Near-integrability of low dimensional periodic Klein-Gordon lattices

The low dimensional periodic Klein-Gordon lattices are studied for integrability. We prove that the periodic lattice with two particles and certain nonlinear potential is non integrable. However, in the cases of up to six particles, we prove that their Birkhoff-Gustavson normal forms are integrable, which allows us to apply KAM theory.


Introduction
In this article we deal with the periodic Klein-Gordon (KG) lattice (see for example [1] and references therein) described by the Hamiltonian (1.1) The constant C > 0 measures the interaction to nearest neighbor particles (with unit masses) and V (x) is a non-linear potential.
We study the integrability of (1.1). When C = 0 the Hamiltonian is separable and, hence integrable. There exist plenty of periodic or quasi-periodic solutions in the dynamics of (1.1). It is natural to investigate whether this behavior persists for C small enough (see e.g. [2]). Here we do not assume that C is small.
We are interested in the behavior at low energy, that is why the following main assumptions are in order: Remark 1. Such type of potentials are frequently used in the literature [1]). As it can be seen below the choice of a simplifies considerably the calculations.
We can also assume that C = 1 which can be achieved by rescaling of t. Then our Hamiltonian takes the form Our first result concerns the Hamiltonian with two degrees of freedom (q 2 = q 0 ), i.e. It simply says that the corresponding Hamiltonian system is integrable only when it is linear.
Theorem 1. The periodic KG lattice with n = 2 is non-integrable unless b = 0.
The above result tells us that it is highly unlikely to expect integrability for n > 2 (see also the discussion in the end of the paper).
Motivated by the works of Rink [3,4], who presented the periodic FPU chain as a perturbation of an integrable and KAM non-degenerated system, namely the truncated Birkhoff-Gustavson normal form of order 4 in the neighborhood of an equilibrium, our aim is to verify whether this can be done for the low dimensional KG lattices.
One should note that the Rink's result is due to the special symmetry and resonance properties of the FPU chain and should not be expected for lower-order resonant Hamiltonian systems (see e.g. [5]).
We summarize our second result in the following Theorem 2. The truncated normal forms H = H 2 + H 4 of the periodic KG lattices up to six particles are completely integrable. In particular, these normal forms are KAM nondegenerated excepting the case of six particles.
As a consequence from this result, we may conclude for the low dimensional KG lattices when KAM theory applies, that there exist many quasi-periodic solutions of small energy on a long time scale (see section 2 and for more detailed explanation [3]) and chaotic orbits are of small measure.
The paper is organized as follows. In section 2 some notions and facts used in the paper are given. In section 3 we calculate the Birkhoff-Gustavson normal forms for the cases up to six particles and show that they are integrable. We finish with some concluding remarks as well as some possible lines of further study.
The proof of Theorem 1 is based on the Ziglin-Morales-Ramis theory and since it is more algebraic in nature, it is carried out in the Appendix.

Resonances and normalization
In this section we recall briefly some notions and facts about integrability of Hamiltonian systems, action-angle variables, perturbation of integrable systems and normal forms. More complete exposition can be found in [6,7,8].
Let H be an analytic Hamiltonian defined on a 2n dimensional symplectic manifold. The corresponding Hamiltonian system isẋ = X H (x). (2.1) It is said that a Hamiltonian system is completely integrable if there exist n independent integrals is the Poisson bracket. On a neighborhood U of the connected compact level sets of the integrals M c = {F j = c j , j = 1, . . . , n} by Liouville -Arnold theorem one can introduce a special set of symplectic coordinates, I j , ϕ j , called action -angle variables. Then, the integrals F 1 = H, F 2 , . . . , F n are functions of action variables only and the flow of X H is simplė Therefore, near M c , the phase space is foliated with X F i invariant tori over which the flow of X H is quasi -periodic with frequencies (ω 1 (I), . . . , ω n (I)) = ( ∂H ∂I 1 , . . . , ∂H ∂In ). The map is called frequency map. Consider a small perturbation of an integrable Hamiltonian H 0 . According to Poincaré the main problem of mechanics is to study the perturbation of quasi-periodic motions in the system given by the Hamiltonian KAM -theory [9,10,11] gives conditions on the integrable Hamiltonian H 0 which ensures the survival of the most of the invariant tori. The following condition, usually called Kolmogorov's condition, is that the frequency map should be a local diffeomorphism, or equivalently on an open and dense subset of U. We should note that the measure of the surviving tori decreases with the increase of both perturbation and the measure of the set where above Hessian is too close to zero. In the neighborhood of an equilibrium (0, 0) we have the following expansion of H H = H 2 + H 3 + H 4 + . . . , We assume that H 2 is a positively defined quadratic form. The frequency ω = (ω 1 , . . . , ω n ) is said to be in resonance if there exists a vector k = (k 1 , . . . , k n ), k j ∈ Z, j = 1, . . . , n, such that (ω, k) = k j ω j = 0, where | k| = | k j | is the order of resonance.
With the help of a series of canonical transformations close to the identity, H simplifies. In the absence of resonances the simplified Hamiltonian is called Birkhoff normal form, otherwise -Birkhoff-Gustavson normal form.
Often to detect the behavior in a small neighborhood of the equilibrium, instead of the Hamiltonian H one considers the normal form truncated to some order It is known that the truncated to any order Birkhoff normal form is integrable [8]. The truncated Birkhoff-Gustavson normal form has at least two integrals -H 2 andH. Therefore, the truncated normal form of two degrees of freedom Hamiltonian is integrable.
In order to obtain estimates of the approximation by normalization in a neighborhood of an equilibrium point we scale q → εq, p → εp. Here ε is a small positive parameter and ε 2 is a measure for the energy relative to the equilibrium energy. Then, dividing by ε 2 and removing tildes we get Provided that ω j > 0 it is proven in [12] thatH is an integral for the original system with error O(ε m−1 ) and H 2 is an integral for the original system with error O(ε) for the whole time interval. If we have more independent integrals, then they are integrals for the original Hamiltonian system with error O(ε m−2 ) on the time scale 1/ε. The first integrals for the normal form H are approximate integrals for the original system, that is, if the normal form is integrable then the original system is near integrable in the above sense.
Returning to the Hamiltonian of the periodic KG lattice (1.2) we see that its quadratic part H 2 is not in diagonal form Here L n is the following n × n matrix The eigenvalues of L n are of the form Ω k = a + ω 2 k , ω k = 2 sin kπ n . In order to obtain the corresponding eigenvectors y k , following [3] we define and if n is even, Further, for 1 ≤ k < n/2, we define y k and y n−k via their coordinates It is easily checked that {y 1 , . . . , y n } is an orthonormal basis of R n , consisting of eigenvectors of L n . Let Y be the n × n matrix formed by the vectors y k as columns, then Y T Y = Id, Y −1 L n Y = Ω := diag(Ω 1 , . . . , Ω n ). The symplectic Fourier-transformation q = Yq, p = Yp brings H 2 in diagonal form The variables (q,p) are known as phonons.
We need one more definition. Definition. ( [3]) It is said that ω ∈ R n satisfies the property of internal resonance if for any k ∈ Z n with (k, ω) = 0, |k| = 4, we have k j = −k n−j when 1 ≤ j < n/2.
In the following table we list the frequencies Ω k of some low dimensional periodic KG lattice: As it is seen from the table we almost always have internal resonances. The assumption on a prevents the appearance of more complicated resonances.

Low-dimensional lattices
In this section we calculate the normal forms for the periodic KG lattices with particles up to six. We do not use ε in the forth degree expression, but keep in mind that we are close to the equilibrium. Of course, it is assumed that b = 0.

Two particles
This case is easy. It is well known that the truncated to any order normal form of a two degrees of freedom Hamiltonian is integrable. It remains only to verify the KAM condition.
We have already brought the quadratic part of the Hamiltonian in diagonal form. It is important that it is written in the phonons (q,p).
Further, we perform a scalingq which preserves the symplectic form. The Hamiltonian (3.1) becomes a(a + 4) +q 4 2 a .
Usually at this place one makes the following change of variables Since the frequencies Ω 1 = √ a + 4, Ω 2 = √ a are incommensurable, the only resonant terms which remain are The other terms can be removed via symplectic near-identity change. Therefore, the normal form of (3.1) up to order 4 or, returning to the (q,p) coordinates we get This normal form is clearly integrable with quadratic first integrals I j =p 2 j +q 2 j , j = 1, 2. Finally, introducing symplectic polar coordinates, which are action -angle variables It can easily be checked that the Kolmogorov's condition is valid.

Three particles
First, we make use of the phonons (q,p) (as explained in Section 2) to transform the quadratic part of the Hamiltonian (1.2) n = 3 in diagonal form Further, the scalingq After removing the non-resonant terms, the normal form of the (3.4) up to order four H = or, returning to (q,p) we get This normal form is integrable with the following quadratic first integrals In order to introduce action-angle variables, we need to find the set of regular values of the energy momentum map EM : (q,p) → (F 1 , G 1 , I 3 ).

Four particles
It turns out that the normal form of the periodic KG lattice in the case of four particles is surprisingly simple, no matter of the internal resonance. After transforming the quadratic part in diagonal form and scalinḡ the Hamiltonian (1.2) n = 4 takes the form a(a + 4) .
There is an internal resonance between the frequencies Ω 1 and Ω 3 . In variables (z j , w j ), j = 1, 2, 3, 4, the generators of the normal form are z j w j , and z 1 w 3 , z 3 w 1 .
However, as it is seen from (3.8) the variablesq 1 andq 3 do not couple, so the last two generators do not appear in the normal form as if there are no resonances. After removing the non-resonant terms, the normal form of (3.8) up to order four H = H 2 + H 4 reads

Five particles
There is a close resemblance between the two cases of tree and five particles. We transform the quadratic part of the Hamiltonian (1.2) n = 5 using the phonons (q,p) and scale in the usual way. Further, we use the notations Ω j for short. Recall that that is, Ω contains two internal resonances. The generators of the normal form in (z, w) variables are z j w j , j = 1, . . . , 5 and z 1 w 4 , z 4 w 1 , z 2 w 3 , z 3 w 2 .
Then the truncated up to order 4 normal form H = H 2 + H 4 for (1.2) n = 5 reads or, written in the coordinates (q,p) is As it is seen, the above normal form is integrable with the following first integrals In a similar way as in [4] and above treated case of tree particles, in the domain where the first integrals are independent, we can introduce action-angle variables where F j , G j , j = 1, 2 are as above and ϕ 5 := arctanp 5 /q 5 .
As above one can verify that (F 1 , F 2 , G 1 , G 2 , I 5 , φ 1 , φ 2 , ψ 1 , ψ 2 , ϕ 5 ) are canonical coordinates. Then the truncated up to order 4 normal form as a function of the action variables is It is straightforward to be checked up that the Kolmogorov's condition is valid.

Concluding remarks
This paper presents partial results on integrability of normal forms of the periodic KG lattices. We study the normal forms because the original systems are non-integrable. This is proven rigorously in the case of two degrees of freedom (Theorem 1) and that is one of the differences with the FPU chain. One can carry out the non-integrability proof for n = 3 in the same line, but with more efforts. There is a technical difficulty to carry through that proof in the higher dimensions, however. The variational equation (VE) does not split nicely and one needs either a tool to deal with higher dimensional (NVE) or another particular solution with the (VE) along it suitable enough. Nevertheless, we claim that the periodic KG lattice is non-integrable for all n ≥ 2.
The calculation of the normal forms goes in the standard way, because we treat only low dimensions. It is also facilitated by the assumption an a. It is easy to see that there are plenty of resonances when a ∈ Q. The result in Theorem 2 allows us to view the periodic KG Hamiltonian (1.2) as a perturbation of a non-degenerate Liouville integrable Hamiltonian, namely the truncated up to order four Birkhoff-Gustavson normal form. One can also verify the other KAM condition, known as Arnold-Moser's condition [6].
A closer look at the resonant relations in the above cases shows up that there are no third order resonant terms. This suggests that we can also study a potential V (x) of the form The normalization for the periodic KG lattices in the cases up to six particles results in H 3 = 0. Therefore, Theorem 2 remains valid also for the potentials (4.1).
Finally, we do not address here the symmetric invariant manifolds in the KG lattices because they can be retrieved from [22].
In any case, the results of this paper serve to understand the lattices with many particles. We intend to study them with the approach of Rink [4], based on using the symmetry properties, which we also enjoy here, to construct suitable normal forms. Proposition 1. The Hamiltonian system corresponding to (A.1) admits a particular solution q 0 1 (t) = sn( a + 4 + b/2t, k), p 0 where sn is the Jacobi elliptic function with the module k = are the periods of (A.2). Here K, K ′ are the complete elliptic integrals of the first kind. In the parallelogram of the periods, the solution (A.2) has two simple poles Denoting by ξ (1) j = dp j , j = 1, 2 the variational equations (VE) (written as second order equations) areξ 2 ), then (A.5) stands for normal variational equation (NVE), to be more specific ξ (1) 2 + a + 3bsn 2 ( 4 + a + b/2t, k) ξ This equation has regular singularities at t 1,2 , that is, it is a Fuchsian one. From the expansion of the sn in the neighborhood of the pole t 1 , we have where c is an arbitrary constant and It is not difficult to see from (A.6) and (A.7) that the indicial equation [20] at t = t 1 r(r − 1) − 6 = 0 has roots r 1,2 = −2, 3 (compare with the expansions below). Therefore, the monodromy around t 1 is trivial and can not serve as an obstacle to integrability [21] (similarly for the monodromy around t 2 ).
In fact, we can say more about the identity component of the Galois group of (VE) (A.4, A.5). Notice that each of the variational equations (VE) is a Lamé equation in Jacobi form. Proof. The analysis is facilitated by the fact that (VE) is slitted into two second order differential equations with the coefficients in the field of elliptic functions.
Let us start with the first equation (A.4) (the consideration of the second equation is similar). It is straightforward that one solution of (A.4) is ξ and similarly for (A.4) One can see that these expansions are in fact convergent since t 1 is a regular singular point (cf. [20]). Hence, the fundamental matrix X(t) of (VE) is Now, let us consider the higher variational equations along the particular solution (A.2). We put q 1 = q 0 1 (t) + εξ 1 + ε 2 ξ 1 + ε 3 ξ 1 + . . . , p 1 =q 1 , q 2 = 0 + εξ where ε is a formal parameter and substitute these expressions into the Hamiltonian system governed by (A.1). Comparing the terms with the same order in ε we obtain consequently the variational equations up to any order. The first variational equation is, of course, (A.4), (A.5). For the second variational equation we haveξ 1 + K 1 , ξ 2 + K 2 , (A. 13) where 2 . (A.14) In this way we can obtain a chain of linear non-homogeneous differential equationṡ ξ (k) = A(t)ξ (k) + f k (ξ (1) , . . . , ξ (k−1) ), k = 1, 2, . . . , (A. 15) where A(t) is the linear part of the Hamiltonian equations along the particular solution and f 1 = 0. The above equation is called k-th variational equation (VE k ). If X(t) is a fundamental matrix of (VE 1 ), then the solutions of (VE k ), k > 1 can be found by Let us study the local solutions of (VE 2 ). In our case f 2 = (0, K 1 , 0, K 2 ) T and from (A.11) we get 1,2 K 1 , ξ We are looking for a component of X −1 (t)f 2 with a nonzero residuum at t = t 1 . This would imply the appearance of a logarithmic term. Indeed, the residue at t = t 1 of ξ 2,1 K 2 with the specific representatives is Since c is an arbitrary parameter, we choose it in such a way that the expression in the square brackets does not vanish for a > 0. There are many such values of c, say c = 1. Recall that by assumption b = 0. We have obtained a nonzero residuum at t = t 1 , which implies the appearance of a logarithmic term in the solutions of (VE 2 ). Then its Galois group is solvable but not abelian. Hence, we conclude the non-integrability of the Hamiltonian system (1.3).