Figure 1

Top: the dynamical cactus, repulsive particles (here dipoles), on the surface of a cylinder. $a$ is the distance between consecutive rings and $\Omega$ is their angular shift. The dashed line is the so-called “generative spiral” [10]. Bottom: for a dipole-dipole interaction, energy $V$ versus screw angle $\Omega$ (semilogarithmic plot) for halving values of the parameter $a/R$ starting from 0.5 (lowest line), in arbitrary units $ϵ$.Reuse & Permissions

Figure 2

(Color online) The equivalent Newtonian picture, standard for Klein-Gordon-like solitons or for tunneling problems [5]: a soliton is a solution of Eq. (5) that corresponds to a trajectory $\omega \left(s\right)$ [red solid line (light gray)] of a point particle that starts and ends in two equally valued maxima [horizontal dotted red lines (light gray)] of the equivalent potential energy $\Phi \left(\omega \right)=\frac{1}{2}{v}^{2}I{\left(\omega -\tilde{\omega }\right)}^2-V\left(\omega \right)$ (black solid line). As the kinetic energy must be zero on those maxima, it takes the Newtonian trajectory an infinite amount of time $s$ (i.e., “time” in the Newtonian picture, space in the soliton picture) to reach the “top of the hill” with asymptotic speed equal to zero [5]. Unlike the Klein-Gordon-like case, the presence of the quadratic term $\frac{1}{2}{v}^{2}I{\left(\omega -\tilde{\omega }\right)}^2$ (dashed line), which accounts for the kinetic energy of the domains, slightly shifts the asymptotic value of the kinks (red dotted lines) away from stable static structures ${\omega }_1,{\omega }_2$ (black dotted lines), the minima of the real potential energy $V\left(\omega \right)$ (dashed line). Instead, the soliton connects spirals of angle ${\overline{\omega }}_1={\omega }_1+{\delta }_1$, ${\overline{\omega }}_2={\omega }_2+{\delta }_2$, which in general are not stable structures of $V$. The extra quadratic term also allows solitons among nondegenerate configurations and brings the speed of the soliton $v$ into the picture. [Everything is in arbitrary units. The soliton is obtained by numerical integration of the Eq. (5) using the energy profile portrayed in the figure.]Reuse & Permissions

Figure 3

(Color online) Dynamical simulation for the collision or conversion of two solitons, emitted from free boundaries of a dynamical cactus. Top panel: screw angle vs space (in number of rings) and time (in seconds); note the elastic wave preceding the soliton. Second panel: energy per particle (mJ) vs screw angle (rad) for our system (${\omega }_1=1.79\text{rad}$, ${\omega }_2=1.43\text{rad}$, ${\omega }_3=2.40\text{rad}$); the two kinks—before collision—connect a high (inner: ${\omega }_1$) to a low (outer: ${\omega }_2$) energy domain; after collision a low (inner: ${\omega }_3$) to a high (outer: ${\omega }_2$). Third panel: angular speed $\left(s^{-1}\right)$ vs space at a given time: the speed of the soliton can be extracted as $v=\Delta \dot{\theta }/\Delta \omega =22.1{\text{s}}^{-1}$ (predicted: $23.4{\text{s}}^{-1}$). Fourth panel: the precursor; plot of the screw angle vs space at different times while the soliton and its preceding wave advance. The amplitude ${\delta }_1$ of the precursor is predicted via Eq. (13) as ${\delta }_1={\omega }_1-{\overline{\omega }}_1=-0.043\text{rad}$, in excellent agreement with numerical observations. The simulation uses the experimental density and the magnetic interaction as in Ref. [8]. (See supplementary materials in Ref. [8] for an animation.)Reuse & Permissions

Figure 4

(Color online) Equivalent Newtonian diagram (see caption of Fig. 2 for explanation) for symmetric solitons between degenerate spirals. As the quadratic term raises all the maxima of $-V\left(\omega \right)$, solitons with the same speed but connecting different domains are possible under different applied torques. The smallest kink [solid red line (light gray)] and a larger kink [solid blue line (dark gray)] are shown.Reuse & Permissions

Figure 5

(Color online) Equivalent Newtonian diagram (see caption of Fig. 2 for explanation) for pulses. Top: a traveling pulse among degenerate structures is possible because the nonzero speed $v$ raises the maximum of Φ. Bottom: theoretical [red dashed (light gray)] and experimental [blue solid (dark gray)] frozen-in pulse on the measured energy curve of the dynamical cactus [8].Reuse & Permissions

Figure 6

(Color online) Equivalent Newtonian diagram for both axially constrained and unconstrained soliton. In blue dashed (dark gray) curve, the axially constrained soliton among the stable structures ${\omega }_1$, ${\omega }_2$ is shown. The two red solid curves (light gray) depict the soliton in the axially unconstrained case, which is longer, and its corresponding drop in density $\lambda$.Reuse & Permissions