Dark breathers in granular crystals

We present a study of the existence, stability, and bifurcation structure of families of dark breathers in a one-dimensional uniform chain of spherical beads under static load. A defocusing nonlinear Schrödinger equation (NLS) is derived for frequencies that are close to the edge of the phonon band and is used to construct targeted initial conditions for numerical computations. Salient features of the system include the existence of large amplitude solutions that emerge from the small amplitude solutions described by the NLS equation, and the presence of a nonlinear instability that, to the best of the authors’ knowledge, has not been observed in classical Fermi-Pasta-Ulam lattices. Finally, it is also demonstrated that these dark breathers can be detected in a physically realistic experimental settings by merely actuating the ends of an initially at rest chain of beads and inducing destructive interference between their signals.

Figure 1

Top: A site-centered dark breather solution with ${\omega }_b=1.9$ in the strain variable $y_n=u_{n+1}-u_n$ (left), renormalized strain variable $\mu -|y_n|$, where $\mu$ is the breather amplitude (middle), and in displacement variables $u_n$ (right). The markers indicate where the solution takes values on the lattice. The solid line is shown for clarity. Bottom: Same as top row but for the bond-centered solution.

Figure 2

Left: Renormalized $l^2$ norm of the site-centered solution versus ${\omega }_b$ with a vertical center of $c=0$. The vertical line shows the edge of the phonon band $\omega =2$. The upper branch of the curve reaches past the edge of the phonon band and increases indefinitely. Right: Same as left, but for the bond-centered solution. Regions where a real instability is present are indicated by a dashed red line (see Sec. 7 for details).

Figure 3

Left: A site-centered dark breather solution with ${\omega }_b=1.9$ and a nonzero center $c=0.13$ in the strain variable $y_n$. Middle: The same solution but in the displacement variables $u_n$. Right: Renormalized $l^2$ norm of the site-centered solution versus ${\omega }_b$ with a vertical center of $c=0.13$. The vertical line shows the edge of the phonon band $\omega =2$. In contrast to the $c=0$ case shown in Fig. 2, the bottom branch terminates at a value below the edge of the phonon band.

Figure 4

Floquet multipliers of the strongly (left) and weakly (right) nonlinear solutions with $c=0$ versus frequency $\omega$ for the site-centered (top) and bond-centered (bottom) types. The Floquet multipliers for ${\omega }_b=1.95$ in the complex plane are shown in the respective insets.

Figure 5

Manifestation of the real-instability of the strongly nonlinear bond-centered solution for ${\omega }_b=2.5$. The left panel shows the evolution of the $n=6$ node. The right panel is a contour plot of the time evolution of the entire solution. The color intensity corresponds to the magnitude of the strain $y_n$. The checkered pattern is a consequence of the time periodicity of the solution. Notice the density dip of the solution near $n=0$, and how this structure re-arranges itself in a way reminiscent of a moving (dark breather) pattern after $t\approx 25$.

Figure 6

Left: A site-centered dark breather solution with ${\omega }_b=1.99$ in the strain variable $y_n$ with fixed boundary conditions $y_0=y_{N+1}=0$ with $N=101$. Right: The same solution but in the displacement variables $u_n$.

Figure 7

Left: Analytical approximation (solid line) with an envelope function that is the NLS dark soliton (dashed line), and a numerically exact dark breather (markers) for ${\omega }_b=1.9$ and $c=0$. Right: The amplitude of the NLS approximation (solid line), numerical breather with fixed $c=0$ (dashed-dot line), and numerical breather with varying $c$ (dashed line).

Figure 8

Top: contour plots of the space-time evolution of an exact weakly nonlinear, bond-centered dark breather with a vertical shift of $c=0$ and frequency ${\omega }_b=1.99$ (left) and an initially zero chain actuated out of phase at the boundaries with frequency ${\omega }_b=1.99$ (right). The color intensity corresponds to the magnitude of the strain $y_n$. Bottom: A zoom of the corresponding top panels between times 90 and 130.

Figure 9

Left: The driven chain at $t\approx 110$ (markers) and an exact dark breather (solid line) for ${\omega }_b=1.99$. Top right: Evolution of the middle bead for the driven chain (dashed line) and exact dark breather (solid line). Bottom right: The $l^{\infty }$ norm of the difference between the driven chain and exact dark breather for time points such that $y_{\mathrm{middle}}\left(t\right)$ is at a maximum.

Figure 10

Top: Contour plots of the space-time evolution of the driven chain with the same parameters as in Fig. 8 and with a dissipation coefficient of $\tau =100$ (left) and $\tau =10$ (right). Color intensity corresponds to the magnitude of the strain. Bottom: Comparison between the driven chain (markers) and exact breather solution (solid line) for $\tau =100$ (left) and $\tau =10$ (right). Here the times were chosen to yield the closest resemblance.