Flaming $2\pi$ kinks in parametrically driven systems

Macroscopic extended systems with dissipation and injection of energy can exhibit particlelike solutions. Dissipative kinks with an oscillatory cloak and a family of localized states that connect uniform symmetric states in a magnetic wire forced with a transversal oscillatory magnetic field and in a parametrically driven damped pendula chain are studied. The oscillatory cloak is composed of evanescent waves emitted at the kink position and generated by a resonant mechanism. These waves mediate the kink interaction and generate a family of localized states.

Figure 1

Flaming $2\pi$ kink in a parametrically driven magnetic wire obtained from numerical simulation of Eq. (1) with $h\left(t\right)=H_0+h_{0}cos\left(\omega t\right), H_0=2, h_0=0.8, \omega =6, \beta =10$, and $\alpha =0.02$. (a) Schematic representation of magnetization m(z) upon a driven magnetic wire. (b) Magnetization components of the flaming magnetic kink. (c) Spatiotemporal evolution of the flaming kink. The horizontal dashed line in the diagrams shows the time when the images of (b) were obtained. (d) Corresponding $\varphi$ profile in the spherical representation of the magnetization. $\{\mathrm{\Gamma },\sigma ,\lambda \}$ are the amplitude, wavelength, and steepness of the evanescent wave, respectively. (e) Corresponding $\theta$ profile in the spherical representation of the magnetization.

Figure 2

Flaming $2\pi$ kink in a vertically driven chain of coupled pendula obtained from numerical simulation of Eq. (2) with ${\omega }_0^2\left(t\right)={\omega }_0^2+\gamma cos\left(\omega t\right), {\omega }_0^2=1, \gamma =0.3, \omega =1.4$, and $\mu =0.1$. (a) Schematic representation of a vertically driven dissipative chain of coupled pendula. (b) Spatiotemporal evolution of a motionless flaming kink. (c) Profile of a flaming kink; $\{\mathrm{\Gamma },\sigma ,\lambda \}$ are the amplitude, wavelength, and steepness of the evanescent wave, respectively.

Figure 3

Characterization of a flaming $2\pi$ kink for a vertically driven chain of pendula with $\beta H_0=1$. (a) Phase space as a function of frequency and amplitude forcing with $\mu =0.1$. The tongues account for the strong resonances ($1:1$ and $1:2$). The flaming $2\pi$ kinks are observed in the dark region. (b) Amplitude of evanescent waves $\mathrm{\Gamma }$ as a function of forcing frequency by different dissipation coefficients and forcing intensities (c). Wavelength $\sigma$ (d) and steepness $\lambda$ (e) of evanescent waves as a function of frequency forcing $\omega$.

Figure 4

Localized flaming states in a vertically driven chain of coupled pendula obtained from numerical simulation of Eq. (2) with the same parameters as in Fig. 2. (a) Schematic representation of a bound state composed of a flaming $2\pi$ kink and a $2\pi$ antikink. Profile at a given time (b) and spatiotemporal temporal evolution (c) of a localized flaming structure. (d) Phase space of the flaming $2\pi$ kink position $\{\mathrm{\Delta },\dot{\mathrm{\Delta }}\}$ and their stationary localized flaming states, respectively.

Figure 5

Flaming localized states in a magnetic wire forced with a transversal oscillatory magnetic field. (a) Arrow representation of a bound state composed of a flaming $2\pi$ kink and a flaming $2\pi$ antikink at a given time. (b) Cartesian components of the magnetization for three particlelike states with different widths. For every magnetization component, the magnetization profile is shown above the corresponding spatiotemporal diagram. The horizontal line in the diagram shows the instant when the profiles were obtained. For this figure, we used the same parameters as in Fig. 1.