Rotobreathers in a chain of coupled elastic rotators

Rotobreathers in the chain of coupled linearly elastic rotators are analyzed. Each rotator is a particle connected by a massless elastic rod with a frictionless pivot; it has two degrees of freedom, length and angle of rotation. The rods of the rotators and the elastic bonds between the nearest rotators are linearly elastic, and the nonlinearity of the system is of a purely geometric nature. It is shown that long-lived rotobreathers can exist if the stiffness of the rods is high enough to create a relatively wide gap in the phonon spectrum of the chain. The frequency of angular rotation of the rotobreather cannot be above the optical band of the phonon spectrum and is in the spectrum gap. Generally speaking, the rotation of the rotobreather is accompanied by radial oscillations; however, one can choose such initial conditions so that the radial oscillations are minimal. Some parameters of rotobreathers with minimal radial vibrations are presented on the basis of numerical simulations. The results obtained qualitatively describe the behavior of physical systems with coupled rotators.

Figure 1

Schematic of a chain of coupled elastic rotators numbered by the index $n$. A rotator is a pointwise particle of mass $M$ connected by a massless, linear elastic rod of stiffness $k$ to a frictionless pivot. Particles rotate parallel to the $(y,z)$ plane about common rigid spoke, which is parallel to the $x$ axis. Each particle is coupled to the nearest neighbors by linear elastic bonds of stiffness $K$. The rods and bonds have equilibrium lengths $r_0$ and $R_0$, respectively, and current lengths $r_n$ and $R_n$, respectively.

Figure 2

Phase diagram of the chain of rotators. Regime I is realized for $R_0\le a=1$, regime III for $R_0\ge \sqrt{a^2+4r_0^2}=\sqrt{1+4r_0^2}$, and regime II in between. Ground states in these regimes are described in the text.

Figure 3

(a) Example of phonon dispersion curves for the ground state in regime I, Eqs. (11) and (12), for $r_0=0.5, R_0=0.8$, and $k=5$. (b) Maximal acoustic frequency, ${\omega }_{\varphi }^{\max }$, and minimal and maximal optic frequencies, ${\omega }_r^{\min }$ and ${\omega }_r^{\max }$, as the functions of $k$.

Figure 4

(a) Example of phonon dispersion curves for the ground state in regime III, Eqs. (20) and (21), for $r_0=0.5, R_0=2\sqrt{2}$, and $k=5$. (b) Maximal acoustic frequency, ${\omega }_{\varphi }^{\max }$, and minimal and maximal optic frequencies, ${\omega }_r^{\min }$ and ${\omega }_r^{\max }$, as the functions of $k$.

Figure 5

Results for regime I. (a) Time evolution of the total energy of the chain with absorbing boundary conditions and central rotator initially excited with $r^{*}=0.6$ and ${\dot{\varphi }}^{*}=1.6$. (b)–(d) Trajectories of the excited rotator on the $(y,z)$ plane for time intervals specified for each case. In (d), a periodic motion of the rotator is observed, since the trajectory is closed. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 6

Results for regime I. Properties of three rotobreathers obtained with the initial length $r^{*}=0.6$ and three different values of the initial angular velocity, ${\dot{\varphi }}^{*}=1.55$, 1.6, and 1.7. In (a), trajectories of the central rotator on the $(y,z)$ plane are shown by the blue, red, and green lines, respectively. In (c), the length of the central rotator as the function of time is shown by the blue, red, and green lines, respectively. All three rotobreathers have very close angular frequency ${\mathrm{\Omega }}_{\varphi }=1.27$ and radial frequency ${\mathrm{\Omega }}_r=4{\mathrm{\Omega }}_{\varphi }$, as shown in (b). The dashed lines in (b) show the borders of the phonon spectrum with a very narrow optic band. It can be seen that ${\mathrm{\Omega }}_r$ lies above the optic band and ${\mathrm{\Omega }}_{\varphi }$ in the gap of the spectrum. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 7

Results for regime I. Quasiperiodic rotobreather with minimal radial vibrations obtained with the initial parameters $r^{*}=0.6$ and ${\dot{\varphi }}^{*}=1.75$. (a) Nearly circular trajectory of the central rotator on the $(y,z)$ plane. (b) Radial frequency ${\mathrm{\Omega }}_r$ and angular frequency ${\mathrm{\Omega }}_{\varphi }$ as the functions of time in the regime of quasiperiodic motion. These frequencies are equal in this case. Dashed lines show the borders of the acoustic and optic bands of the phonon spectrum. (c) Length of the central rotator as the function of time. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 8

Results for regime I. Properties of three quasiperiodic rotobreathers obtained with the initial conditions $r^{*}=0.6$ and three values of the initial angular velocity, ${\dot{\varphi }}^{*}=2.3$, 2.4, and 2.6. In (a), trajectories of the central rotator on the $(y,z)$ plane are shown by the blue, red, and green lines, respectively. In (c), the length of the central rotator as the function of time is shown by the blue, red, and green lines, respectively. All three rotobreathers in the regime of quasiperiodic motion have very close radial frequency ${\mathrm{\Omega }}_r\approx 5.62$ and angular frequency ${\mathrm{\Omega }}_{\varphi }\approx 1.86$, as shown in (b). The dashed lines in (b) show the borders of the phonon spectrum. It can be seen that ${\mathrm{\Omega }}_r$ lies above the optic band and ${\mathrm{\Omega }}_{\varphi }$ in the gap of the spectrum. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 9

Results for regime I. Time evolution of angular coordinates [panels (a)–(c)] and radial coordinates [panels (d)–(g)] for rotobreather excited with $r^{*}=0.6$ and ${\dot{\varphi }}^{*}=2.6$. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 10

Results for regime I. Members of the families of quasiperiodic rotobreathers having closed trajectories with intersections on the $(y,z)$ plane. The rotobreathers were excited with $r^{*}=0.6$ and (a) ${\dot{\varphi }}^{*}=3.55$ and (b) ${\dot{\varphi }}^{*}=5.03$. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 11

Results for regime I. Characteristics of the rotobreathers with minimal radial vibrations as the functions of the initial rotator length $r^{*}$. (a) Initial angular velocity ${\dot{\varphi }}^{*}$ (the red line) and equal radial and angular frequencies of the rotobreather ${\mathrm{\Omega }}_r={\mathrm{\Omega }}_{\psi }$ (blue line). Dashed lines show the borders of the phonon spectrum. (b) Energy of the rotobreather. The inset in (b) shows the ratio of the rotobreather energy to the energy initially given to the system. Model parameters: $r_0=0.5, R_0=0.8$, and $k=20$.

Figure 12

Results for regime III. (a) Time evolution of the total energy of the chain with absorbing boundary conditions and central rotator initially excited with $r^{*}=1.2$ and ${\dot{\varphi }}^{*}=10$. (b)–(d) Trajectories of the excited rotator on the $(y,z)$ plane for time intervals specified for each case. In (d), a periodic motion of the rotator is observed, since the trajectory is closed. Model parameters: $r_0=0.5, R_0=2\sqrt{2}=2.828$, and $k=20$.

Figure 13

Results for regime III. (a) Initial energy given to the system $H^{*}$ (blue solid line) and energy of the rotobreathers $H$ (open symbols) as the functions of the initial angular velocity of the central rotator ${\dot{\varphi }}^{*}$ for the fixed initial length of the central rotator $r^{*}=1.2$. (b) Frequencies of radial and rotational motion, ${\mathrm{\Omega }}_r$ and ${\mathrm{\Omega }}_{\varphi }$, as the functions of the rotobreather energy presented by the blue triangles and red circles, respectively. Dashed lines show the borders of the phonon spectrum. The green lines and symbols in (a) and (b) show the energies of the four families of rotobreathers. Model parameters: $r_0=0.5, R_0=2\sqrt{2}=2.828$, and $k=20$.

Figure 14

Results for regime III. The length of the central rotator as the function of time for quasiperiodic rotobreathers excited with the initial parameters $r^{*}=1.2$ and different values of ${\dot{\varphi }}^{*}$: (a) 4, (b) 10, (c) 13.5, and (d) 16. Panels (a) to (d) present rotobreathers of groups 1 to 4, respectively. Model parameters: $r_0=0.5, R_0=2\sqrt{2}=2.828$, and $k=20$.