Nonreciprocal Coupling Induced Self-Assembled Localized Structures

Chains of coupled oscillators exhibit energy propagation by means of waves, pulses, and fronts. Nonreciprocal coupling radically modifies the wave dynamics of chains. Based on a prototype model of nonlinear chains with nonreciprocal coupling to nearest neighbors, we study nonlinear wave dynamics. Nonreciprocal coupling induces a convective instability between unstable and stable equilibrium. Increasing the coupling level, the chain presents a propagative pattern, a traveling wave. This emergent phenomenon corresponds to the self-assembly of localized structures. The pattern wavelength is characterized as a function of the coupling. Analytically, the phase diagram is determined and agrees with numerical simulations.

Figure 1

Nonreciprocal coupled chain of pendulums and front propagation. (a) Schematic representation of a chain of pendulums coupled with a nonreciprocal material. ${\theta }_i(t)$ is the angle formed by the pendulum and the vertical axis in the $i$ position at time $t$. Yellow cylinder accounts for a nonreciprocal metamaterial. Instantaneous profile and spatiotemporal evolution of $\pi$ kink obtained for Eq. (2) with $\omega =1$, $D=4$, $\alpha =1$ (b), and $\alpha =2.5$ (c).

Figure 2

Phase diagram of the overdamped Frenkel-Kontorova Eq. (2) with $\omega =1$. In zone I, the upright pendulums invade the upside-down ones. This process is reversed in zone II. The blue curve, Eq. (4), is the analytical absolute convective instability between the upright and upside-down pendulums. The purple ($D=\alpha$) and orange curves account for the monotonous to nonmonotonous front transition, using Eq. (3). The star symbol (☆) accounts for the critical point $(\alpha =1/2,D=1/2)$ where the critical curves converge. Red and green curves separate the localized structures’ self-assembly region. The red curve was obtained using formula $D=1/2$. The green curve is achieved through divergence of the self-assembly wavelength. All circles are obtained by means of numerical simulations.

Figure 3

Nonmonotonous fronts of the overdamped Frenkel-Kontorova Eq. (2) with $\omega =1$. Profile (a) and spatiotemporal evolution (b) of a nonmonotonous front propagates from upside-down pendulums into upright ones for $D=1$ and $\alpha =1.25$. $v$ and $v^{\prime }$ account for the speeds of different fronts. Profile (c) and spatiotemporal evolution (d) of a nonmonotonous front propagate from upright pendulums into upside-down pendulums for $D=0.25$ and $\alpha =0.325$.

Figure 4

Self-assembly of localized structures and the wavelength surface map for the overdamped Frenkel-Kontorova Eq. (2) with $D=0.4$, $\alpha =0.5$, and $\omega =1$. (a) Spatiotemporal evolution and respective profiles in two instants of time, $t_1=120$ and $t_2=395$. $v$ accounts for the traveling wave velocity. (b) Wavelength surface map for the $D$ and $\alpha$ parameter space.

Figure 5

Self-assembly of localized structures obtained for the overdamped Frenkel-Kontorova Eq. (2) with $D=0.25$, $\omega =1$, and $\alpha =0.425$. (b) Spatiotemporal evolution of Eq. (2) with initial condition top panel (a) and final state bottom panel (c). $v$ stands for the speed of the ensemble of localized structures.