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Solitary phase waves in a chain of autonomous oscillators

In the present paper we study phase waves of self-sustained oscillators with a nearest neighbor dispersive coupling on an infinite lattice. To analyze the underlying dynamics we approximate the lattice with a quasi-continuum, QC. The resulting partial differential model is then further reduced to the Gardner equation which predicts many properties of the underlying solitary structures. Using an iterative procedure on the original lattice equations we determine the shapes of solitary waves, kinks, and the flat-like solitons, that we refer to as flatons. Direct numerical experiments reveal that the interaction of solitons and flatons on the lattice is notably clean. All in all we find that both the QC and the Gardner equation predict remarkably well the discrete patterns and their dynamics.

Interacting limit cycle oscillators play a fundamental role in synchronization studies. When the coupling is small, systems dynamics reduces to that of the oscillators phases. In many set-ups, as in the seminal Kuramoto model, the interaction is dissipative and leads ultimately to synchrony of all phases. Yet, in many experimental set-ups the coupling is dispersive with the resulting phase equations being conservative. We focus on a conservative phase dynamics on a one-dimensional lattice and demonstrate the existence of a very robust dynamics of solitary waves. A crucial role in the understanding of the dynamics is played by its quasi-continuum approximation via a partial differential equation which provides a remarkably accurate description of the underlying phenomena on the lattice.

I. INTRODUCTION
Dynamics of networks of oscillators have gathered in recent years considerable attention.
The dynamics of even the simplest network architectures, like global coupling in a population 1 or a local coupling on a regular lattice (see e.g., 2 ), is highly nontrivial even for the simplest Kuramoto-Sakaguchi type of interactions 3 . In the latter case, one addresses the phase dynamics of oscillators coupled via their first harmonics, with an additional phase shift. This phase shift determines the relative importance of dissipative, diffusion-type, and conservative (dispersive) interactions. For the diffusion-type coupling, the interaction results in a global synchronization of a homogeneous lattice (in a inhomogeneous lattice with random oscillator frequencies, the diffusive coupling should be strong enough to ensure synchrony 4 .) Homogeneous oscillator lattices with a purely conservative coupling follow a very different path for their phase dynamics, which is, surprisingly enough, Hamiltonian leading to a formation of a nontrivial waves, like compactons or kovatons [5][6][7][8] .
In the present paper we extend our previous work and address both analytically and numerically a dispersive variant of the Kuramoto-Sagakuchi chain. Studying the solitary structures we find solitons in a bounded range of velocities. At range's edge solitons collapse and kink/anti-kink emerge. However, close to the transition's threshold, we find a narrow strip of velocities wherein solitons undergo a structural change and rather than grow with amplitude they widen and turn into flat-top solitons, referred to as flatons. Notably, interaction between solitons and flatons is found to be remarkably clean. A purely dispersive coupling of self-sustained oscillators, of the type studied here, is relevant in micromechanical oscillators studied both theoretically 9-13 and more recently were explored experimentally 14 .
Note that the long chains we focus on are not easily accessible experimentally. The more realistic short chains where boundary effects matter will be a subject of future studies.

II. THE BASIC MODEL
Consider a chain of self-sustained, autonomous, oscillators with a nearest-neighbor coupling, described via their complex amplitudes A n : The amplitudes in (1) were normalized with the equilibrium amplitude of a single oscillator being unity, whereas µ, assumed to be large, governs the relaxation rate to the equilibrium, so that the limit cycle oscillations are strongly stable. Contrary to the large dissipation of the local amplitude dynamics, the present coupling is assumed to be purely conservative as found in nano-electro-mechanical setups addressed in 9-13 and recently realized experimentally 14 . In the µ → ∞ limit one may neglect the changes in amplitude's modulus and set A n = e iϕn , where ϕ n is the phase of the oscillator. This leads to a phase chain model In chain (2), any linear phase profile ϕ n = (π/2 − α)n is uniformly rotating (a so-called twisted state 15 ). To study the deviations from this plane wave we introduce the phase difference θ n = ϕ n+1 − ϕ n + α − π/2, and rescale the time εt → t, to obtain the following basic model As is clear from its derivation, Eq. (3) which is the basis of our studies, describes phase waves on the top of the plane wave, whereas α defines the slope of the background linear phase profile. The very particular α = π/2 case was addressed in Refs. [5 and 6].
Among the features of Eq. (3) we note the dispersion relation ω = 2 cos α sin k, −π ≤ k ≤ π of its linear waves and the aforementioned "sonic vacuum" when α = π/2 5,6 , for linear waves are absent and the lattice becomes essentially nonlinear. Finally, we note the conservation laws valid on an infinite chain.

A. General Features
In spite of their innocuous appearance, Eqs. (3) describe a complicated nonlinear system which defies a direct analysis. As in previous works 5,6,8 , to gain insight into its dynamics we shall adopt a quasi-continuous description wherein the discrete system is replaced with a continuous formulation which keeps a trace of its discrete origin. To this end we approximate the chain as: θ j (t) → θ(x, t), and f j+1 − f j−1 → 2(f x + 1 6 f xxx ) which yields 1 2 As there is no small parameter in the problem, Eq. (6) cannot, in the strict mathematical sense, be considered as an asymptotic description of the discrete problem and its utility can Similarly to the discrete system, Eq. (6) is also invariant under θ → 2α − θ. Thus, if θ 1 is solution so is θ 2 = 2α − θ 1 . This is a cyclic property with θ 2 leading back to θ 1 and as in the discrete case the trivial θ = 0 solution yields θ = 2α whereas θ = α is mapped into itself.
Separating the linear convection, we rewrite (6) as where Note the non-monotone nature of C qc (θ); it attains its maximal value at θ = α and vanishes at both θ = 0 and θ = 2α, which merely reflects its invariance under (4).
Eq. (6) conserves four local quantities where Q(θ) = sin(θ ′ − α)dθ ′ = cos(α−θ). I 1 and I 2 are in a direct correspondence with the corresponding discrete conservation quantities. Notably, the QC admits also a Lagrangian (for more details see 8 ) where Consequently, the QC conserves also the momentum v 2 dx, and in the original variables

B. The Gardner Approximation
To unfold the key phenomena we begin with a weakly non-linear regime, θ << 1, wherein Note that whereas, on one hand, we have neglected the nonlinear corrections to the third derivative which, insofar that θ − α = ±π/2, has only a minor quantitative impact, on the other hand, to preserve the crucial non-monotone nature of convection we have carried its expansion to the third order. Using the Galilean invariance to dispense with convection's linear part, after normalization Eq. (13) begets the celebrated Gardner equation Both C G and Eq. (14) are invariant under u → 1 − u, which echoes the invariance of the original lattice and its QC rendition under (4). Its solitons, traveling with speed λ, satisfy an ordinary differential equation with respect to s = x − λt: Note that due to the defocusing effect of the cubic term, the potential peaks at u = 3 4 [1 + 1 − 8λ/9], and comes down as the speed increases. Consequently, the resulting solitons have a bounded range of admissible propagation speeds: 0 < λ < 1. At the limiting velocity λ = 1, potential's peaks touches the u-axis and the soliton solution (16) flattens into a constant = 1. This is a singular limit at which both kink and an anti-kink form Close to the edge of solitons upper velocity range, there is a narrow strip of velocities where solitons undergo a structural change and rather than grow with amplitude they begin to widen and their top flattens. To extract these features from Eq. (16), let with soliton's amplitude being u max = 1 − ǫ. The extent of soliton's widening is expressed via x 1/2 , where soliton's amplitude has decreased by half; Thus, amplitude(velocity) changes ∼ 1 − ǫ (∼ 1 − ǫ 2 ) which are pretty much numerically unobservable, cause solitons to widen as ∼ ln 1/ǫ. We shall refer to the flat-like solitons as flatons.
The proximity of flatons velocities to the edge of the admissible speeds range enables to approximate them by a kink-antikink pair placed at 2x 1/2 ≫ 1 from each other: and provides an upper bound to all flatons.

C. Analysis of the Traveling Waves
We now proceed to unfold the solitary wave structure of QC, Eq. (6). Seeking travelling and, as their small amplitude regime indicates, traveling waves call for 2 cos α < λ. Integrating Eq. (21) we have where the potential P qc reads A typical potential landscape for α = π/4 is displayed in Fig. 1 for three values of λ.
As in the weakly nonlinear case, the bounded potential sets an upper bound at which propagation is possible, corresponding to potential's top descending toward the θ-axis at θ = 2α with λ = 2 sin α/α, where the soliton flattens into a constant and kink/anti-kink emerge. Consequently, 2 cos α = λ min < λ < λ max = 2 sin α α determines the interval of admissible velocities of solitary waves.
On the basis of the weakly nonlinear regime, we anticipate that as λ → λ max = 2 sin(α)/α, solitons turn into flatons which, as illustrated in Fig. 2, is indeed the case.
Note the structural singularity of Eq. (22) at θ − α = ±π/2, denoted by the vertical lines on the potential landscape, where Eqs. (6) and (23) degenerate, setting α = π/2 as the highest admissible value of parameter α with the corresponding maximal amplitude θ = 2α = π, and the maximal speed of the kink being λ max = 4 π . In the special α = π/2 case, rather than a sequence of flatons approaching the kink limit, a kovaton emerges which, since Eq. (6) becomes singular both at θ = 0 and at the top θ = π, is and strictly compact there (see 5,6 for a full elaboration of this case). Also, unlike flatons where every width corresponds to a different speed, however minutely different, all kovatons travel at exactly the same limiting velocity of their corresponding kink with their width being chosen at will.
Seeking travelling waves θ n (t) = Θ(t − an) of Eq.(25) where a = 1/λ is the inverse velocity, we obtain an advance-delay equatioṅ We integrate Eq. (26) once to obtain with the integration constant chosen to assure that Θ = 0 is a solution. In what follows (27) will be a starting point for the following iterative procedure.
To determine the kink we solve Eq. (27) iteratively: Starting from an initial ansatz Θ (0) (x), having a proper asymptotic behavior at x → ±∞, these iterations converge and yield the kink profiles shown in Fig. 3.

B. Solitary Waves
In order to apply the iterative procedure due to V. Petviashvili 16,17 to solitary waves, to avoid convergence to the trivial solution we modify it by introducing an intermediate normalization step where || ⋄ || stands for any norm (in our implementation the L 1 -norm was used). Also, to assure a faster convergence the exponent 1 < γ ≤ 2 was adjusted to the assumed α and a It is instructive to represent the solitary waves in terms of the original phases ϕ n rather than in the phase differences θ n (cf. Eqs. (2), (3)). This is done in Fig. 4, where we have adopted the reference frame with ω = 0.

V. DIRECT SIMULATIONS OF THE CHAIN
We now proceed to present the results of our direct simulations of the chain (25). We address two basic initial-value problems.

1.
A > 2α; as seen in Fig. 5 (a,b), a kink forms. It is followed by a characteristic linear profile connecting downstream A with kink's amplitude 2α. Behind kink's leading profile few solitons emerge which, due to the invariance (4), point down from the top plateau 2α.
Being much slower than the kink they lag far behind the emerging dispersionless profile.
2. α A 2α. As seen in Fig. 5 (c,d), a kink forms, but now t is followed by an oscillating domain with a nearly triangular envelope, which intermediates between the downstream A and kink's frontal amplitude 2α. The waves-train can be viewed as a sequence of negative solitons, though in Fig. 5 (c,d) only the first few pulses have became a truly isolated pointingdown solitons.

3.
A α (see Fig. 5 (e,f)). Kink does not form. Instead, there is a wave packet embedded within a triangular envelope. In the course of the evolution the leading waves continue to separate from each other to became true solitons with the leading amplitude ≈ 2A propagating with a velocity as given via P qc (λ; π 2 , 2A) = 0.
We now append the above phenomenological description of the numerical simulations with an analysis based on the remarkable proximity we have found between the dynamics on the lattice, its QC rendition, and the Gardner equation. To this end we shall refer to the recent study of the Gardener equation by Kamchatnov et al [18] and to an earlier, seminal work of Gurevitch and Pitaevski [19]. The comparison is based on their analysis of the signalling problem for Eq. (14) u(x, 0) =

    
A for x < 0, 0 for x > 0. (33) We start noting (the parenthesis refer to the corresponding QC case) that the convection ) has three key points: u = 0 (θ = 0) and u = 1 (θ = 2α) where it vanishes, and a turning point where it attains its maximal value which separates the two domains of monotonicity at u = 1/2 in Gardner's case and θ = α in the QC.
With the upstream being trivial, according to Ref. [18] there are three regimes according to the position of the downstream amplitude A vs. the three key points: We now detail the dynamics of Gardner's equation vis a vis the numerical results, in parenthesis, in the various regimes. 3) 1 < A (QC: 2α < A). This is a dispersionless regime and rather than a modulated periodic wave we have a rarefaction wave which intermediates between the downstream A and a kink at the front, propagating with the highest admissible velocity, see Fig.(10) in 18 . Exactly the same effect is observed on panels (a-b) of the chain, with 2α being kink's amplitude.

B. Evolution of an initial pulse
This is arguably the most basic numerical experiment. We follow the evolution of an initially single pulse excitation θ n (0) =

C. Interaction of solitary waves
The essence of our findings is summarized in Fig. 7 which displays collision of two solitons and a collision of a flaton with a soliton: both in Fig. 7 and other numerical experiments we have carried, the interaction on the lattice of solitary waves, whether solitons or flatons is remarkably clean and the interacting entities reemerge without visible distortion or radiation.

VI. SUMMARY
In this paper we have explored emergence and interaction of nonlinear traveling waves in a phase oscillators chain. We have found, both for the quasi-continuous rendition and its flaton's tails decay exponentially, kovaton's tails decay at a doubly-exponential rate which reflects the fact that in the QC limit it has a strictly compact support. Finally, we reiterate the both remarkable role of the Gardner equation (14), which was deduced at a second stage of approximation of the chain, in unfolding the various facets of the dynamics and the actual affinity between its patterns and the patterns both observed on the chain and predicted by its QC rendition.