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A modulated nonlinear system: breathers and time crystals

Masayuki Kimura, Juan F. R. Archilla, Yusuke Doi, Víctor J. Sánchez-Morcillo

Abstract

In this work we study space-time modulated system using as a specific example a system of cantilevers with their on-site potential provided by electromagnets fed with DC and AC currents. The system equations are deduced and the effect of the modulation on the dispersion bands is examined. We adapt the theory of breather existence and stability to space-time modulation. We concentrate the numerical calculations in time-modulated system for which we have obtained different types of breathers for a wide range of frequencies that can be classified in two different classes (i) the modulation frequency is an integer multiple of the breather frequency and (ii) the breather frequency is an integer multiple of the modulation frequency. The first class, corresponds to the formation of time crystals as there is no entropy production. The construction of a similar system would be of interest to study the properties of dynamic metamaterials.

A modulated nonlinear system: breathers and time crystals

Abstract

In this work we study space-time modulated system using as a specific example a system of cantilevers with their on-site potential provided by electromagnets fed with DC and AC currents. The system equations are deduced and the effect of the modulation on the dispersion bands is examined. We adapt the theory of breather existence and stability to space-time modulation. We concentrate the numerical calculations in time-modulated system for which we have obtained different types of breathers for a wide range of frequencies that can be classified in two different classes (i) the modulation frequency is an integer multiple of the breather frequency and (ii) the breather frequency is an integer multiple of the modulation frequency. The first class, corresponds to the formation of time crystals as there is no entropy production. The construction of a similar system would be of interest to study the properties of dynamic metamaterials.

Paper Structure

This paper contains 23 sections, 48 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Physical system and dynamical equations
  3. Effect of space-time modulation on the linear dispersion
  4. Dispersion bands modification with only space modulation
  5. Effect of only time modulation on the phonon bands
  6. Breakup of the time-invariance: first approximation
  7. Dispersion bands for space-time modulation
  8. Breathers with space-time modulation
  9. Theory of exact traveling breathers
  10. Application to space-time harmonically modulated systems
  11. Breather obtention in the time-modulated system
  12. Stability of breathers in the time-modulated system
  13. Extended autonomous symplectic system
  14. Time crystals: breathers with time modulation and $\Omega=n_T \omega_\text{b}$
  15. Period doubling $\Omega=2 \omega_\text{b}$
  16. ...and 8 more sections

Figures (12)

  • Figure 1: System of cantilevers mechanically coupled by a rod. (b) Side view of a cantilever. A permanent magnet is attached at the tip of the cantilever. An electromagnet is placed below the permanent magnet. (c) Definition of magnetic charges. Modified with permission from Ref. kimura2009a
  • Figure 2: Numerical simulations after thermalization with modulation frequency and wave number $\Omega=0.02$ and $h=0.5$ : ( Top:) Energy density plot ( Bottom) 2D FFT.
  • Figure 3: ( Top:) Numerical dispersion bands for $h=2\pi/3$; ( Bottom:) Theoretical degeneracy raising of the dispersion bands due to space modulation. Note that the Brillouin zone is $[0,2\pi/3]$. See the text for explanation.
  • Figure 4: (Top) Contour plot of the 2DFFT for $\delta=0.05$, $\Omega=0.1$, $\kappa=0.1$ (Bottom) Identical parameters except $\Omega=0.01$
  • Figure 5: Contour plot of the FFT2 for $\delta=0.05$, $\Omega=0.05$, $\kappa=0.1$ and the theoretical dispersion relation lines: Note that there is an area to the right where the harmonics are very close and become mixed.
  • ...and 7 more figures