Oscillating decorated interfaces in parametrically driven systems

Macroscopic systems forced by the temporal modulation of their parameters exhibit complex interfaces between symmetric states. Here we investigate the origin of the transition from a flat to an oscillating decorated interface. Based on a model that describes a magnetic plane under the influence of an oscillating magnetic field and an extended Josephson junction under the influence of an alternating current, we derive a simple model that accounts for the interface dynamics. Analytically this model allows us to reveal that this transition is a parametric resonance between the frequencies of interface modes and the forcing. Numerical simulations of magnetic systems, extended Josephson junctions, and our simplified model show quite good agreement.

Figure 1

Decorated interfaces in parametrically driven systems. (a) Domain wall in a parametrically driven magnetic plane obtained from numerical simulation of Eq. (1) with $h\left(t\right)=H_0+h_{0}cos\left(\omega t\right)$, $H_0=1$, $h_0=0.4$, $\omega =2.94$, $\beta =6$, and $\alpha =0.1$. (b) Schematic representation of an extended Josephson junction, which is composed of two superconductors separated by an insulating strip under the influence of an alternating superconducting current. The inset shows a numerical simulation of Eq. (2) with ${\omega }_0=1$, $\gamma =0.4$, $\omega =1.6$, $\beta =1$, and $\mu =0.1$.

Figure 2

Stability diagram of the domain walls or $2\pi$ kinks for the sine-Gordon model (2) for $\mu =0.1$ and $\beta =1$. The horizontal and vertical axes represent, respectively, the frequency and amplitude of the forcing. Regions $1\text{:}2$ and $1\text{:}1$ account for the Arnold tongues (parametric resonances). The dashed (solid) bottom curve represents the supercritical (subcritical) transition from a flat to an oscillating decorated interface.

Figure 3

Spatiotemporal dynamics of decorated interfaces. (a) Decorated interface at a given time and spatiotemporal diagram of the interface position $P(y,t)$ for the sine-Gordon model (2) for ${\omega }_0=1$, $\gamma =0.4$, $\omega =1.2$, $\beta =1$, and $\mu =0.1$. (b) Left panel shows the profile and spatiotemporal diagram of the interface position $P(y,t)$ of Eq. (6) for $\mu =0.05$, $\gamma =0.3$, and $\omega =1.8$. Right panel illustrates the relation between the interface wave number and the forcing frequency.

Figure 4

Bifurcation diagram of the amplitude equation (7). The ${\mathrm{\Gamma }}_k$ represent the Arnold tongues, whereas the solid and dashed curves represent the numerical and analytical tongues, respectively. The solid and dashed curves stand for the subcritical and supercritical bifurcations of the flat interface, respectively. The bottom solid curve ${\mathrm{\Phi }}_0$ represents the saddle-node bifurcation of the decorated interface. The $1\text{:}2$ region accounts shows parametric instability of the parametrically driven sine-Gordon model (2).

Figure 5

Oscillating decorated interface of the parametrically driven sine-Gordon model (2) for ${\omega }_0=1.0$, $\beta =1.0$, $\omega =1.12$, $\gamma =1.46$, and $\mu =0.2$.