DISCRETE BREATHERS OF NONLINEAR DIMER LATTICES BRIDGING THE ANTI-CONTINUOUS AND CONTINUOUS LIMITS ANDREW HOFSTRAND HUAIYU LIyAND MICHAEL I. WEINSTEINz

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DISCRETE BREATHERS OF NONLINEAR DIMER LATTICES:

BRIDGING THE ANTI-CONTINUOUS AND CONTINUOUS LIMITS

ANDREW HOFSTRAND∗, HUAIYU LI†,AND MICHAEL I. WEINSTEIN‡

Abstract. In this work, we study the dynamics of an inﬁnite array of nonlinear dimer oscillators

which are linearly coupled as in the classical model of Su, Schrieﬀer and Heeger (SSH). The ratio

of in-cell and out-of-cell couplings of the SSH model deﬁnes distinct phases: topologically trivial

and topologically non-trivial. We ﬁrst consider the case of weak out-of-cell coupling, corresponding

to the topologically trivial regime for linear SSH; for any prescribed isolated dimer frequency, ωb,

which satisﬁes non-resonance and non-degeneracy assumptions, we prove that there are discrete

breather solutions for suﬃciently small values of the out-of-cell coupling parameter. These states are

2π/ωb- periodic in time and exponentially localized in space. We then study the global continuation

with respect to this coupling parameter. We ﬁrst consider the case where ωb, the seeding discrete

breather frequency, is in the (coupling dependent) phonon gap of the underlying linear inﬁnite array.

As the coupling is increased, the phonon gap decreases in width and tends to a point (at which the

topological transition for linear SSH occurs). In this limit, the spatial scale of the discrete breather

grows and its amplitude decreases, indicating the weakly nonlinear long wave regime. Asymptotic

analysis shows that in this regime the discrete breather envelope is determined by a vector gap soliton

of the limiting envelope equations. We use the envelope theory to describe discrete breathers for SSH-

coupling parameters corresponding to topologically trivial and, by exploiting an emergent symmetry,

topologically nontrivial regimes, when the spectral gap is small. Our asymptotic theory shows

excellent agreement with extensive numerical simulations over a wide range of parameters. Analogous

asymptotic and numerical results are obtained for the continuations from the anti-continuous regime

for frequencies, ωb, below the acoustic or above the optical phonon bands.

Key words. discrete lattice dynamical systems, topological states, multiple-scale asymptotics.

AMS subject classiﬁcations. 34A34, 34A33, 34C25

1. Introduction.

1.1. Motivation and background. There is great current interest in the study

of wave propagation through discrete and continuous periodic media, which exhibits

nontrivial topological properties. While it is common for physical systems to support

defect modes concentrated at points or interfaces, these modes are in general not stable

against signiﬁcant perturbations of the structure. However, it has been recognized

that topological characteristics in the bulk (Floquet-Bloch) band structure can give

rise to modes which are robust against large (but localized) perturbations of the

system. The role of band structure topology in wave physics was ﬁrst recognized

in the context of condensed matter physics, e.g. the integer quantum Hall eﬀect

[20] and topological insulators [14]. The hallmark of topological materials is the

presence of topologically protected edge states. These states are localized at interfaces

(line defects, facets), propagate unidirectionally and are robust against localized -

even large - imperfections in the system. Many of the topological wave phenomena

observed in these contexts were subsequently realized in engineered metamaterial

systems in photonics [13,28], acoustics [26], electronics [12] and elasticity [41] which,

in the regime of linear phenomena, are characterized by a linear band structure.

There is very wide interest in technologies based on topologically protected states

∗New York Institute of Technology, New York, NY (ahofstra@nyit.edu).

†Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY

(hl3002@columbia.edu).

‡Department of Applied Physics and Applied Mathematics and Department of Mathematics,

Columbia University, New York, NY (miw2103@columbia.edu).

arXiv:2210.04387v2 [nlin.PS] 3 Apr 2023

2A. HOFSTRAND, H. LI, AND M. I. WEINSTEIN

due to the potential for extraordinarily robust energy and information transfer in

communications and computing.

Such systems can naturally be probed in the nonlinear regime via strong excita-

tion, and so it is of great interest to study whether topological properties persist in the

regime where nonlinear eﬀects are present and whether perhaps diﬀerent topological

phenomena emerge [12,6,7,19,33].

Among the simplest models exhibiting topological phases is the Su-Schrieﬀer-

Heeger (SSH) model [39], a discrete (tight binding) model on one-dimensional lattice

in which two ”atoms” per cell (dimers) are linearly and nearest-neighbor coupled.

Fig. 1.(a) schematic of the inﬁnite dimer lattice considered with A (B) sites in red (blue) and

intra (inter) cell coupling γin (γout ); (b) band structure of the lattice for various ratios |γin /γout |.

A gap in the energy spectrum opens if and only if |γin /γout| 6= 1.

Figure 1displays a schematic of SSH array of dimers. The red sites are called A−

sites and the blue sites are called B−sites. Each A−site has two nearest neighbor

B−sites and each B−site has two nearest neighbor A−sites. In-cell (intra-cell) and

out-of-cell (inter-cell) nearest neighbors are coupled via hopping coeﬃcients γin ∈R

and γout ∈R, respectively: for n∈Z,

EψA

n=γinψB

n+γoutψB

n−1

(1.1)

EψB

n=γinψA

n+γoutψA

n+1.(1.2)

The spectrum of the SSH-Hamiltonian, HSSH, acting in the space of discrete wave

functions, l2(Z), consists of two real intervals (bands), swept out by the two eigenval-

ues E−(k)≤0≤E+(k), as the quasimomentum kvaries over the interval [0,2π], of

the family of Bloch Hamiltonian HSSH (k) = σ1h1(k) + σ2h2(k) (obtained by discrete

Fourier transform, σjdenote Pauli matrices); see Figure 1. A gap in the spectrum

occurs for |γin/γout| 6= 1 and the bands touch in a linear crossing for |γin/γout|= 1.

The two distinct topological phases correspond to |γin/γout|>1 (trivial) and

|γin/γout|<1 (non-trivial), and are identiﬁed with the cases where the Zak phase, a

winding number (about the origin) associated with the variation of the vector ﬁeld

h(k) = (h1(k), h2(k)) as kvaries over S1=R/2πZ, is equal to zero or one. The

topological character is also manifested in the spectrum of states for a terminated

DISCRETE BREATHERS OF NONLINEAR DIMER LATTICES 3

(semi-inﬁnite) structure; there exists a zero energy edge state which decays expo-

nentially into the bulk if and only if one is in the topologically non-trivial phase

(|γin/γout|<1). See, for example, [32,31].

In this paper we study a non-linear variant of the SSH-model, introduced in

[7]. In this model, each “atom” of the array corresponds to a nonlinear mass-spring

oscillator described by a Newtonian law: ¨x=−V0(x), with an even anharmonic

potential, V(x). For illustrative purposes, it will be convenient at times to work with

the speciﬁc potential:

(1.3) V(x) = 3

2x2+Γ

4x4.

The case when Γ >0 is referred to as the case of a hardening nonlinearity and the

case when Γ <0 as a softening nonlinearity.

Within a ﬁxed dimer / cell, the two mass spring systems with amplitudes xAand

xBare linearly coupled via the in-cell coupling coeﬃcient γin :

¨xA=−V0(xA) + γinxB

(1.4)

¨xB=−V0(xB) + γinxA

The coupled system (1.4) is the fundamental unit with which we build up a nonlinear

SSH-network. The system (1.4) will be assumed to have non-resonant and non-

degenerate time-periodic orbits in a sense which we shall make precise in Theorem 2.2

below.

We build an SSH- network of nonlinear dimers by coupling each oscillator to its

out-of-cell nearest neighbors via a second coupling coeﬃcient, γout. This gives the

system:

¨xA

n=−V0(xA

n) + λγoutxB

n−1+γinxB

(1.5)

¨xB

n=−V0(xB

n) + γinxA

n+λγoutxA

n+1, n ∈Z.

In (1.5) we make explicit the B−site terms which interact with the xA

nand the A−

site terms which interact with xB

n.V0consists of both linear and non-linear onsite

contributions. The non-negative parameter, λhas been inserted in order to interpolate

between the anti-continuous limit (λ= 0) and globally coupled models.

More generally, we study

(1.6) ¨xA

n+V0(xA

n)−γinxB

¨xB

n+V0(xB

n)−γinxA

n+λRxA

xBn

=0

0, n ∈Z,

where Ris a bounded linear operator on l2(Z;R2). The coupling operator, R, can

couple sites beyond nearest-neighbor, but its deﬁning matrix elements are assumed

to be exponentially decaying away from the diagonal; see (2.2)). The case of nearest

neighbor interactions (see (1.5)) corresponds to:

RxA

xBn

=−γout xB

n−1

n+1.

Consider the band structure of the linearized dynamics for (1.5) about the zero

state (xA

n=xB

n= 0, n∈Z), determined by the set of non-trivial plane wave states:

4A. HOFSTRAND, H. LI, AND M. I. WEINSTEIN

ei(kn−ωt)ξ, 06=ξ∈C2. In terms of E=ω2−ω2

0vs. k, the band spectrum is a

re-centering about ω2

0=V00(0) of the SSH band spectrum:

(1.7)

E±(k) = ω2±(k)−ω2

0=±s(γin −λγout)2+ 4γinλγout cos2k

2, ω2

0≡V00(0).

Figure 1displays the graphs of these band functions. The two spectral bands are

separated by a gap for |λγout/γin| 6= 1 and touch at a linear crossing for |λγout/γin|= 1.

It is therefore natural to contrast the properties of the λ−parametrized family of

equations (1.5) for the (linearly) topologically distinct regimes |λγout/γin|<1 and

|λγout/γin|>1.

Remark 1.1 (Phonon gaps). In terms of the frequency parameter ω, there are

two pairs of dispersion curves, symmetric about ω= 0, each pair having phonon gap

for |λγout/γin| 6= 1: one about ω0and one about −ω0, each of width ≈ω−1

0||, where

=γin −λγout

1.2. Summary of the article and results. We study the existence and prop-

erties of discrete breathers, solutions of the inﬁnite lattice nonlinear system (1.6),

which are periodic in time and localized on the discrete lattice Z. We outline the key

points of this paper:

1. Existence of discrete breathers; Theorem 2.2.Assume that the anharmonic

potential in (1.6) satisﬁes V(−x) = V(x). Let t7→ X∗(t) denote a non-

resonant and non-degenerate Tb=2π

ωb−periodic solution of the limiting (λ=

0) inﬁnite dimer array, associated with (1.4). Then, for all λsuﬃciently small

and non-zero, there is a unique Tb−periodic solution

t7→ Xλ(t) = xA

n(t)

n(t)n∈Z

with X0=X∗of the globally coupled lattice equations (1.6). This solution

lies in the space H2

Tb, consisting of sequences X(t), which satisfy X(t) =

X(−t), and together with derivatives up to order 2, are Tb−periodic and

square integrable over [0, Tb], and square summable (spatially) over Z:

kXλk2

Tb≡X

n∈ZZTb

j=0 

dtjxA

n(t;λ)

n(t;λ)

2dt < ∞.

Furthermore, the mapping λ7→ kXλkH2

is smooth. For a discussion of

the behavior when the non-resonance hypothesis is violated, see in particular

Remark 2.3 and Figure 3.

Our proof is based on a Poincar´e continuation strategy, used in the pioneer-

ing article [23] on discrete breathers. The richer structure of the building-

block isolated dimer dynamical system (1.4) (anti-continuous limit) allows for

richer behaviors in the global array. Finally recall that for general nonlinear

autonomous dynamical systems the period of the solution varies along the

continuation. Here, the symmetry condition on V(x) enables us to restrict

our study to time-reversible solutions with ﬁxed period. The analysis can be

adapted to more general potentials, V(x), by incorporating the determination

of the discrete breather period as a function of λ.

DISCRETE BREATHERS OF NONLINEAR DIMER LATTICES 5

2. Applications of Theorem 2.2.In Section 3we apply Theorem 2.2 to obtain

discrete breather solutions which are continuations of two classes of solutions

to the isolated dimer dynamical system (1.4): in-phase (Type I) and out-of-

phase (Type II) solutions. We verify the non-resonance and non-degeneracy

assumptions of Theorem 2.2 by a combination of rigorous analysis and nu-

merical computation.

3. Exponential spatial decay of discrete breathers, Theorem 4.1 The breather

solutions obtained via Theorem 2.2 have the square-summable decay behavior

of functions in H2

Tb. Hence they are only guaranteed to decay at inﬁnity in a

mild sense. In Section 4we prove Theorem 4.1, a general result on exponential

spatial decay of the discrete breathers, which applies to those constructed in

Theorem 2.2. Our proof uses ideas underlying Combes and Thomas discrete

operator estimates (see, for example, [2]), and oﬀers a diﬀerent perspective

on the earlier decay results in [23].

4. Numerical simulations of discrete breathers: ranging from the highly discrete

(anti-continuous) regime to the nearly continuum regime. For

0< λ < λ?≡ |γin/γout|,

the phonon spectrum (linearized spectrum about the zero state) has an open

spectral gap centered about the linearized “atomic” frequency, ω0=pV00(0);

see Remark 1.1. The gap width is of order one for λnear zero and shrinks

down to the point ω0as λapproaches λ∗. Using a numerical method, outlined

in Appendix A, we construct solutions corresponding to ﬁne grid of λ−values

starting at λ= 0 and continued, when possible, till very close to λ∗. The

initializing discrete breather, which is supported on the n= 0 dimer, is taken

to be a periodic orbit with frequency, ωbcorresponding to one of the following

cases:

(A) ωb≈ω0in the phonon gap,

(B) ωbjust above the optical branch of the phonon spectrum and

Figure 2presents a summary of our continuation results for in-phase (Type

I) periodic orbits ωbin Case (A); this terminology is introduced in Section 3.

For λ > 0 suﬃciently small, the breather is strongly localized on a few lattice

sites; this behavior is captured by Theorem 2.2. For λless than but near λ?,

where the parameter

≡γin −λγout

is small, the spectral (phonon) gap is small, and we expect the discrete

breather spatial proﬁle to decay very slowly on the lattice length-scale. This

behavior is clearly indicated in panels (d)-(f) of Figure 2.

5. Continuum envelope theory in the small phonon gap regime, λ→λ?As λ

approaches λ?≡ |γin/γout|(→0) the discrete breather has the structure

of a weakly nonlinear wave-packet, whose amplitude decreases and width

increases. In Section 5we consider separately the limits: λ↑λ?(↓0) and

λ↓λ?(↑0), which correspond, respectively, to the vanishing gap limit in

the topologically trivial and topologically non-trivial linear phases.

In each of these scenarios we construct, by multiple scale asymptotic analysis,

weakly nonlinear wave packets comprised of bulk spectral components corre-

sponding to energies near the band crossing (see Figure 1b). This multiple

scale expansion describes discrete breathers as a bifurcation from the phonon

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DISCRETEBREATHERSOFNONLINEARDIMERLATTICES:BRIDGINGTHEANTI-CONTINUOUSANDCONTINUOUSLIMITSANDREWHOFSTRAND,HUAIYULIy,ANDMICHAELI.WEINSTEINzAbstract.Inthiswork,westudythedynamicsofaninnitearrayofnonlineardimeroscillatorswhicharelinearlycoupledasintheclassicalmodelofSu,SchrieerandHeeger(SSH).Theratioof...

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DISCRETE BREATHERS OF NONLINEAR DIMER LATTICES BRIDGING THE ANTI-CONTINUOUS AND CONTINUOUS LIMITS ANDREW HOFSTRAND HUAIYU LIyAND MICHAEL I. WEINSTEINz

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