# Breathers in Hamiltonian PT -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model

- (A1)
- The coupling parameters C and D are small. Therefore, we can introduce a small parameter μ such that both C and D are proportional to ${\mu}^{2}$.
- (A2)
- A resonant periodic force is applied to the common strings for each pair of coupled pendula. Therefore, D is considered to be proportional to $cos(2\omega t)$, where ω is selected near the unit frequency of linear pendula indicating the $1:2$ parametric resonance between the force and the pendula.

## 3. Symmetries and Conserved Quantities

## 4. Breathers (Time-Periodic Solutions)

**Lemma**

**1.**

- (a)
- $\Omega >|\gamma |$ - two symmetric unbounded branches exist for $\pm E>{E}_{0}$,
- (b)
- $\Omega <|\gamma |$ - an unbounded branch exists for every $E\in \mathbb{R}$,
- (c)
- $\Omega <-|\gamma |$ - a bounded branch exists for $-{E}_{0}<E<{E}_{0}$,

**Proof.**

- (a)
- If $\Omega >|\gamma |$, then the Parametrization (25) yields a monotonically increasing map ${\mathbb{R}}^{+}\ni {A}^{2}\mapsto {E}^{2}\in ({E}_{0}^{2},\infty )$ because$$\frac{d{E}^{2}}{d{A}^{2}}=\frac{8(8{A}^{2}+\Omega )}{{(4{A}^{2}+\Omega )}^{3}}\left[2{(4{A}^{2}+\Omega )}^{3}-{\gamma}^{2}\Omega \right]>0$$$${E}^{2}={E}_{0}^{2}+\mathcal{O}({A}^{2})\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}A\to 0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}^{2}=64{A}^{4}+\mathcal{O}({A}^{2})\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}A\to \infty .$$
- (b)
- If $\Omega <|\gamma |$, the Parametrization (25) yields a monotonically increasing map $({A}_{+}^{2},\infty )\ni {A}^{2}\mapsto {E}^{2}\in {\mathbb{R}}^{+}$, where$${A}_{+}^{2}:=\frac{|\gamma |-\Omega}{4}.$$$$2{(4{A}^{2}+\Omega )}^{3}-{\gamma}^{2}\Omega \ge {\gamma}^{2}(2|\gamma |-\Omega )>0,$$$${E}^{2}\to 0\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}{A}^{2}\to {A}_{+}^{2}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}^{2}=64{A}^{4}+\mathcal{O}({A}^{2})\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}A\to \infty .$$
- (c)
- If $\Omega <-|\gamma |$, then the Parametrization (25) yields a monotonically decreasing map $(0,{A}_{-}^{2})\ni {A}^{2}\mapsto {E}^{2}\in (0,{E}_{0}^{2})$, where$${A}_{-}^{2}:=min\left\{\frac{|\Omega |-|\gamma |}{4},\frac{|\Omega |}{8}\right\}$$$$\frac{d{E}^{2}}{d{A}^{2}}=-\frac{8(|\Omega |-8{A}^{2})}{(|\Omega |-4{A}^{2}{)}^{3}}\left[2(|\Omega |-4{A}^{2}{)}^{3}-{\gamma}^{2}|\Omega |\right]<0$$$${E}^{2}={E}_{0}^{2}+\mathcal{O}({A}^{2})\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}A\to 0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{E}^{2}\to 0\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}{A}^{2}\to {A}_{-}^{2}$$

**Remark**

**1.**

**Implicit Function Theorem**(Theorem 4.E in [31]). Let $X,Y$ and Z be Banach spaces and let $F(x,y):X\times Y\to Z$ be a ${C}^{1}$ map on an open neighborhood of the point $({x}_{0},{y}_{0})\in X\times Y$. Assume that

**Theorem**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

- (a)
- If $\Omega >|\gamma |$, the Constraint (37) is not satisfied because the left-hand side$$2{(\Omega +4{A}^{2})}^{3}\ge 2{\Omega}^{3}>2\Omega {\gamma}^{2}$$
- (b)
- If $\Omega <|\gamma |$ and ${A}^{2}\ge {A}_{+}^{2}$, where ${A}_{+}^{2}$ is given by Equation (27), the Constraint (37) is not satisfied because the left-hand side$$2{(\Omega +4{A}^{2})}^{3}\ge 2{(\Omega +4{A}_{+}^{2})}^{3}=2{|\gamma |}^{3}$$
- (c)
- If $\Omega <-|\gamma |$ and ${A}^{2}\le {A}_{-}^{2}$, where ${A}_{-}^{2}$ is given by Equation (28), the Constraint (37) is not satisfied because the left-hand side is estimated by$$2{(4{A}^{2}+\Omega )}^{3}\le 2(4{A}_{-}^{2}-{|\Omega |)}^{3}={min\{-2|\gamma |}^{3},-{|\Omega |}^{3}/4\}$$$$2{(4{A}^{2}+\Omega )}^{3}=-{2|\gamma |}^{3}=-|\Omega |{\gamma}^{2}=\Omega {\gamma}^{2}$$

**Remark**

**2.**

## 5. Stability of Zero Equilibrium

**Proposition**

**1.**

**Proof.**

**Remark**

**3.**

## 6. Variational Characterization of Breathers

**Perturbation Theory for Linear Operators**(Theorem VII.1.7 in [32]). Let $T(\u03f5)$ be a family of bounded operators from Banach space X to itself, which depends analytically on the small parameter ϵ. If the spectrum of $T(0)$ is separated into two parts, the subspaces of X corresponding to the separated parts also depend analytically on ϵ. In particular, the spectrum of $T(\u03f5)$ is separated into two parts for any $\u03f5\ne 0$ sufficiently small.

**Theorem**

**2.**

- If $|E|>{E}_{0}$, the spectrum of ${\mathcal{H}}_{E}^{\u2033}$ in ${\ell}^{2}(\mathbb{Z})$ includes infinite-dimensional positive and negative parts.
- If $|E|<{E}_{0}$ and $\Omega <-|\gamma |$, the spectrum of ${\mathcal{H}}_{E}^{\u2033}$ in ${\ell}^{2}(\mathbb{Z})$ includes an infinite-dimensional negative part and either three or one simple positive eigenvalues for branches (b) and (c) of Lemma 1 respectively.

**Proof.**

- (a)
- ${\mu}_{1},{\mu}_{2},{\mu}_{3}>0$.
- (b)
- ${\mu}_{1},{\mu}_{2},{\mu}_{3}>0$.
- (c)
- ${\mu}_{1}<0$, ${\mu}_{2}>0$, and ${\mu}_{3}<0$.

- (1),(3)
- If $|E|>{E}_{0}:=\sqrt{{\Omega}^{2}-{\gamma}^{2}}$, then ${\mu}_{+}>0$ and ${\mu}_{-}<0$.
- (2),(4)
- If $|E|<{E}_{0}$ and $\Omega <-|\gamma |$, then ${\mu}_{+},{\mu}_{-}<0$.

- Since ${\mathcal{H}}_{E}^{\u2033}$ is Hermitian on ${\ell}^{2}(\mathbb{Z})$, its spectrum is a subset of the real line for every $\u03f5\ne 0$.
- The zero eigenvalue persists with respect to $\u03f5\ne 0$ at zero because the Eigenvector (47) belongs to the kernel of ${\mathcal{H}}_{E}^{\u2033}$ due to the gauge invariance for every $\u03f5\ne 0$.
- The other eigenvalues of $\mathcal{L}$ are isolated away from zero. The spectrum of ${\mathcal{H}}_{E}^{\u2033}$ is continuous with respect to ϵ and includes infinite-dimensional parts near points ${\mu}_{+}$ and ${\mu}_{-}$ for small $\u03f5>0$ (which may include continuous spectrum and isolated eigenvalues) as well as simple eigenvalues near ${\mu}_{1,2,3}$ (if ${\mu}_{1,2,3}$ are different from ${\mu}_{\pm}$).

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

## 7. Spectral and Orbital Stability of Breathers

**Proposition**

**2.**

**Proof.**

**Hamilton–Krein Index Theorem**(Theorem 3.3 in [33]). Let L be a self-adjoint operator in ${\ell}^{2}$ with finitely many negative eigenvalues $n(L)$, a simple zero eigenvalue with eigenfunction ${v}_{0}$, and the rest of its spectrum is bounded from below by a positive number. Let J be a bounded invertible skew-symmetric operator in ${\ell}^{2}$. Let ${k}_{r}$ be a number of positive real eigenvalues of $JL$, ${k}_{c}$ be a number of quadruplets $\{\pm \lambda ,\pm \overline{\lambda}\}$ that are neither in $\mathbb{R}$ nor in $i\mathbb{R}$, and ${k}_{i}^{-}$ be a number of purely imaginary pairs of eigenvalues of $JL$ whose invariant subspaces lie in the negative subspace of L. Let $D={\langle {L}^{-1}{J}^{-1}{v}_{0},{J}^{-1}{v}_{0}\rangle}_{{\ell}^{2}}$ be finite and nonzero. Then,

**Lemma**

**3.**

**Proof.**

**Definition**

**1.**

**Theorem**

**3.**

**Proof.**

**Definition**

**2.**

**Theorem**

**4.**

**Proof.**

- The double zero eigenvalue persists with respect to $\u03f5\ne 0$ at zero because of the gauge invariance of the breather $(U,V)$ (with respect to rotation of the complex phase). Indeed, ${\mathcal{H}}_{E}^{\u2033}(\sigma \Phi )=0$ follows from the result of Theorem 2. The generalized eigenvector is defined by equation $\mathcal{S}{\mathcal{H}}_{E}^{\u2033}\Psi =\sigma \Phi $, which is equivalent to equation ${\mathcal{H}}_{E}^{\u2033}\Psi ={(V,\overline{V},U,\overline{U})}^{T}$. Differentiating Equation (21) in E, we obtain $\Psi ={\partial}_{E}\Phi $. Since $\mathrm{dim}[\mathrm{Ker}({\mathcal{H}}_{E}^{\u2033})]=1$ and$${\langle \sigma \Phi ,\mathcal{S}{\partial}_{E}\Phi \rangle}_{{\ell}^{2}}=\sum _{n\in \mathbb{Z}}{\partial}_{E}\left({U}_{n}{\overline{V}}_{n}+{\overline{U}}_{n}{V}_{n}\right)=\frac{d{Q}_{u,v}}{dE}$$
- Using the same Computation (59), it is clear that ${\lambda}_{0}\in i\mathbb{R}$ for every E along branches (a) and (b) of Lemma 1. Assume that ${\lambda}_{0}\ne \pm {\lambda}_{+}$ and ${\lambda}_{0}\ne \pm {\lambda}_{-}$, which is expressed by the non-degeneracy Condition (62). Then, the pair $\pm {\lambda}_{0}$ is isolated from the rest of the spectrum of the operator $-i\mathcal{S}{\mathcal{H}}_{E}^{\u2033}$ at $\u03f5=0$. Since the eigenvalues $\lambda =\pm {\lambda}_{0}$ are simple and purely imaginary, they persist on the imaginary axis for $\u03f5\ne 0$ because they cannot leave the imaginary axis by the Hamiltonian symmetry of Proposition 2.
- If $|\gamma |<|\Omega |$, $E\ne 0$, and $E\ne \pm {E}_{0}$, the semi-simple eigenvalues $\pm {\lambda}_{+}$ and $\pm {\lambda}_{-}$ of infinite multiplicity are nonzero and located at the imaginary axis at different points for $\u03f5=0$. They persist on the imaginary axis for $\u03f5\ne 0$ according to the following perturbation argument. First, for the central site $n=0$, the spectral Problem (53) can be written in the following abstract form$$\left(S{\mathcal{L}}_{0}(\u03f5)-2\u03f5S-i\lambda I\right){\varphi}_{0}=-\u03f5S({\varphi}_{1}+{\varphi}_{-1})$$$${\varphi}_{0}=-\u03f5{\left(S{\mathcal{L}}_{0}(\u03f5)-2\u03f5S-i\lambda I\right)}^{-1}S({\varphi}_{1}+{\varphi}_{-1})$$$$S{\mathcal{L}}_{n}(\u03f5){\varphi}_{n}+\u03f5S{(\Delta \varphi )}_{n}-i\lambda {\varphi}_{n}=-{\delta}_{n,\pm 1}\u03f5S{\varphi}_{0},\phantom{\rule{1.em}{0ex}}\pm n\in \mathbb{N}$$$$S{\mathcal{L}}_{n}{\varphi}_{n}+\u03f5S{(\Delta \varphi )}_{n}=i\lambda {\varphi}_{n},\phantom{\rule{1.em}{0ex}}\pm n\in \mathbb{N}$$$${(E\pm i\lambda )}^{2}+{\gamma}^{2}-{\left(\Omega -4\u03f5{sin}^{2}\frac{k}{2}\right)}^{2}=0$$$$\lambda =\pm i\left(E\pm \sqrt{{\left(\Omega -4\u03f5{sin}^{2}\frac{k}{2}\right)}^{2}-{\gamma}^{2}}\right)$$In addition to the continuous spectrum given by Equation (67), there may exist isolated eigenvalues near $\pm {\lambda}_{+}$ and $\pm {\lambda}_{-}$, which are found from the second-order perturbation theory [34]. Under the condition $E\ne 0$ and $E\ne \pm {E}_{0}$, these eigenvalues are purely imaginary. Therefore, the infinite-dimensional part of the spectrum of the operator $-i\mathcal{S}{\mathcal{H}}_{E}^{\u2033}$ persists on the imaginary axis for $\u03f5\ne 0$ near the points $\pm {\lambda}_{+}$ and $\pm {\lambda}_{-}$ of infinite algebraic multiplicity.

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Definition**

**3.**

**Lemma**

**4.**

- (a)
- the subspaces of $-i\mathcal{S}{\mathcal{H}}_{E}^{\u2033}$ in ${\ell}^{2}(\mathbb{Z})$ near $\pm {\lambda}_{+}$, $\pm {\lambda}_{-}$, and $\pm {\lambda}_{0}$ have positive, negative, and positive Krein signature, respectively;
- (b)
- the subspaces of $-i\mathcal{S}{\mathcal{H}}_{E}^{\u2033}$ in ${\ell}^{2}(\mathbb{Z})$ near $\pm {\lambda}_{+}$, $\pm {\lambda}_{-}$, and $\pm {\lambda}_{0}$ have negative, positive (if $E>{E}_{0}$) or negative (if $E<{E}_{0}$), and positive Krein signature, respectively;
- (c)
- all subspaces of $-i\mathcal{S}{\mathcal{H}}_{E}^{\u2033}$ in ${\ell}^{2}(\mathbb{Z})$ near $\pm {\lambda}_{+}$, $\pm {\lambda}_{-}$, and $\pm {\lambda}_{0}$ (if ${\lambda}_{0}\in i\mathbb{R}$) have negative Krein signature.

**Proof.**

- (a)
- We can see on panel (a) of Figure 3 that ${\lambda}_{0},{\lambda}_{\pm}$ do not intersect for every $E>{E}_{0}$ and are located within fixed distance $\mathcal{O}(1)$, as $|E|\to \infty $. Note that the upper-most ${\lambda}_{0}$ and ${\lambda}_{+}$ have positive Krein signature, whereas the lowest ${\lambda}_{-}$ has negative Krein signature, as is given by Lemma 4.
- (b)
- We observe on panel (b) of Figure 3 that ${\lambda}_{+}$ intersects ${\lambda}_{0}$, creating a small bubble of instability in the spectrum. The insert shows that the bubble shrinks as $\u03f5\to 0$, in agreement with Theorem 4. There is also an intersection between ${\lambda}_{-}$ and ${\lambda}_{0}$, which does not create instability. These results are explained by the Krein signature computations in Lemma 4. Instability is induced by opposite Krein signatures between ${\lambda}_{+}$ and ${\lambda}_{0}$, whereas crossing of ${\lambda}_{-}$ and ${\lambda}_{0}$ with the same Krein signatures is safe of instabilities. Note that for small E, the isolated eigenvalue ${\lambda}_{0}$ is located above both the spectral bands near ${\lambda}_{+}$ and ${\lambda}_{-}$. The gap in the numerical data near $E={E}_{0}$ indicates failure to continue the breather solution numerically in ϵ, in agreement with the proof of Theorem 1.
- (c)
- We observe from panel (c) of Figure 3 that ${\lambda}_{0}$ and $-{\lambda}_{-}$ intersect but do not create instabilities, since all parts of the spectrum have the same signature, as is given by Lemma 4. In fact, the branch is both spectrally and orbitally stable as long as ${\lambda}_{0}\in i\mathbb{R}$, in agreement with Theorem 3. On the other hand, there is ${E}_{s}\in (0,{E}_{0})$, if $\Omega \in (-2\sqrt{2}|\gamma |,-|\gamma |)$, such that ${\lambda}_{0}\in \mathbb{R}$ for $E\in (0,{E}_{s})$, which indicates instability of branch (c), again, in agreement with Theorem 3.

**Lemma**

**5.**

**Proof.**

**Remark**

**10.**

## 8. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A schematic picture for the chain of coupled pendula connected by torsional springs, where each pair is hung on a common string.

**Table 1.**A summary of results on breather solutions for small ϵ. Here, IB is a narrow instability bubble seen on panel (b) of Figure 3.

Parameter Intervals | $|\mathit{E}|>{\mathit{E}}_{\mathbf{0}}$ | $|\mathit{E}|<{\mathit{E}}_{\mathbf{0}}$ | ||
---|---|---|---|---|

$\mathbf{\Omega}>|\mathit{\gamma}|$ | $\mathbf{\Omega}<-|\mathit{\gamma}|$ | $\mathbf{\Omega}<-|\mathit{\gamma}|$ | $\mathbf{\Omega}<-|\mathit{\gamma}|$ | |

Existence on Figure 2 | point 1 on branch (a) | point 2 on branch (b) | point 3 on branch (b) | point 4 on branch (c) |

Continuum | Sign-indefinite | Sign-indefinite | Negative | Negative |

Spectral stability | Yes | Yes (IB) | Yes (IB) | Depends on parameters |

Orbital stability | No | No | Yes if $|{\lambda}_{0}|>|{\lambda}_{\pm}|$ | Yes if spectrally stable |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chernyavsky, A.; Pelinovsky, D.E.
Breathers in Hamiltonian PT -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force. *Symmetry* **2016**, *8*, 59.
https://doi.org/10.3390/sym8070059

**AMA Style**

Chernyavsky A, Pelinovsky DE.
Breathers in Hamiltonian PT -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force. *Symmetry*. 2016; 8(7):59.
https://doi.org/10.3390/sym8070059

**Chicago/Turabian Style**

Chernyavsky, Alexander, and Dmitry E. Pelinovsky.
2016. "Breathers in Hamiltonian PT -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force" *Symmetry* 8, no. 7: 59.
https://doi.org/10.3390/sym8070059