# Near-Integrability of Periodic Klein-Gordon Lattices

## Abstract

**:**

## 1. Introduction

**Theorem**

**2.**

- (i)
- completely integrable and KAM nondegenerate for N odd;
- (ii)
- completely integrable for N even.

**Remark**

**1.**

**Theorem**

**3.**

## 2. Resonances and Symmetries

**near integrable**in the above sense.

**phonons**. Denote for short

**Assertion.**The only possible tuples $(k,{k}^{\prime},{k}^{\u2033},{k}^{\u2034})$, which satisfies (4) are those that can be derived from $(k,{k}^{\prime},N-k,N-{k}^{\prime})$ by permutations.

**Remark**

**2.**

**Theorem**

**4.**

## 3. Proof of Theorem 1

**Theorem**

**5.**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**1**.

**Proposition**

**1.**

**Remark**

**3.**

**Proof.**

**Remark**

**4.**

## 4. The Birkhoff Normal Form

**Theorem**

**7.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Proof**

**of**

**Theorem**

**2.**

## 5. KG Lattice with Fixed Endpoints

**Theorem**

**8.**

**Proof.**

## 6. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Christov, O.
Near-Integrability of Periodic Klein-Gordon Lattices. *Symmetry* **2019**, *11*, 475.
https://doi.org/10.3390/sym11040475

**AMA Style**

Christov O.
Near-Integrability of Periodic Klein-Gordon Lattices. *Symmetry*. 2019; 11(4):475.
https://doi.org/10.3390/sym11040475

**Chicago/Turabian Style**

Christov, Ognyan.
2019. "Near-Integrability of Periodic Klein-Gordon Lattices" *Symmetry* 11, no. 4: 475.
https://doi.org/10.3390/sym11040475