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`Advances in Mathematical PhysicsVolume 2018, Article ID 7023696, 12 pageshttps://doi.org/10.1155/2018/7023696`
Research Article

Near-Integrability of Low-Dimensional Periodic Klein-Gordon Lattices

Faculty of Mathematics and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria

Received 30 September 2017; Accepted 24 December 2017; Published 22 January 2018

Academic Editor: Alkesh Punjabi

Copyright © 2018 Ognyan Christov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 nonintegrable. 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 in most cases.

1. Introduction

In this article we deal with the periodic Klein-Gordon (KG) lattice (see, e.g., [1] and references therein) described by the HamiltonianThe constant measures the interaction to nearest neighbor particles (with unit masses) and is a nonlinear potential.

We study the integrability of (1). When the Hamiltonian is separable and, hence, integrable. There exist plenty of periodic or quasi-periodic solutions in the dynamics of (1). It is natural to investigate whether this behavior persists for small enough (see, e.g., [2]). Here we assume that is neither very small nor too large.

We are interested in the behavior at low energy; that is why the following main assumptions are in order:

Remark 1. Such types of potentials are frequently used in the literature [1]. As it can be seen below the choice of simplifies considerably the calculations.

We can also assume that which can be achieved by rescaling of . Then our Hamiltonian takes the formOur first result concerns the Hamiltonian with two degrees of freedom ; that is,It simply says that the corresponding Hamiltonian system is integrable only when it is linear.

Theorem 2. The periodic KG lattice with is nonintegrable unless .

The above result suggests that generically the periodic KG lattice is nonintegrable for . Still there might be some special cases of integrability, while the opposite is not proven rigorously.

Motivated by the works of Rink and Verhulst [3, 4], who presented the periodic FPU chain as a perturbation of an integrable and KAM nondegenerated 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 Rink and Verhulst’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 concerning the integrability of truncated resonant normal forms of the periodic KG lattices up to six particles in the following.

Theorem 3. The truncated normal form of the periodic KG lattice is(i)completely integrable and KAM nondegenerated for ;(ii)completely integrable and KAM nondegenerated when for all but one;(iii)completely integrable for .

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. We discuss also the appearance of some additional resonant relations. In Section 3 we calculate the Birkhoff-Gustavson normal forms for the cases up to six particles and show that they are integrable in most cases. We finish with some concluding remarks as well as some possible lines of further study.

The proof of Theorem 2 is based on the Ziglin-Morales-Ramis theory and since it is more algebraic in nature, it is carried out in the Appendix.

2. 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 [68].

Let be an analytic Hamiltonian defined on a -dimensional symplectic manifold. The corresponding Hamiltonian system isIt is said that a Hamiltonian system is completely integrable if there exist independent integrals in involution, namely, for all and , where is the Poisson bracket. On a neighborhood of the connected compact level sets of the integrals by Liouville-Arnold theorem one can introduce a special set of symplectic coordinates, , called action-angle variables. Then, the integrals are functions of action variables only and the flow of is simpleTherefore, near , the phase space is foliated with invariant tori over which the flow of is quasi-periodic with frequencies .

The mapis called frequency map.

Consider a small perturbation of an integrable Hamiltonian . 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 [911] gives conditions on the integrable Hamiltonian 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 equivalentlyon an open and dense subset of . 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 a neighborhood of the equilibrium we have the following expansion of : We assume that is a positively defined quadratic form. The frequency is said to be in resonance if there exists a vector , such that , where is the order of resonance.

With the help of a series of canonical transformations close to the identity, simplifies. In the absence of resonances the simplified Hamiltonian is called Birkhoff normal form and, otherwise, Birkhoff-Gustavson normal form, which may contain combinations of angles arising from resonances.

Often to detect the behavior in a small neighborhood of the equilibrium, instead of the Hamiltonian 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— and . Therefore, the truncated normal form of two-degree-of-freedom Hamiltonian is integrable.

In order to obtain estimates of the approximation by normalization in a neighborhood of an equilibrium point we scale Here is a small positive parameter and is a measure for the energy relative to the equilibrium energy. Then, dividing by and removing tildes we get Provided that it is proven in [12] that is an integral for the original system with error and is an integral for the original system with error for the whole time interval. If we have more independent integrals, then they are integrals for the original Hamiltonian system with error on the time scale .

The first integrals for the normal form 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 (4) we see that its quadratic part is not in diagonal form:Here is the following matrix:The eigenvalues of are of the form . In order to obtain the corresponding eigenvectors , following [3] we defineand if is even,Further, for , we define and via their coordinatesIt is easily checked that is an orthonormal basis of , consisting of eigenvectors of . Let be the matrix formed by the vectors as columns; then . The symplectic Fourier-transformation brings in diagonal formThe variables are known as phonons.

We need one more definition.

Definition 4 (see [3]). It is said that satisfies the property of internal resonance if for any with ,  , we have when .

In Table 1 we list the frequencies of some low-dimensional periodic KG lattice.