Lyapunov Center Theorem of Infinite Dimensional Hamiltonian Systems

Junxiang Xuand Qi Li

School of Mathematics,Southeast University,Nanjing,Jiangsu 210096,China

Abstract. In this paper we reformulate a Lyapunov center theorem of infinite dimensional Hamiltonian systems arising from PDEs.The proof is based on a modified KAM iteration for periodic case.

Key Words: Hamiltonian systems,KAM iteration,small divisors,lower dimensional tori.

1 Introduction

The KAM theory of infinite dimensional Hamiltonian systems arising from PDEs has been studied widely. Earlier works in this area are due to Wayne, Craig, Bourgain,Kuksin, P¨oschel, Eliasson and etc. [4,6–8,11,13,15]. More recently, the infinite dimensional KAM theory has been developed and there are many infinite dimensional KAMtype theorems. For some related results, see [1–3,5,9,10,12,14,18] and the references therein. We also refer to [17] for a survey on both finite and infinite dimensional KAM theory.

However,most of previous works mainly focus on quasi-periodic case,since periodic case can be regarded as a special quasi-periodic case. But Craig and Wayne constructed periodic solutions of the nonlinear wave equations and nonlinear Schr¨odinger equations with periodic boundary value by the Nash-Moser methods in weaker small divisor conditions, see [6,7]. Their results give an infinite dimensional version of the Lyaponuv center theorem.

In this paper, we want to extend the finite dimensional Lyapunov center theorem to infinite dimensional Hamiltonian systems. The result can be regarded as a complement of infinite dimensional KAM theorems. Our results can also be applied to some partial differential equations,but here we do not pursue this problem.

2 Main results

Consider a nearly integrable infinite dimensional Hamiltonian H = N+P. The normal form

where T is the usual 1-torus,R is the 1-dimensional real space and the Hilbert space

with

The frequency ω ∈O = (β1,β2) is regarded as parameter and the normal frequenciesΩ1,Ω2,··· are Lipschitz-continuous in the parameter ω.

The small perturbation P = P(θ,I,u,v) is analytic in (θ,I,u,v) and Lipschitzcontinuous in ω, here and below, the dependence of P in the parameter ω is usually implied and not written explicitly only for simplicity if there is no any confusion.

The Hamiltonian system of H is

where Ω=(Ω1,Ω2,···)and Ωu=(Ω1u1,Ω2u2,···).

If P=0,the system(2.1)is integrable and admits a family of invariant tori

on which there are periodic trajectories

If f(θ)is analytic on the trip domain D(s) = {θ||Imθ| ≤s},denote the super norm of f by

Define a little stronger norm by

If f also depends on ω ∈O,define the Lipschitz norm of f by

where

where we write f(θ)= f(θ;ω)for simplicity. Define by

Define

If w(θ,I,u,v) is an analytic map from D(s,r) to ℓa,¯pand Lipschitz-continuous in ω,define

Denote the Hamiltonian vector field of P by XP=(PI, −Pθ, Pv, −Pu).Define

Assumption 2.1(Asymtotics of Normal Frequencies). Suppose

where b > 0,¯b > 0,d > 1 and ¯d < d are constants. Moreover, there exist positive integers ¯N,j0>0 and a small positive number ¯α,such that for ∀j ≥j0,|Ωj+¯N−Ωj|≥¯α hold.

Assumption 2.2(Regularity of Perturbation). Let ¯p ≥p. Suppose P is analytic in(θ,I,u,v)on D(s,r)and Lipschitz-continuous in ω on O and XPis an analytic mapping from D(s,r)to Γa,¯pand Lipschitz-continuous in ω on O.

Theorem 2.1(Infinite Dimensional Lyapunov Center Theorem). Let

If Assumptions 2.1 and 2.2 hold,then there exist a positive integer J >0 and a sufficiently small constant γ>0 independent of є,such that for sufficiently small 0< α ≤1,if

and є ≤γα2,then there exists a nonempty Cantor-like subset Oα⊂O, a Lipschitz continuous family of tori embeddings Φ : T×Oα→Γa,¯pand a Lipschitz-continuous mapping ω∗: Oα→R, such that for ω ∈Oα, the map Φ restricted to T×{ω} is a real analytic embedding of a rotational torus with frequency ω∗(ω)for the Hamiltonian H at ω. Thus,for ω ∈Oα,Φ(ω∗t+θ0;ω)is a periodic trajectory of the system(2.1). Moreover,

where Φ0is the trivial embedding

and

where c is independent of є,α.

The measure estimates of OOαis based on the following theorem.

Theorem 2.2. If Assumption 2.1 holds and ωk: ω ∈O →R satisfying

then there exists J >0,such that

where Rkj(α)={ω||kωk±Ωj|≤α}and c is a constant independent of α.

Remark 2.1. It is well known that small divisor conditions are necessary for KAM theorems. However, in periodic case the small divisor condition is very special. In finite dimensional case,the small divisor condition can hold for the parameter on a interval,so the invariant tori exist continuously depending on the frequency ω. But this is not true for infinite dimensional Hamiltonian systems. Since there are infinitively many normal frequencies, the small divisor conditions can only hold on a Cantor-like subset even in the periodic case,which causes much difficulty in the estimates of Lipschitz-norms with respect to the parameter ω. Moreover, the smoothness of the parameter is in the sense of Whitney [9]. Furthermore, the Lipschitz-norms with respect to the parameter ω can result in some loss of regularity about the angle variable θ, so the traditional implicit function theorem is not valid for our problem,the proof is still based on KAM iteration.

Remark 2.2. The asymtotics of normal frequencies that Ωj≥bjdwith d > 1 is very important for small divisor conditions |kω±Ωj| ≥α. In fact, if 0 < d ≤1, the above small divisor conditions may not hold.

3 Proof of main theorems

At first we make the complex conjugate coordinates change

then the associated symplectic form is dθ ∧dI+idz ∧d¯z.In the new coordinates,

where〈·,·〉is the inner product in the Hilbert space ℓ2= ℓ0,0. Below,without confusion we use the same notations”ℓa,p”and”Γa,¯p”to denote their complexification respectively.Thus,D(s,r)is a complex neighbourhood of T×{0}×{0}×{0}in Γa,p.

The proof of Theorem 2.1 is based on a KAM iteration. We construct a sequence of symplectic maps Φ0,Φ1,···Φn−1such that the zero-th and the first order coefficients of H ◦Φ0◦Φ1◦···Φn−1with respect to I,z and ¯z become smaller and smaller and finally vanish. In this paper, the special small divisor conditions |kω±Ωj| ≥α do not cause the loss of smoothness, so the zero-th and the first order coefficients of H ◦Φ0◦Φ1◦···Φn−1with respect to I,z,¯z can become smaller and smaller. This is different from quasi-periodic case.

Below,in all estimates of KAM step we always use the same c to denote the constants of estimates which are independent of KAM steps.

Outline of KAM steps

Let H = N+P,where N = ωI+〈Ωz,¯z〉,P= R+ ¯P,where

where q = (q1,q2,···) ∈N∞and|q| = |q1|+|q2|+··· and ¯q has the same meaning as q. Let

Note that XP=(PI,−Pθ,iP¯z,−iPz).Thus we have

where

Let

3.1 Solving linear homological equation

The linear homological equation is as follows:

where{·,·}is the Poisson product, ˆN and F are functions to be solved.

By(3.2)we have

By(3.1)and(3.3a),it follows easily

In the same way we have

Thus we have

If

from(3.3b)that

Thus we obtain

For the Lipschitz-norms we have

If|Ωj|≥2β2|k|,then|kω+Ωj|≥|Ωj|.By Assumption 2.1,it follows

and so

Using Lemma A.1 and(3.1b)we have

In the same way as above,by(3.1c)we have

Similarly,for F001we have

3.2 Coordinates transformations

By integrating the first equation we have

where||·||s−3ρis the norm with respect to the new variable θ+. For the Lipschitz norm,also by Lemma A.2 we have

By Gronwall’s inequality we have

Now we consider the transformations for z and ¯z. Since

by Lemma A.2 and(3.10a)we have

In the same way as the above and using(3.10b)it follows that

By(3.17),(3.18)and Lemma A.1 we have

Moreover,by(3.10c),(3.10d)and Lemma A.2 it follows easily

Similarly,we have

Now we estimate the transformation of variable I,which is a little more complicated. By integrating the second equation of(3.11)we have

By Gronwall’s inequality we have

Now we estimate Lipschitz norm. By Lemma A.2 and using(3.4)and(3.6)we have

Since

by Lemma A.2 and using(3.9a),(3.9b),(3.17),(3.18),if є2≤αr,we have

Similarly we have

Thus,

By Gronwall’s inequality we have

By(3.22)and(3.23)and using Lemma A.1 we have

Without loss of generality we only estimate the second term and the forth term,since the other terms can be estimated similarly or even more simple. If

it is easy to see that

Using Lemma A.2,(3.5)and(3.15)we have

By Lemmas A.2 and A.1,and using(3.9a),(3.15),if є2≤αρ2,we have

By(3.25)and combining all the above estimates we have

By Gronwall’s inequality it follows that

To estimate Lipschitz-norm of Dθ+It,we still only consider the second term and the forth term in(3.25)without loss of generality.By Lemma A.2 and combining(3.5),(3.15),(3.16),(3.22)and(3.23),it follows easily that

In the same way as the above,if є2≤αρ2,by Lemmas A.1 and A.2 and using(3.9a),(3.9b)and(3.15)-(3.18)we have

By(3.25)and using the above estimates we have

By Gronwall inequality it follows that

Since

in the same way as the above and by(3.5)it follows that

For the derivatives of Itwith respect to z+,we have

In the same way as the above,by(3.5)and(3.7)it follows that

so,

For Lipschitz norm,by(3.5),(3.8)and Lemma A.1,we have

so

Similarly we have

Let s+= s −5ρ,r+=r −2δ. Combining above estimates we have

3.3 Estimates of new perturbation

Now we use Lemma 3.1 to estimate the Hamiltonian vector field of the new perturbation P+= P1+P2+P3,where

To estimate the new perturbation we need the following lemma.

Proof. Let

Then

By Lemma A.2 we have

Again,by(3.17),(3.21b)and(3.22)we have

So

Thus,we complete the proof.

Now we first consider P1. Note that

Since

by Lemma 3.1 and using(3.15),(3.24a),(3.19a)and(3.21c)it follows that

Moreover,by(3.15),(3.26),(3.20a),(3.21e),it is easy to see that

Thus,we have

To estimate Lipschitz norms,we first estimate the following norms:

Since

by Lemma 3.1 and using(3.14),(3.23),(3.18)and(3.21b)it follows that

By Lemma A.1 and Cauchy’s inequality it follows that

In the same way it follows that

If є2≤αρ2,combining all the above estimates we have

In the same way as the estimates of◦Φ, by Lemma A.2 and Cauchy’s inequality it follows that

Also we have

Thus,

where |||·|||a,¯pis the operator norm from ℓa,¯pto ℓa,¯p. By the generalized Cauchy’s inequality of analytic function on Banach space it follows that

Also note that

By (3.33), (3.15), (3.16), (3.24a) (3.24b), (3.20a), (3.20b), (3.21e), (3.21f) and combining(3.36),(3.38),(3.40)and(3.42a),we have

In the same way as the above,by(3.37),(3.39),(3.41)and(3.42b)we have

Since

by Lemma 3.1 and(3.29)we have

Moreover,by(3.38),(3.40),(3.29)and(3.30)it follows that

At I+=0,z+=0, ¯z+=0,we have

Similarly we have

At I+=0,z+=0, ¯z+=0,we have

Thus by(3.34),(3.45),(3.49)and(3.52a)we have

By(3.43),(3.46),(3.50)and(3.52b)we have

only consists of the lower order terms of P1with respect to I+,z+, ¯z+,and

consists of the higher order terms. By(3.35),(3.47),(3.51a)and(3.53a)we have

By(3.44),(3.48),(3.51b)and(3.53b)we have

Now we estimate P2. By the construction of F and using (3.4), (3.5), (3.7), (3.9a) and(3.9b)it follows easily that

Let

We only estimate Q2and Q3,since Q1and Q4can be estimated similarly. Since

by (3.58), (3.5), (3.9a), (3.10c), (3.19a), (3.21c) and (3.24a) and using Lemma 3.1 and Cauchy’s inequality it follows that if є2≤αr,then

Since

by(3.1b),(3.1c),(3.10a),(3.10c)and using Lemmas 3.1 and A.2 it follows that

Thus we have

In the same way as the above it follows that

Thus we have

Now we estimate P3. Since the two terms can be estimated similarly, we only consider the first one. By Lemma A.2 we have

Since

by Lemma A.2 it follows that

Thus,

Similarly,we have

KAM iteration

Now we go through the KAM iteration. Let

Then,by the previous estimates we have

Convergence of the KAM iteration

For m=1,we have

Then we have

Obviously,it follows that

By the induction assumption,we have

Now we verify all conditions of(3.64). By the estimates of the KAM steps,we can choose the sufficiently small γ such that the following inequalities hold:

Since

it follows that|ω∗−ω|≤cє and

Then

Moreover,

Then,the Φ in Theorem 2.1 is given by Φ(θ;ω)= Φ∗(θ,0,0,0;ω).

Thus,we complete the proof.

Appendix

In this section we give some lemmas which are useful in KAM steps.

Lemma A.1. If f(θ)is a 2π-periodic analytic function on D(s),then

Proof. The proof is easy and we omit the details.

Lemma A.2. Let Θ : θ+→θ(θ+)is a real analytic map from D(s −ρ)to D(s),which depends on the parameter ω and satisfies that for θ+∈D(s −ρ)

where є< ρ ≤1.Let f(θ)be a 2π-periodic function depending on a parameter ω. If f is analytic in θ on D(s)and Lipschitz continuous in ω on O,then

Acknowledgements

This study was funded by the National Natural Science Foundation of China(Nos.11871146 and 11671077).