Near-integrability of periodic Klein-Gordon lattices

TL;DR

This paper proves the non-integrability of the periodic Klein-Gordon lattice but shows that its resonant normal form is integrable, leading to the existence of many quasi-periodic solutions at low energy levels.

Contribution

It demonstrates the non-integrability of the periodic KG lattice and establishes the integrability of its resonant normal form, revealing invariant tori and quasi-periodic solutions.

Findings

The KG lattice is non-integrable in Liouville sense.

The resonant Birkhoff normal form of the KG Hamiltonian is integrable.

Existence of many invariant tori and quasi-periodic solutions at low energy levels.

Abstract

In this paper we study the Klein-Gordon (KG) lattice with periodic boundary conditions. It is an $N$ degrees of freedom Hamiltonian system with linear inter-site forces and nonlinear on-site potential, which here is taken to be of the $\phi^4$ form. First, we prove that the system in consideration is non-integrable in Liuville sense. The proof is based on the Morales-Ramis theory. Next, we deal with the resonant Birkhoff normal form of the KG Hamiltonian, truncated to order four. Due to the choice of potential, the periodic KG lattice shares the same set of discrete symmetries as the periodic Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is integrable. To do this we utilize the results of B. Rink on FPU chains. If $N$ is odd this integrable normal form turns out to be KAM nondegenerate Hamiltonian. This implies the existence of many invariant tori at low-energy level in the dynamics of the periodic KG lattice, on which the motion is quasi-periodic. We also prove that the KG lattice with Dirichlet boundary conditions (that is, with fixed endpoints) admits an integrable, KAM nondegenerated normal forth order form, which in turn shows that almost all low-energetic solutions of KG lattice with fixed endpoints are quasi-periodic.

Equations

$$H=\sum_{j\in\mathbb{Z}}\Big{[}\frac{p_{j}^{2}}{2}+\frac{C}{2}(q_{j+1}-q_{j})^{2}+V(q_{j})\Big{]},\quad p_{j}=\dot{q}_{j}.$$

$$u_{tt}=u_{xx}+V(u).$$

$$V(x)=\frac{a}{2}{x}^2+\frac{\beta }{4}{x}^4,a>0,$$

$$H=\sum_{j\in\mathbb{Z}/N\mathbb{Z}}\Big{[}\frac{p_{j}^{2}}{2}+\frac{1}{2}(q_{j+1}-q_{j})^{2}+\frac{a}{2}(q_{j})^{2}+\frac{\beta}{4}(q_{j})^{4}\Big{]},\quad p_{j}=\dot{q}_{j}.$$

$$H=\sum _j=\frac{1}{2}{p}_j^2+W(q_{j+1}-q_j),p_j={\dot{q}}_j,(q_0,p_0)=(q_n,p_n),$$

$$\dot{x}=X_H(x).$$

$${\dot{I}}_j=0,{\dot{\phi }}_j=\frac{\partial H}{\partial I_j},j=1,\dots ,n.$$

$$(I_1,I_2,\dots ,I_n)\to \left(\frac{\partial H}{\partial I_1},\frac{\partial H}{\partial I_2},\dots ,\frac{\partial H}{\partial I_n}\right)$$

$$H=H_0(I)+\epsilon H_1(I,\phi ),\epsilon <<1.$$

$$\det \left(\frac{{\partial }^2{H}_0}{\partial I_i\partial I_j}\right)\ne 0$$

$$H$$

$$H_2$$

$$\overline{H}=H_2+\dots +{\overline{H}}_m.$$

$$\overline{H}=H_2+\epsilon {\overline{H}}_3+\dots +{\epsilon }^{m-2}{\overline{H}}_m.$$

$$H_2=\frac{1}{2}{p}^{T}p+\frac{1}{2}{q}^T{L}_{N}q.$$

$$L_N:=\left(\begin{array}{ccccc}2+a& -1& & & -1\\ -1& 2+a& -1& & \\ & & & & \\ & \ddots & \ddots & \ddots & \\ & & & & \\ & & -1& 2+a& -1\\ -1& & & -1& 2+a\end{array}\right).$$

$$H_2=\frac{1}{2}{P}^{T}P+\frac{1}{2}{Q}^T\Omega Q,$$

$$c_{kj}:=\cos \left(\frac{2kj\pi }{N}\right),s_{kj}:=\sin \left(\frac{2kj\pi }{N}\right).$$

$$q_j=$$

$$p_j=$$

$$Q_j$$

$$Q_{N-j}$$

$$Q_N$$

$$Q_{\frac{N}{2}}=\frac{1}{\sqrt{N}}\sum _{k=1}^N(-1{)}^k{q}_k,P_{\frac{N}{2}}=\frac{1}{\sqrt{N}}\sum _{k=1}^N(-1{)}^k{p}_k.$$

$$H_2=\sum _{k=1}^N\frac{{\omega }_k}{2}(P_k^2+Q_k^2).$$

$${\omega }_k=\sqrt{1+4{\sin }^2\frac{k\pi }{N}}.$$

$$\sqrt{1+4{\sin }^2\frac{k\pi }{N}}+\sqrt{1+4{\sin }^2\frac{k^{\prime }\pi }{N}}=\sqrt{1+4{\sin }^2\frac{k^{\prime \prime }\pi }{N}}+\sqrt{1+4{\sin }^2\frac{k^{\prime \prime \prime }\pi }{N}}.$$

$$R:$$

$$\mapsto (q_2,q_3,\dots ,q_N,q_1;p_2,p_3,\dots ,p_N,p_1)$$

$$S:$$

$$\mapsto -(q_{N-1},q_{N-2},\dots ,q_1,q_N;p_{N-1},p_{N-2},\dots ,p_1,p_N).$$

$$H=\sum _{k=1}^N\frac{1}{2}{\omega }_k(P_k^2+Q_k^2)+H_4(Q_1,\dots ,Q_N).$$

$$z_k=Q_k+i{P}_k,{\xi }_k=Q_k-i{P}_k,$$

Full text

First, we prove that the system in consideration is non-integrable in Liuville sense. The proof is based on the Morales-Ramis theory.

Next, we deal with the resonant Birkhoff normal form of the KG Hamiltonian, truncated to order four. Due to the choice of potential, the periodic KG lattice shares the same set of discrete symmetries as the periodic Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is integrable. To do this we utilize the results of B. Rink on FPU chains.

If $N$ is odd this integrable normal form turns out to be KAM nondegenerate Hamiltonian. This implies the existence of many invariant tori at low-energy level in the dynamics of the periodic KG lattice, on which the motion is quasi-periodic.

We also prove that the KG lattice with Dirichlet boundary conditions (that is, with fixed endpoints) admits an integrable, KAM nondegenerated normal forth order form, which in turn shows that almost all low-energetic solutions of KG lattice with fixed endpoints are quasi-periodic.

1 Introduction

Let us introduce the Klein-Gordon (KG) lattice described by the Hamiltonian

[TABLE]

The constant $C>0$ measures the interaction to nearest neighbor particles (with unit masses) and $V(x)$ is a non-linear potential. This lattice appears, for instance, as a spatial discretization of the Klein-Gordon equation

[TABLE]

Both models subjected to different boundary conditions are used to describe a wide variety of physical phenomena: crystal dislocation, localized excitations in ionic crystals (see e.g. [15], [9]). In particular, the model (1.1) with the Morse potential $V(x)=D(e^{-\alpha x}-1)^{2}$ is applied to studying the thermal denaturation of DNA [17].

Our aim is to study the regular behavior of the trajectories of (1.1) and in particular, to study the complete integrability of the corresponding Hamiltonian system. 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. [10]). At this point it is worth mentioning that in anticontinuous limit $C\to 0$, there exist self-localized periodic oscillations called also discrete breathers [11], [5]). Here we assume that $C$ is neither very small nor too large and put $C=1$, which can be achieved by rescaling $t$.

We are interested mainly in the behavior at low energy, so we take quartic ($\phi^{4}$) potential

[TABLE]

which is frequently used in the research on the subject (see [15] and the literature therein).

First, periodic boundary conditions for (1.1) are assumed. Then one gets system with $N$ degrees of freedom, described by the Hamiltonian

[TABLE]

Our first result concerns the integrability of the Hamiltonian system governed by (1.4).

Theorem 1**.**

The periodic KG lattice with Hamiltonian (1.4) is integrable only when $\beta=0$.

Motivated by the works of Rink [18, 19], 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, we would like to verify whether this can be done for KG lattices.

As in the case of periodic FPU chain the properties of the periodic KG lattice near the equilibrium strongly depend on the parity of the number of the particles $N$. Assume, in addition, that $a=1$ (see an explanation for this choice in the next section). Then we have the following

Theorem 2**.**

The fourth order normal form $\overline{H}=H_{2}+\overline{H}_{4}$ of the periodic KG lattice is:

(i) completely integrable and KAM non-degenerate for $N$ odd;

(ii) completely integrable for $N$ even.

Remark 1. The statement of Theorem 2 will be made more precise in section 3. Note that the cases with $N=2$ and $N=4$ particles are rather exceptions to the general situation. The corresponding first integrals are quadratic and the KAM conditions are trivially checked.

As a consequence from this result, we may conclude for the periodic KG lattices when KAM theory applies, that there exist many quasi-periodic solutions of small energy on a long time scale and chaotic orbits are of small measure.

Next, Dirichlet boundary conditions for (1.1) are considered. Due to Rink [21] in the case of FPU chain such a system can be viewed as an invariant symplectic submanifold of a periodic FPU chain. This approach also works for KG lattice and the corresponding result is as follows.

Theorem 3**.**

The fourth order normal form $\overline{H}=H_{2}+\overline{H}_{4}$ of KG lattice with fixed endpoints is completely integrable and KAM nondegenerate.

This result and KAM theorem show the existence of large-measure set of low-energy quasi-periodic solutions of KG lattice with fixed endpoints.

Related works: Recall that the FPU chain with $n$ particles (with unit masses) and with periodic boundary conditions can be described by a Hamiltonian system with the Hamiltonian

[TABLE]

where $W$ is a real potential of the kind $W(x)=\frac{1}{2!}x^{2}+\frac{\alpha}{3!}x^{3}+\frac{\beta}{4!}x^{4}.$ When $\alpha\neq 0,\beta=0$ the chain is called an $\alpha$-chain. Accordingly, when $\alpha=0,\beta\neq 0$ the chain is known as a $\beta$-chain.

Probably T. Nishida [16] and J. Sanders [22] are among the first who have calculated the normal form of the FPU chain with fixed endpoints and with periodic boundary conditions, respectively. By imposing some very strong non-resonant assumptions, they verify the KAM theory conditions, but in general resonances do exist.

We already mentioned the works of Rink who has proved rigorously that the periodic FPU Hamiltonian is a perturbation of a nondegenerate Liouville integrable Hamiltonian, namely the normal form of order 4, [19]. Furthermore, he has described the geometry of even FPU lattice in [20], and finally rigorously has proved the Nishida’s conjecture stating that almost all low-energetic motions in FPU with fixed endpoints are quasi-periodic [21].

One should note that the Rink’s results are consequence of the special symmetry and resonance properties of the FPU chain and should not be expected for lower-order resonant Hamiltonian systems (see e.g. [2]).

However, several open problems remain: some of them purely computational and some of them of more philosophical nature. Henrici and Kappeler [7, 8] managed to solve practically all these problems, and generalized the results of Rink, by applying special sets of canonical variables, originally were designed for the Toda chain.

Our study on the normal forms of the KG lattice (in particular, Theorems 2 and 3) is related to the results of Rink on the normal form of the periodic FPU chain and use his approach. Notice that there are some differences between the potentials defining the models describing FPU and KG lattices. Moreover, the periodic FPU chain with three particles is integrable while Theorem 1 shows that this is not the case for the periodic KG lattice.

Our goal is to see whether these differences affect the integrability and eventually the dynamics of the system corresponding to the truncated normal form. In view of the wide applicability of the KG models, we think that this study is naturally motivated.

The paper is organized as follows. In section 2 we recall some known facts about Liouville integrable systems, action-angle variables, KAM conditions and normal forms. We also consider the resonances and discrete symmetries of the considered Hamiltonian system. In section 3 we give the truncated to order four normal form for the periodic KG lattice and consider the integrability of this normal form for $N$ odd and even, respectively. For any of the cases a more detailed description of the commuting first integrals is given. This proves Theorem 2. Section 4 is devoted to the KG lattice with fixed endpoints. Making use of the result from the previous section, the truncated normal form of the corresponding Hamiltonian is derived and Theorem 3 is proved. We finish with some remarks and possible directions to extend our results.

The proof of Theorem 1 is based on the Morales-Ramis theory and since it is more algebraic in nature, it is carried out in the Appendix.

2 Resonances and symmetries

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 [1].

Let $H$ be an analytic Hamiltonian defined on a $2n$ dimensional symplectic manifold. The corresponding Hamiltonian system is

[TABLE]

It is said that a Hamiltonian system is completely integrable if there exist $n$ independent integrals $F_{1}=H,F_{2},\ldots,F_{n}$ in involution, namely $\{F_{i},F_{j}\}=0$ for all $i$ and $j$, where $\{,\}$ is the Poisson bracket. On a neighborhood $U$ of the connected compact level sets of the integrals $M_{c}=\{F_{j}=c_{j},j=1,\ldots,n\}$ by Liouville - Arnold theorem one can introduce a special set of symplectic coordinates, $I_{j},\varphi_{j}$, called action - angle variables. Then, the integrals $F_{1}=H,F_{2},\ldots,F_{n}$ are functions of action variables only and the flow of $X_{H}$ is simple

[TABLE]

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 $(\omega_{1}(I),\ldots,\omega_{n}(I))=(\frac{\partial H}{\partial I_{1}},\ldots,\frac{\partial H}{\partial I_{n}})$.

The map

[TABLE]

is called frequency map.

Consider a small perturbation of an integrable Hamiltonian $H_{0}(I)$. According to Poincaré the main problem of mechanics is to study the perturbation of quasi-periodic motions in the system given by the Hamiltonian

[TABLE]

KAM - theory 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

[TABLE]

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 a neighborhood of the equilibrium $(q,p)=(0,0)$ we have the following expansion of $H$

[TABLE]

We assume that $H_{2}$ is a positively defined quadratic form. The frequency $\omega=(\omega_{1},\ldots,\omega_{n})$ is said to be in resonance if there exists a vector $k=(k_{1},\ldots,k_{n}),\,k_{j}\in\mathbb{Z},j=1,\ldots,n$, such that $(\omega,k)=\sum k_{j}\omega_{j}=0$, where $|\,k|=\sum|\,k_{j}|$ 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, 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 $H$ one considers the normal form truncated to some order

[TABLE]

It is known that the truncated to any order Birkhoff normal form is integrable [1]. The truncated Birkhoff-Gustavson normal form has at least two integrals - $H_{2}$ and $\overline{H}$. Therefore, the truncated resonant 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 $q\to\varepsilon\tilde{q},\,p\to\varepsilon\tilde{p}.$ Here $\varepsilon$ is a small positive parameter and $\varepsilon^{2}$ is a measure for the energy relative to the equilibrium energy. Then, dividing by $\varepsilon^{2}$ and removing tildes we get

[TABLE]

Provided that $\omega_{j}>0$ it is proven in [24] that $\bar{H}$ is an integral for the original system with error $O(\varepsilon^{m-1})$ and $H_{2}$ is an integral for the original system with error $O(\varepsilon)$ for the whole time interval. If we have more independent integrals, then they are integrals for the original Hamiltonian system with error $O(\varepsilon^{m-2})$ on the time scale $1/\varepsilon$.

The first integrals for the normal form $\overline{H}$ 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 (1.4) we see that its quadratic part $H_{2}$ is not in diagonal form

[TABLE]

Here $L_{N}$ is the following $N\times N$ matrix

[TABLE]

The eigenvalues of $L_{N}$ are of the form $\omega_{k}^{2}=a+4\sin^{2}\frac{k\pi}{N}$. There is a symplectic Fourier-transformation $q=MQ,p=MP$ which brings $H_{2}$ in diagonal form

[TABLE]

where $M^{-1}L_{N}M=\Omega:=\mathrm{diag}(\omega_{1}^{2},\ldots,\omega_{N}^{2})$. The variables $(Q,P)$ are known as phonons. Denote for short

[TABLE]

Then the transform $(q,p)\to(Q,P)$ in coordinate form is

[TABLE]

From these formulas one can easily get the explicit form of the matrix $M$. Later on, we will make use of the inverse transform: for $1\leq j<\frac{N}{2}$ we have

[TABLE]

and if $N$ is even

[TABLE]

Finally, by simple scaling $Q_{k}\to\frac{1}{\sqrt{\omega_{k}}}Q_{k},\,P_{k}\to\sqrt{\omega_{k}}P_{k}$ we get the desired form of $H_{2}$

[TABLE]

Next, we need to study the possible relations between the frequencies $\omega_{k}$. In general, $\omega_{k}=\omega_{j}$ holds if only if $j=N-k$. The resonances $\omega_{k}:\omega_{N-k}=(1:1)$ (known also as internal resonances [18]) are important for the construction of the normal form.

The assumption on $a$ does not prevent the appearance of more complicated resonances. It is clear that there are plenty of resonances when $a\in\mathbb{Q}$. Moreover, there are certain irrational values of $a>0$ for which fourth order resonances exist in the low-dimensional periodic KG lattices (see [3]). Such values of $a$ are difficult to control in the higher dimensions, that is why from now on we put $a=1$

[TABLE]

Note that another resonant relation 2 : 2 : 1 appears when $N=3s$. The other possible fourth order relations are

1. $\omega_{k}=3\omega_{k^{\prime}}$         2) $\omega_{k}=\omega_{k^{\prime}}+\omega_{k^{\prime\prime}}+\omega_{k^{\prime\prime\prime}}$

2. $\omega_{k}+\omega_{k^{\prime}}=2\omega_{k^{\prime\prime}}$  4) $\omega_{k}+\omega_{k^{\prime}}=\omega_{k^{\prime\prime}}+\omega_{k^{\prime\prime\prime}}$

for some $1\leq k,k^{\prime},k^{\prime\prime},k^{\prime\prime\prime}\leq N$. It is straightforward that there are no such $k,k^{\prime},k^{\prime\prime},k^{\prime\prime\prime}$ to fulfill 1) and 2) since $1\leq\omega_{k}\leq\sqrt{5}$. Since 3) is a particular case of the last relation, let us consider 4). Suppose the tuple $(k,k^{\prime},k^{\prime\prime},k^{\prime\prime\prime})$ is a solution of

[TABLE]

Then such is the tuple $(N-k,N-k^{\prime},N-k^{\prime\prime},N-k^{\prime\prime\prime})$ and any combination between two of them. Then in searching for these tuples we can consider only the cases where $k+k^{\prime}+k^{\prime\prime}+k^{\prime\prime\prime}\equiv 0\mod{N}$.

Assertion. The only possible tuples $(k,k^{\prime},k^{\prime\prime},k^{\prime\prime\prime})$, which satisfies 4) are those that can be derived from $(k,k^{\prime},N-k,N-k^{\prime})$ by permutations.

Remark 2. So far we have no rigorous proof for that claim, but we believe so. Direct computations show that this assertion is true for low dimensional cases $N=2,\ldots,6$. Numerical simulations are confirmative.

Finally, in order to keep symmetry in the formulas we continue to write $\omega_{N}$ and $\omega_{N/2}$ instead of their particular values $1$ and $\sqrt{5}$.

The Hamiltonian (1.4) of the periodic KG lattice possesses discrete symmetries. Two of them, important for the dynamics and exactly the same as in the periodic FPU chain, are the linear mappings $R,S:T^{*}\mathbb{R}^{N}\to T^{*}\mathbb{R}^{N}$ defined by (see [19, 21])

[TABLE]

and

[TABLE]

It is easily seen that $R$ can serve as a generator of a group $\langle R\rangle$, isomorphic to the cyclic group of order $N$ and $S$ as a generator of $\langle S\rangle$, isomorphic to the cyclic group of order two. Note that $R$ and $S$ are canonical transformations $R^{*}(dq\wedge dp)=S^{*}(dq\wedge dp)=dq\wedge dp$ and they leave the Hamiltonian invariant $R^{*}H=(H\circ R)=S^{*}H=H$. Moreover, they leave the Hamiltonian vector field $X_{H}$ invariant, which implies that they commute with the flow of $X_{H}$. It is observed in [19] that the subgroup $\langle R,S\rangle:=\{Id,R,\ldots,R^{N-1},SR,\ldots,SR^{N-1}\}$ of the symmetry group of $H$, with the relations $R^{N}=S^{2}=Id,SR=R^{-1}S$, is isomorphic to the $N$th dihedral group.

For Hamiltonian systems with symmetry, we have

Theorem 4**.**

(see [4] and [6]). Let $H=H_{2}+H_{3}+\ldots$ be the expansion of $H$ in a neighborhood of equilibrium and $G$ be a group of linear symplectic symmetries of $H$. Then a normal form $\overline{H}=H_{2}+\ldots+\overline{H}_{m}$ for $H$ can be constructed in such a way that $\overline{H}$ is also $G$-symmetric.

3 The Birkhoff normal form

In phonon coordinates, the Hamiltonian (1.4) is

[TABLE]

Further, we introduce the complex variables

[TABLE]

which are not symplectic, but are natural in the construction of the normal form. In these variables $H_{2}$ reads

[TABLE]

Next, we are looking for the monomials $z^{\Theta}\xi^{\theta}$, $\Theta,\theta$ being multi-indices, which commute with $H_{2}$, i.e. $ad_{H_{2}}(z^{\Theta}\xi^{\theta})=0$. These monomials are called resonant monomials and cannot be removed in the process of normalization. We then get

[TABLE]

with

[TABLE]

Hence, the resonant monomials are ones with $\nu(\Theta,\theta)=0$. Therefore, modulo the Remark 2 from the previous section we have that the set of multi-indices $(\Theta,\theta)$ for which $|\Theta|+|\theta|=4$ and $\nu(\Theta,\theta)=0$ is contained in the set given by the relations

[TABLE]

This means that $\overline{H}_{4}$ is generated by

[TABLE]

However, we want a normal form which is invariant under $R^{*}$ and $S^{*}$. To obtain such we define for $1\leq j<\frac{N}{2}$

[TABLE]

$a_{N}:=\frac{1}{2}z_{N}\xi_{N}=\frac{1}{2}(Q_{N}^{2}+P_{N}^{2})$ and if $N$ is even $a_{\frac{N}{2}}:=\frac{1}{2}z_{\frac{N}{2}}\xi_{\frac{N}{2}}=\frac{1}{2}(Q_{\frac{N}{2}}^{2}+P_{\frac{N}{2}}^{2})$.

These quantities are known as Hopf variables and they satisfy the relations

[TABLE]

and in these variables $H_{2}$ is

[TABLE]

The nontrivial Poisson brackets between these quantities are

[TABLE]

It is observed in [19] that $a_{k}$, $a_{\frac{N}{2}}$ and $a_{n}$ are invariant under $R^{*}$ and $S^{*}$ and also the products $a_{k}a_{l},a_{N}a_{k},b_{k}b_{l},1\leq k,l<\frac{N}{2}$ and if $N$ is even $a_{\frac{N}{2}}a_{k},1\leq k\leq\frac{N}{2}$ and $d_{k}d_{\frac{N}{2}-k}-c_{k}c_{\frac{N}{2}-k},1\leq k\leq\frac{N}{4}$, so $\overline{H}_{4}$ must be a linear combination of the above four order terms. Indeed, we have

Theorem 5**.**

The truncated up to order four normal form for the periodic KG lattice is

[TABLE]

[TABLE]

In the above formula the terms with subscripts $\frac{N}{2}$, $\frac{N}{4}$ appear if $\frac{N}{2}\in\mathbb{N}$, $\frac{N}{4}\in\mathbb{N}$, respectively.

The calculation of the above normal form is long, tedious, but straightforward, that’s why it is not presented here. Instead, we proceed with two important corollaries, which prove the assertions in Theorem 2.

Corollary 1**.**

When $N$ is odd, the truncated normal form $\overline{H}=H_{2}+\overline{H}_{4}$ of the periodic KG lattice is Liouville integrable with the quadratic integrals $a_{j},b_{j},\,1\leq j\leq\frac{N-1}{2}$ and $a_{N}$. Moreover, this normal form is KAM nondegenerate.

Proof. $H_{2}$ is a linear combination of $a_{j}$ and $a_{N}$. When $N$ is odd, $\overline{H}_{4}$ becomes

[TABLE]

and it is clear that $a_{j},b_{j},\,1\leq j\leq\frac{N-1}{2}$ and $a_{N}$ commute with $\overline{H}_{4}$ and with each other.

In order to introduce action-angle variables we follow the scheme from [19], slightly adjusted to our case. We need to find the set of regular values of the energy momentum map

[TABLE]

Denote it by $U_{r}=\{(a_{j},b_{j},a_{N})\in\mathbb{R}^{N},\,a_{j}>0,|b_{j}|<a_{j},\,a_{N}>0\}$. Then for all $(a_{j},b_{j},a_{N})\in U_{r}$ the level sets of $EM^{-1}(a_{j},b_{j},a_{N})$ are diffeomorphic to N-tori.

Let $\mathrm{arg}:\mathbb{R}^{2}\setminus\{(0,0)\}\to\mathbb{R}/2\pi\mathbb{Z}$ be the argument function $\mathrm{arg}(r\cos\Phi,r\sin\Phi)\to\Phi$. Define the following set of variables $(a_{j},b_{j},a_{N},\phi_{j},\psi_{j},\varphi_{N})$ $a_{j},b_{j},a_{N}$ as above and

[TABLE]

Using the formula $d\mathrm{arg}(x,y)=\frac{xdy-ydx}{x^{2}+y^{2}}$, one can verify that $(a_{j},b_{j},a_{N},\phi_{j},\psi_{j},\varphi_{N})$ are indeed canonical coordinates $\sum dP_{j}\wedge dQ_{j}=\sum da_{j}\wedge d\phi_{j}+db_{j}\wedge d\psi_{j}+da_{N}\wedge d\varphi_{N}$. Since $a_{j},b_{j}$ and $a_{N}$ are quadratic functions in the phase variables, they can be extended to global action variables.

Finally to check the nondegeneracy condition, we compute the Hessian of $\overline{H}_{4}$ with respect to $a_{j},a_{N},b_{k}$. Denote $\lambda_{j}:=1/\omega_{j}$. Then

[TABLE]

[TABLE]

Clearly $\frac{\partial^{2}\overline{H}_{4}}{\partial b_{j}\partial b_{k}}$ is nondegenerate. After some algebra, one can check that

[TABLE]

i.e., it is also nondegenerate.

$\hfill\square$

Thus, the periodic KG lattice (1.4) with an odd number of particles can, after normalization, be viewed as a perturbation of a nondegenerate integrable Hamiltonian system, namely its fourth order normal form. Therefore, by the KAM theorem almost all low-energy solutions of (1.4) are periodic or quasi-periodic and live on invariant tori.

Corollary 2**.**

When $N$ is even, the truncated normal form $\overline{H}=H_{2}+\overline{H}_{4}$ of the periodic KG lattice is Liouville integrable with the quadratic integrals $a_{k},\,1\leq k\leq\frac{N}{2}$ and $a_{N}$, $b_{k}-b_{\frac{N}{2}-k},\,1\leq k<\frac{N}{4}$ and $c_{\frac{N}{4}}$ (if $\frac{N}{4}\in\mathbb{N}$) and the quartics

[TABLE]

Proof. This follows from simple calculations of all Poisson brackets using (3.8).

Let us deal with the exceptional cases. First, in the case of $N=2$ particles the frequencies are incommensurable, i.e., there are no resonances. From (5) we obtain

[TABLE]

where $a_{k}=\frac{1}{2}(Q_{k}^{2}+P_{k}^{2}),\,k=1,2$. The action variables $a_{k}$ can be extended to global action-angle variables $(a_{k},\varphi_{k})$ usually called symplectic polar coordinates. Then the KAM condition is immediate.

Further, for the case with $N=4$ particles, we get from (5) and (3.6)

[TABLE]

Denote $I_{k}=\frac{1}{2}(Q_{k}^{2}+P_{k}^{2}),\,k=1,\ldots,4$. Note that $a_{1}=I_{1}+I_{3},\,c_{1}=I_{1}-I_{3},\,a_{2}=I_{2},\,a_{4}=I_{4}$. Then $\overline{H}_{4}$ becomes

[TABLE]

The actions $I_{k}$ can be extended to global action-angle variables $(I_{k},\varphi_{k})$ (symplectic polar coordinates) and the KAM condition is straightforward (compare with [3]). $\hfill\square$

Proof of Theorem 2. Part (i) is proved by Corollary 1, whereas part (ii) comes after Corollary 2. $\hfill\blacksquare$

4 KG lattice with fixed endpoints

In this section we consider the KG lattice with $n$ ($n\geq 3$, not necessarily even) particles and with fixed endpoints

[TABLE]

It was realized in [21] that such the FPU lattice with the fixed boundary conditions can be viewed as an invariant subsystem of the periodic FPU lattice with $N=2n+2$ particles. This invariant subsystem is obtained by the fixed point set of the compact group $\langle S\rangle=\{Id,S\}$. Since $S$ is also a symmetry for the periodic KG Hamiltonian this constriction is applicable here and we will describe it briefly.

Define the set

[TABLE]

By the definition of $S$ (2) we get

[TABLE]

from where it follows that $q_{0}=q_{n+1}=q_{0}=q_{n+1}=0$. $Fix\langle S\rangle$ is a symplectic manifold with symplectic form obtained from $dq\wedge dp$ with restriction on $Fix\langle S\rangle$. One can takes as coordinates on $Fix\langle S\rangle$ $(q_{1},\ldots,q_{n},p_{1},\ldots,p_{n})$. Then the lattice with fixed endpoints and $n$ particles is described on $Fix\langle S\rangle$ by the Hamiltonian $H_{Fix\langle S\rangle}$.

Let us consider the periodic KG Hamiltonian with $N=2n+2$ particles. From the definition of the phonon particles (2), we can obtain that $S$ acts on them in the following way

[TABLE]

Hence,

[TABLE]

which is a symplectic manifold isomorphic to $T^{*}\mathbb{R}^{n}$. We can take as coordinates on $Fix\langle S\rangle$ $(x_{k},y_{k})=(Q_{2n+2-k},P_{2n+2-k}),\,k=1,\ldots,n$.

We have already constructed an $S$-invariant truncated normal form $\overline{H}$ for the periodic KG lattice. It was realized in [21] that in order to obtain the normal form of the Hamiltonian $H_{Fix\langle S\rangle}$, one needs to restrict the symmetric normal form $\overline{H}$ to $Fix\langle S\rangle$, that is,

[TABLE]

Clearly, $b_{k}=d_{k}=0,\,1\leq k\leq n,\,a_{N}=a_{\frac{N}{2}}=0$ and $c_{k}=-a_{k},\,1\leq k\leq n$. Introduce

[TABLE]

Then we have

Theorem 6**.**

The KG lattice with $n$ particles and fixed boundary conditions has the fourth order normal form $\overline{H}=H_{2}+\overline{H}_{4}$, where $H_{2}=\sum_{k=1}^{n}\omega_{k}I_{k},\quad\omega_{k}:=\sqrt{1+4\sin^{2}\frac{k\pi}{2n+2}}$ and

[TABLE]

In the above formula the term with subscripts $\frac{n+1}{2}$ appears if $n$ is odd.

This normal form is completely integrable with the quadratic first integrals $I_{k},\,k=1,\ldots,n$ and it is KAM nondegenerate.

Proof. It follows from Theorem 5 and the explanations above that the quantities $I_{k}$ are Poisson commuting, so the complete integrability is clear. The variables $I_{k}$ can be extended to global action-angle variables $(I_{k},\varphi_{k})$ - symplectic polar coordinates. It remains to verify the nondegeneracy of the normal form $\overline{H}$. Denote the Hessian of $\overline{H}$ with $\mathcal{H}$ and let $\lambda_{j}=1/\omega_{j}$ as before. We have

[TABLE]

where $\triangle_{n}=diag(\lambda_{1},\ldots,\lambda_{n})$ and $F_{n}$ is an $n\times n$ matrix which for $n$ even, respectively odd takes the form

[TABLE]

Then

[TABLE]

After some linear algebra $F_{n}$ is shown to be nonsingular, from where the nondegeneracy of $\overline{H}$ follows.

$\hfill\blacksquare$

As a consequence of the above result, we conclude that almost all low-energy solutions of the KG lattice with fixed endpoints $H_{Fix\langle S\rangle}$ are quasi-periodic and live on invariant tori.

5 Concluding Remarks

In present paper, we deal with the integrability of the KG lattices. First, we study the periodic KG lattice with $N$ particles (1.4) and quartic potential

[TABLE]

We have shown that this Hamiltonian system is integrable in Liouville sense only when $\beta=0$. For this we use Differential Galois theory and Morales-Ramis approach.

The considered system enjoys the same important discrete symmetries $R,S$ (2), (2) as in the periodic FPU chain. Following [19] we construct an $R,S$-symmetric resonant forth order normal form $\overline{H}$. This normal form happens to be Liouville integrable. It is similar to the normal form of the periodic FPU $\beta$-chain, but it is natural to be expected. Hence, the periodic KG lattice can be considered as a perturbation of its integrable Bikhoff normal form.

If $N$ is odd the integrals of the normal form are quadratic. The global action-angle variables can be introduced and it turns out that this normal form is KAM nondegenerated. This proves the existence of many quasi-periodic solutions in the dynamics of the periodic KG lattice at low-energy level.

The resonant normal form with $N$ even admits certain set of quartic integrals in addition to the quadratic ones. Probably, it would be interesting to explore the geometry of the system, defined by this normal form. One can assume that the things are similar to the even periodic FPU chain [20] up to small modifications due to the extra degree of freedom.

Next, we consider the KG lattice with fixed endpoints. Such a system can be considered as an invariant symplectic submanifold of a larger periodic KG lattice. For this the discrete symmetry $S$ is utilized. Then the normal form in this case is easy to get from the previous result, and hence, it is integrable. Further, KAM theorem applies which implies that almost all low-energetic solutions are quasi-periodic.

Finally, we notice that the results of this paper do not provide an answer to one of the most important problems: what happens in the dynamics of the KG lattice when the number of particle becomes larger and larger.

Let us emphasize again the importance of discrete symmetries, and in particular, the symmetry $S$ in the carrying out of the above analysis. This leads to the following question: what happens when we drop the symmetry $S$? We can ask the same thing in a different way: Can the results of this paper be extended for the KG lattice with the potential

[TABLE]

which is more relevant in studying the the dynamic of low-energetic solutions in the DNA model?

It is clear that the non-integrability result of Theorem 1 can easily be extended in the same lines as in the Appendix. The formal computation of the normal form would be more difficult, because we have to transform away the third order terms. However, straightforward calculations for the low-dimensional periodic KG lattices with $a=1$ and $N=2,\ldots,6$ show that resonant third order terms do not appear in the corresponding normal forms (see [3]). Hence, these normal forms remain integrable for the latter potential. In view of the applications to the DNA models, it is clearly of some interest to calculate these normal forms in the general case.

Acknowledgements. This work is partially supported by grant DN 02-5 of Bulgarian Fond ”Scientific Research”.

Appendix A Proof of Theorem 1

The proof of Theorem 1 is based on Ziglin-Morales-Ruiz-Ramis theory. The main result of this theory merely says that if a Hamiltonian system is completely integrable then the identity component of the Galois group of the variational equation along certain particular solution is abelian. The necessary facts and results about differential Galois theory and its relations with the integrability of Hamiltonian systems, can be found in [12, 13, 14, 23].

In the applications if one finds out that the identity component of the Galois group is non-commutative, then this implies non-integrability. However, if this component turns out to be abelian, one needs additional steps to prove non-integrability as it is carried out below.

It is also assumed that throughout this appendix all variables are complex: $t\in\mathbb{C},q_{j}\in\mathbb{C},p_{j}\in\mathbb{C},j=1,\ldots,N$, but we keep the parameters $a,\beta$ real. The proof goes in the following lines. We obtain a particular solution and write the variational equation along this solution. It appears that the identity component of its differential Galois group is abelian. In order to obtain an obstacle to the integrability, we study the higher variational equations. Although higher variational equations are not actually homogeneous equations, they can be put in the differential Galois context. Their differential Galois groups are in principle solvable. One possible way to show that some of them is not abelian is to find a logarithmic term in the corresponding solution (see [13, 14]). We obtain such a logarithmic term in the solution of the second variational equation when $\beta\neq 0$. Then the non-integrability of the Hamiltonian system follows.

Proof. Suppose $\beta\neq 0$. The following Proposition is immediate.

Proposition 1**.**

The Hamiltonian system corresponding to (1.4) admits a particular solution

[TABLE]

where $sn$ is the Jacobi elliptic function with the module $\kappa=\sqrt{\frac{-\beta/2}{a+\beta/2}}$.

$\hfill\square$

Remark 3. It is assumed that $\beta>0$ which is not restrictive. In any case, the solution is expressed via Jacobi elliptic functions and one can proceed in the same way.

It is straightforward that $T_{1}=\frac{4K}{\sqrt{a+\beta/2}}$ and $T_{2}=\frac{2iK^{\prime}}{\sqrt{a+\beta/2}}$ are the periods of (A.1). Here $K,K^{\prime}$ are the complete elliptic integrals of the first kind. In the parallelogram of the periods, the solution (A.1) has two simple poles

[TABLE]

From the expansion of the $sn$ in the neighborhood of the pole $t_{1}$ we get

[TABLE]

where $\gamma_{3}$ is an arbitrary constant and $\gamma_{5}$ amounts to

[TABLE]

Denoting by $\xi^{(1)}_{j}=dq_{j},\eta^{(1)}_{j}=dp_{j},j=1,2$ the variational equations (VE) (written as second order equations) are

[TABLE]

Proposition 2**.**

The identity component of the differential Galois group of (VE) (A.5) is abelian.

Proof. To see this we first denote

[TABLE]

Then (VE) can be written as

[TABLE]

The structure of the matrix $K_{N}(q)$ is similar to that of $L_{N}$ in (2.6). This suggest using the linear transform

[TABLE]

with already defined matrix $M$, which decouples the system (A.7)

[TABLE]

where $D_{N}(q):=M^{-1}K_{N}(q)\,M=diag(\Lambda_{1}^{2},\ldots,\Lambda_{N}^{2})$, with $\Lambda_{j}^{2}:=V^{\prime\prime}(q)+4\sin^{2}\frac{\pi j}{N},\,j=1,\ldots,N$. In coordinate form the above system can be written as

[TABLE]

After changing the independent variable $\tau:=\sqrt{a+\beta/2}\,t,\,^{\prime}=d/d\tau$ we can see that each of these equations is a Lamé equation in Jacobi form.

[TABLE]

with $\tilde{n}=2$. The identity component of the Galois group of such equations is known to be isomorphic to $\begin{pmatrix}1&0\\ \nu_{j}&1\end{pmatrix},\nu_{j}\in\mathbb{C}$. Therefore, the identity component $G^{0}$ of the Galois group of (VE) is represented by the matrix group

[TABLE]

and it is clearly abelian.

$\hfill\square$

Now, let us consider the higher variational equations along the particular solution (A.1). We write

[TABLE]

where $\varepsilon$ is a formal parameter and substitute these expressions into the Hamiltonian system governed by (1.4). Comparing the terms with the same order in $\varepsilon$ we obtain consequently the variational equations up to any order.

The first variational equation is, of course, (A.5) $(\mathrm{VE}_{1})=(\mathrm{VE})$. For the second variational equation we have

[TABLE]

Denote

[TABLE]

Then the second variational equation can be written as

[TABLE]

In this way we can obtain a chain of linear non-homogeneous differential equations

[TABLE]

where $f^{(1)}=0$. The above equation is usually called $k$-th variational equation $(\mathrm{VE}_{k})$ (here is is written in the form of second-order equation). As it was mentioned above higher variational equations are solvable. Indeed, if $\Xi(t)$ is a fundamental matrix of $(\mathrm{VE}_{1})$, then the solutions of $(\mathrm{VE}_{k}),k>1$ can be found by quadratures

[TABLE]

where

[TABLE]

Let us study the local solutions of $(\mathrm{VE}_{2})$. We make the linear change (with above introduced matrix $M$)

[TABLE]

The system (A.15) becomes

[TABLE]

where

[TABLE]

that is, in these coordinates $(\mathrm{VE}_{2})$ decomposes into $N$ second order differential equations. Therefore, it is enough to show that the identity component of the Galois group of one of them is non-commutative, which implies non-commutativity of the Galois group of $(\mathrm{VE}_{2})$, and hence, non-integrability of the Hamiltonian system under consideration.

To keep the things simple, we take as a solution of $(\mathrm{VE}_{1})$ (A.10)

[TABLE]

and $y^{(1)}_{N}\neq 0$ satisfying

[TABLE]

In what follows we need the expansions around $t_{1}$ of the fundamental system of the solutions with unit Wronskian for (A.21). We get

[TABLE]

Then the fundamental matrix $\Xi_{N}(t)$ of (A.21) can be written as

[TABLE]

Now, we study the local solutions around $t_{1}$ only of the last equation in the system (A.19) taking into account (A.20), (A.21), (2.8) and (2)

[TABLE]

In this case (A.18) becomes

[TABLE]

We are looking for a component of above vector with a nonzero residuum at $t=t_{1}$. This would imply the appearance of a logarithmic term. After some calculations making use of (A.3) and (A), the residue at $t=t_{1}$ of the second component $-y^{(1)}_{N,1}\frac{3\beta}{\sqrt{N}}\,q\,(y^{(1)}_{N})^{2}$ with the specific representatives turns out to be

[TABLE]

Since $\gamma_{3}$ is an arbitrary parameter, we choose it in such a way that the expression in the square brackets of (A.25) does not vanish for $a>0$. There are many such values of $\gamma_{3}$, for instance, we can take $\gamma_{3}=1$. Recall that by assumption $\beta\neq 0$. We have obtained a nonzero residuum at $t=t_{1}$, which implies the appearance of a logarithmic term in the solutions of $(\mathrm{VE}_{2})$. Then its Galois group is solvable but not abelian. Hence, we conclude the non-integrability of the Hamiltonian system (1.4).

$\hfill\blacksquare$