 1, r = 1, σ = 1, ρ = 0.

Also from the change of variables introduced in (13), we have

H4= −

π

256[A1,1 I 2

1 + A1,2 I1 I2 + A2,2 I 22 ],

where A1,1 corresponds to the indices k = 1, l = 1, r = 2, σ = ρ = 1, A2,2 corresponds to the indicesk = l = 2, r = 4, σ = ρ = 1 and A1,2 contain the rest of the indices (σ, ρ, r, l, k, kr

σ , lrρ) ∈ R such that

they contribute to neither A1,1 nor A2,2. For j = 1, 2, we use the notation x j = w1, j (a, b), y j = w2, j (b) andz j = w5, j (b, c).

Due to the fact that the exact value of A1,1, A2,2 and A1,2 imply very long calculations easily done with theMathematica, we will only indicate how to obtain A1,1, A1,2 and A2,2 without providing their explicit formulaeexcept for A1,1. The reader should obtain without problem (using, for example, Mathematica) the explicit values ofA1,2, A2,2 and det(A) (see below).

Then, again from (24) we have

A1,1 = G1,1,1,1,2[2A1,01,2(1)B

0,11,2 (1)C

1,1,11,2,1 + A0,1

1,2(1)B1,01,2 (1)[C

−1,1,−11,2,1 + C1,−1,1

1,2,1 ]

+C1,01,2(1)C

0,11,2(1)[C

1,1,−11,2,1 − C1,−1,−1

1,2,1 ]] =−8 A1,1

x1x3/22 y1√

y2z2(4√

x1 y1z1 +√

x2 y2z2), (25)

where

A1,1 = 72y21√

x1 y1x2 y2z1 − 2(1+ 7d + 10d2)x3/21 x3/2

2√

y1 y2z2 + x21(2√

x1 y1x2 y2z1

+ (1+ 2d + 2d2)x2 y2z2)+ 2x1 y1(12√

x1 y1x2 y2z1 + x2((1+ 5d)2x2 + 3y2)z2).

Furthermore,

A2,2 = G2,2,2,2,4[2A1,02,4(1)B

0,12,4 (1)C

1,1,12,4,2 + A0,1

2,4(1)B1,02,4 (1)[C

−1,1,−12,4,2 + C1,−1,1

2,4,2 ]

+C1,02,4(1)C

0,12,4(1)[C

1,1,−12,4,2 − C1,−1,−1

2,4,2 ]]. (26)

Now, we divide the term A1,2 into the following sums,

A1,2 = S1 + S2 + S3 + S4 + S5 + S6 + S7 + S8, (27)

where S1 corresponds to the indices (r, k, l, σ, ρ) = (1, 1, 1, 0, 0), S2 corresponds to the indices (r, k, l, σ, ρ) =(3, 1, 1, 1, 1), S3 corresponds to the indices (r, k, l, σ, ρ) = (1, 2, 2, 1, 1), S4 corresponds to the indices(r, k, l, σ, ρ) = (3, 2, 2, 1, 1), S5 corresponds to the indices (r, k, l, σ, ρ) = (1, 1, 2, 0, 1), S6 corresponds to theindices (r, k, l, σ, ρ) = (3, 1, 2, 1, 1), S7 corresponds to the indices (r, k, l, σ, ρ) = (3, 2, 1, 1, 1) and S8 correspondsto the indices (r, k, l, σ, ρ) = (1, 2, 1, 1, 0). Thus, using Eq. (24) we can make more explicit the sums Si , i = 1, . . . , 8.Doing so, we get

S1 = G1,1,2,2,1[2A1,01,1(0)B

0,11,1 (0)C

1,1,11,1,2 + A0,1

1,1(0)B1,01,1 (0)[C

−1,1,−11,1,2 + C1,−1,1

1,1,2 ]

+C1,01,1(0)C

0,11,1(0)[C

1,1,−11,1,2 − C1,−1,−1

1,1,2 ]]

S2 = G1,1,2,2,3[2A1,01,3(1)B

0,11,3 (1)C

1,1,11,3,2 + A0,1

1,3(1)B1,01,3 (1)[C

−1,1,−11,3,2 + C1,−1,1

1,3,2 ]

+C1,01,3(1)C

0,11,3(1)[C

1,1,−11,3,2 − C1,−1,−1

1,3,2 ]]

S3 = G2,2,1,1,1[2A1,02,1(1)B

0,12,1 (1)C

1,1,12,1,1 + A0,1

2,1(1)B1,02,1 (1)[C

−1,1,−12,1,1 + C1,−1,1

2,1,1 ]

+C1,02,1(1)C

0,12,1(1)[C

1,1,−12,1,1 − C1,−1,−1

2,1,1 ]]

S4 = G2,2,1,1,3[2A1,02,3(1)B

0,12,3 (1)C

1,1,12,3,1 + A0,1

2,3(1)B1,02,3 (1)[C

−1,1,−12,3,1 + C1,−1,1

2,3,1 ]

+C1,02,3(1)C

0,12,3(1)[C

1,1,−12,3,1 − C1,−1,−1

2,3,1 ]]

S5 = G1,2,2,1,1[2A1,02,1(1)B

0,11,1 (0)C

1,1,11,1,2 + A0,1

2,1(1)B1,01,1 (0)[C

−1,1,−11,1,2 + C1,−1,1

2,1,1 ]

+C1,01,1(0)C

0,12,1(1)C

1,1,−12,1,1 − C0,1

1,1(0)C1,02,1(1)C

1,−1,−12,1,1 ]


S6 = G1,2,2,1,3[2A1,02,3(1)B

0,11,3 (1)C

1,1,11,3,2 + A0,1

2,3(1)B1,01,3 (1)[C

−1,1,−11,3,2 + C1,−1,1

2,3,1 ]

+C1,01,3(0)C

0,12,3(1)C

1,1,−12,3,1 − C0,1

1,3(0)C1,02,3(1)C

1,−1,−12,3,1 ]

S7 = G2,1,1,2,3[2A1,01,3(1)B

0,12,3 (1)C

1,1,11,3,2 + A0,1

1,3(1)B1,02,3 (1)[C

−1,1,−11,3,2 + C1,−1,1

2,3,1 ]

+C1,02,3(0)C

0,11,3(1)C

1,1,−12,3,1 − C0,1

2,3(0)C1,01,3(1)C

1,−1,−12,3,1 ]

S8 = G2,1,1,2,1[2A1,01,1(0)B

0,12,1 (1)C

1,1,12,1,1 + A0,1

1,1(1)B1,02,1 (1)[C

−1,1,−12,1,1 + C1,−1,1

1,1,2 ]

+C1,02,1(0)C

0,11,1(1)C

1,1,−11,1,2 − C0,1

2,1(0)C1,01,1(1)C

1,−1,−11,1,2 ]. (28)

To prove the second statement of the proposition, we introduce the matrix

A =

(A1,1 A1,2/2A1,2/2 A2,2

).

A direct computation using Eqs. (25)–(28) and Mathematica implies that det(A) is an analytic function in theparameters a, b, c, d which is not zero and thus for almost every a, b, c, d ∈ S we get that det(A) 6= 0. Thus,the theorem is proved.

References

[1] J. Bona, M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D 15 (1998) 191–224.[2] J. Bona, M. Chen, J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Derivation

and linear theory, J. Nonlinear Sci. 12 (2002) 283–318.[3] J. Bona, G. Pritchard, L. Scott, An evaluation of a model equation for water waves, Philos. Trans. R. Soc. Lond. Ser. A 302 (1981) 457–510.[4] A Calsina, X. Mora, J. Sola-Morales, The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic

singular limit, J. Differential Equations 102 (1993) 244–304.[5] M. Groves, J. Toland, On variational formulations for steady water waves, Arch. Ration. Mech. Anal. 137 (1997) 203–226.[6] G. Iooss, K. Kirchgassner, Water waves for small surface tension: An approach via normal form, Proc. Roy. Soc. Edinburgh Sect. A

122 (1992) 267–299.[7] K. Kirchgassner, Wave-solutions of reversible systems and applications, J. Differential Equations 45 (1982) 113–127.[8] A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci. 10 (1988) 51–66.[9] A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, in: Lect. Notes in Math., 1489, Springer, 1991.

[10] A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations 110 (1994) 322–355.[11] C. Siegel, J. Moser, Lectures on Celestial Mechanics, in: Grundlehren der mathematischen Wissenschaften, 187, Springer, 1971.[12] C. Valls, Existence of quasi-periodic solutions for elliptic equations on a cylindrical domain, Comm. Math. Helvetici 81 (2006) 783–800.[13] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. II, Comm. Pure Appl. Math. 29 (1976)

49–111.

Documents

Existence of quasiperiodic solutions for the second-order approximation Boussinesq system