Figure 1

Snapshots of time evolution of phase distributions upon soliton-antisoliton collision in the unperturbed ($\alpha =\gamma =0$) double-junction CSGE for (a) direct collision of a soliton and an antisoliton in the junction 1 and (b) indirect collision of a soliton in the junction 1 and an antisoliton in the junction 2. Thick and thin lines represent ${\varphi }_1$ and ${\varphi }_2$, respectively. It is seen that both the horizontal (a) and vertical (b) breathers are decaying due to emission of plasma waves, even in the absence of perturbations.Reuse & Permissions

Figure 2

Detailed view of the time evolution of ${\varphi }_2$ (thin lines) for the horizontal breather from Fig. 1a. The phase ${\varphi }_1$ (thick lines) is shown close to the moments of collisions, marked by arrows. It is seen that emission of two wave fronts occurs at every collision: the fast front (marked by blue dashed lines) has an in-phase symmetry ${\varphi }_1={\varphi }_2$ and propagates with the fast velocity $c_1$, the slow front (marked by green dotted lines) has an out-of-phase symmetry ${\varphi }_1=-{\varphi }_2$ and propagates with the slow velocity $c_N$.Reuse & Permissions

Figure 3

Time dependence of the total $E_{\text{tot}}$, electric $E_e$, and magnetic $E_m$ energies for a horizontal breather in junction $i=5$ of the unperturbed $N=10$ CSGE system. (a) Without radiative losses at the edges $Z=\infty$, (b) with radiative losses $Z={10}^5\Omega$. Panel (c) shows a long-time evolution of $E_{\text{tot}}\left(t\right)$ in the presence of radiative losses. A peculiar logarithmic time decay is seen.Reuse & Permissions

Figure 4

Phase amplitude of the horizontal breather as a function of the breather frequency for perturbed SG ($N=1$) and CSGE with $N=2$ and 10 for different values of the damping $\alpha$ and for $\gamma =0$. The dashed line represents the breather solution Eq. (19) for the SG equation.Reuse & Permissions

Figure 5

Instantaneous voltage (velocity) $V\left(x\right)=\partial {\varphi }_{1,2}/\partial t$ profiles for a driven soliton motion in a dissipative $\alpha =0.01$ CSGE with $N=2$ for increasing driving currents (forces) (a) $\gamma =0.01$, (b) $\gamma =0.015$, and (c) $\gamma =0.03$. Time is counted with respect to the first collision $t_1$ with the left edge of the system. It is seen that the fast in-phase and the slow out-of-phase waves are emitted upon the collision $t=t_1$. In panel (b), the soliton velocity $u$ is close to the velocity of the out-of-phase wave, and the corresponding front is no longer seen ahead of the soliton. Further increase of $u$ in panel (c) leads to profound Cherenkov-type radiation behind the soliton.Reuse & Permissions

Figure 6

Current (driving force)-voltage (velocity) characteristics of a shuttling single soliton in a moderate size systems $L=5$ for (a) a single junction (SG, $N=1$) and (c) a double junction stack (CSGE, $N=2$), appearance of a fine structure of the zero-field step due to interference with emitted plasma waves is clearly seen. Panel (e) shows the continuation of $I-V$ characteristics at large bias for $N=2$ and $\alpha =0.02$. Panels (b), (d), and (f) show the corresponding emission powers. It is seen that in the double junction system the emission at the velocity matching part of the zero-field step [point A in (c) and (d)] is at minimum, unlike the single junction case (a) and (b), and the maximum emission occurs at point B, corresponding to the in-phase geometrical resonance and the Fiske step in the $I-V$. Above the ZFS, the system switches to another strongly emitting resonance [point C in (e) and (f)], which represents a breather-type self-oscillation.Reuse & Permissions