^{1}and Arkady Pikovsky

^{1}

### Abstract

We study waves in a chain of dispersively coupled phase oscillators. Two approaches—a quasicontinuous approximation and an iterative numerical solution of the lattice equation—allow us to characterize different types of traveling waves: compactons, kovatons, solitary waves with exponential tails as well as a novel type of semicompact waves that are compact from one side. Stability of these waves is studied using numerical simulations of the initial value problem.

The topic of this paper unifies two principal directions of nonlinear science: coupled self-sustained oscillators and soliton theory. Coupled autonomous self-sustained oscillators appear in different fields of science, and demonstrate a variety of fascinating phenomena. In this study we demonstrate that particularities of the coupling for a rather simple setup—all oscillators are identical, periodic, and form a regular lattice with a nearest neighbor coupling—can lead to highly nontrivial wave structures. Remarkably, a dispersive coupling of the phases of dissipative oscillators results in a simple lattice equation which is equivalent to a Hamiltonian one. We show that different types of traveling waves exist in this lattice: compactons (ultralocalized waves with a compact support), usual solitary waves with exponential tails, as well as the corresponding kink-type solutions. These waves appear from rather general initial conditions and exist for long times. After very long transients they are destroyed due to inelastic collisions and evolve into phase chaos.

We thank P. Rosenau for helpful discussions and DFG for support.

I. INTRODUCTION

II. BASIC MODEL

III. TRAVELING WAVES IN THE QUASICONTINUUM

A. Solitary waves

B. Kinks

C. Periodic waves

IV. TRAVELING WAVES IN THE LATTICE

A. Fixed point analysis

B. Numerical determination of traveling waves

C. Solitary waves

D. Kinks

E. Periodic waves

F. Solitary waves with periodically decaying tails

V. NUMERICAL SIMULATIONS OF THE INITIAL VALUE PROBLEM

A. Evolution of an initial pulse

B. Transition to chaos in a finite lattice

VI. CONCLUSIONS

### Key Topics

- Oscillators
- 16.0
- Coupled oscillators
- 13.0
- Bifurcations
- 12.0
- Eigenvalues
- 11.0
- Chaos
- 5.0

## Figures

(Top panel) Soliton solution for the coupling , which arise out of the background . The form of the soliton was calculated numerically with the help of Eq. (28) and the velocity is . Here and in the following figures, bold dots show the soliton on the lattice and the dashed line the solution of QCA (5) which has an additional offset for better visibility. (Bottom panel) The soliton in the logarithmic scale.

(Top panel) Soliton solution for the coupling , which arise out of the background . The form of the soliton was calculated numerically with the help of Eq. (28) and the velocity is . Here and in the following figures, bold dots show the soliton on the lattice and the dashed line the solution of QCA (5) which has an additional offset for better visibility. (Bottom panel) The soliton in the logarithmic scale.

(Top panel) The shape of the compacton for the coupling . The background is and the velocity was set to . Markers show the compacton on the discrete lattice and the dashed line is the corresponding solution in the quasicontinuum approximation. (Bottom panel) The same plot in the logarithmic scale. The super exponentially decaying tails are clearly visible.

(Top panel) The shape of the compacton for the coupling . The background is and the velocity was set to . Markers show the compacton on the discrete lattice and the dashed line is the corresponding solution in the quasicontinuum approximation. (Bottom panel) The same plot in the logarithmic scale. The super exponentially decaying tails are clearly visible.

The values of uniform background states that can be connected by kinks, in the QCA for the particular coupling and various values of .

The values of uniform background states that can be connected by kinks, in the QCA for the particular coupling and various values of .

(Top panel) The shape of the kink for , the velocity is . (Center panel) The kink in the logarithmic scale. (Bottom panel) The kink shown from its top in the logarithmic scale. Markers show the kink on the lattice and the dashed line is the QCA with an additional offset for better visibility.

(Top panel) The shape of the kink for , the velocity is . (Center panel) The kink in the logarithmic scale. (Bottom panel) The kink shown from its top in the logarithmic scale. Markers show the kink on the lattice and the dashed line is the QCA with an additional offset for better visibility.

(Top panel) The wave form of the compact kink solution between and for the coupling . Markers show the wave form on the discrete lattice and the dashed line represents the quasicontinuum approximation of the kink. (Bottom panel) The sine of the kink in the logarithmic scale.

(Top panel) The wave form of the compact kink solution between and for the coupling . Markers show the wave form on the discrete lattice and the dashed line represents the quasicontinuum approximation of the kink. (Bottom panel) The sine of the kink in the logarithmic scale.

(Top panel) The shape of a kink with one exponential and one compact tail. The coupling for this specific wave is with . The position of the kink point and the wave velocity can be obtained from Eqs. (31) and (23) , their values are and . (Middle panel) The kink in the logarithmic scale. The compact tail of the kink is near clearly visible. (Bottom panel) The kink shown from its top in logarithmic scale, the exponential tail becomes visible.

(Top panel) The shape of a kink with one exponential and one compact tail. The coupling for this specific wave is with . The position of the kink point and the wave velocity can be obtained from Eqs. (31) and (23) , their values are and . (Middle panel) The kink in the logarithmic scale. The compact tail of the kink is near clearly visible. (Bottom panel) The kink shown from its top in logarithmic scale, the exponential tail becomes visible.

The shape of a periodic wave on the lattice. The offset is , the wavelength is and the velocity .

The shape of a periodic wave on the lattice. The offset is , the wavelength is and the velocity .

Phase space of Eq. (10) for and (a) and (b) . The critical velocity is . It is clearly visible that below the critical velocity the fixed point is a center, hence periodic waves exist in Eq. (4) . Above one can observe solitonic solutions: the background is a saddle point and a homoclinic orbit exists.

Phase space of Eq. (10) for and (a) and (b) . The critical velocity is . It is clearly visible that below the critical velocity the fixed point is a center, hence periodic waves exist in Eq. (4) . Above one can observe solitonic solutions: the background is a saddle point and a homoclinic orbit exists.

(a) Wave speed in dependence of the imaginary eigenvalue , see Eq. (26) . The dots mark possible bifurcation points to real eigenvalues. (b) Wave speed vs the real and imaginary part of the eigenvalues, with . Only the positive parts of the real and imaginary axes are shown.

(a) Wave speed in dependence of the imaginary eigenvalue , see Eq. (26) . The dots mark possible bifurcation points to real eigenvalues. (b) Wave speed vs the real and imaginary part of the eigenvalues, with . Only the positive parts of the real and imaginary axes are shown.

Solitary wave with periodic and exponentially decaying tails. and . The plot was generated with the help of the traveling wave scheme, Eq. (28) for the lattice equation; it has no QCA counterpart.

Solitary wave with periodic and exponentially decaying tails. and . The plot was generated with the help of the traveling wave scheme, Eq. (28) for the lattice equation; it has no QCA counterpart.

Evolution of different initial pulses for the coupling and . The initial conditions were set according to Eq. (32) . (a) and , (b) and , and (c) and . The dashed line shows the initial condition and the solid line the lattice at the time .

Evolution of different initial pulses for the coupling and . The initial conditions were set according to Eq. (32) . (a) and , (b) and , and (c) and . The dashed line shows the initial condition and the solid line the lattice at the time .

Evolution of different initial pulses for the coupling and the background . The initial conditions were set according to Eq. (32) . (a) and , (b) and , (c) and , (d) and , and (e) and . The dashed line shows the initial condition and the solid line the lattice at the time . Furthermore the position of the kink point at is shown.

Evolution of different initial pulses for the coupling and the background . The initial conditions were set according to Eq. (32) . (a) and , (b) and , (c) and , (d) and , and (e) and . The dashed line shows the initial condition and the solid line the lattice at the time . Furthermore the position of the kink point at is shown.

Transition to chaos. The background is and the lattice contains sites with periodic boundary conditions. The field is shown in a gray scale vs time (horizontal axis) and space (vertical axis). (Upper plot) A kink and several solitons emerge out of a pulse. The time interval is . (Lower plot) Emergence of chaos after a collision of two solitons which creates an soliton–antisoliton pair. The time interval is .

Transition to chaos. The background is and the lattice contains sites with periodic boundary conditions. The field is shown in a gray scale vs time (horizontal axis) and space (vertical axis). (Upper plot) A kink and several solitons emerge out of a pulse. The time interval is . (Lower plot) Emergence of chaos after a collision of two solitons which creates an soliton–antisoliton pair. The time interval is .

Transient times to chaos for different backgrounds . The length of the lattice is and the initial -pulse is for and else. In order to obtain an average of the transient times we also varied the width of the initial pulse from to and calculated the transient time as the average over the transient times for different initial pulses. The line is an exponential fit and the transient times scales with .

Transient times to chaos for different backgrounds . The length of the lattice is and the initial -pulse is for and else. In order to obtain an average of the transient times we also varied the width of the initial pulse from to and calculated the transient time as the average over the transient times for different initial pulses. The line is an exponential fit and the transient times scales with .

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