Competition between drift and spatial defects leads to oscillatory and excitable dynamics of dissipative solitons

We have reported in Phys. Rev. Lett. 110, 064103 (2013) that in systems which otherwise do not show oscillatory dynamics, the interplay between pinning to a defect and pulling by drift allows the system to exhibit excitability and oscillations. Here we build on this work and present a detailed bifurcation analysis of the various dynamical instabilities that result from the competition between a pulling force generated by the drift and a pinning of the solitons to spatial defects. We show that oscillatory and excitable dynamics of dissipative solitons find their origin in multiple codimension-2 bifurcation points. Moreover, we demonstrate that the mechanisms leading to these dynamical regimes are generic for any system admitting dissipative solitons.

Figure 1

Homoclinic snaking for $k_0^2=0.5,a=1.2$, and $g=1$. The solid (dashed) lines represent stable (unstable) DSs. The vertical dashed line shows the value of $r$ chosen in the analysis in Sec. 3. The insets show the spatial profile of the DSs with an odd number of peaks, corresponding to the solid black circle on each stable branch.

Figure 2

(a) Bifurcation diagram (maximum ${\left|\right|u\left|\right|}_{\sup }$) as a function of the strength of the spatial defect $h$ for $c=0$. (b) Examples of the steady-state solutions (black solid lines) corresponding to the labeled branches, together with the profiles of the spatial defect (red dashed lines). ${\mathrm{SN}}_1$ and ${\mathrm{SN}}_3$ represent saddle-node bifurcations.

Figure 3

(a) Bifurcation diagram (maximum ${\left|\right|u\left|\right|}_{\sup }$) as a function of the strength of the spatial defect $h$ for $c=0.0015$. (b) Examples of the steady-state solutions (black solid lines) corresponding to the labeled branches, together with the profiles of the spatial defect (red dashed lines).

Figure 4

Bifurcation diagram (maximum ${\left|\right|u\left|\right|}_{\sup }$) as a function of the strength of the spatial defect $h$ for (a) $c=0.05$, (b) $c=0.12$, (c) $c=0.162$, and (d) $c=0.4$. Crosses indicate the extrema (maxima and minima) of the DS oscillatory amplitude. The main dynamical regions A–E are labeled in red.

Figure 5

Two-parameter ($c$ vs $h$) phase diagram of the system for $a=1.2$ and $r=-0.2$. The bifurcation lines and regions A–E are explained in the main text.

Figure 6

Zoom of the two-parameter phase diagram shown in Fig. 5, such that the overlap between regions A and D, as well as A and E is visible. $a=1.2$ and $r=-0.2$.

Figure 7

Contour plots of $u(x,t)$ showing the spatiotemporal evolution of a train of DSs for $h=0.04$ (a), $h=0.06$ (b), and $h=0.08$ (c) for $c=0.12$ and $L=209$. Above the contour plots the spatial profile $u(x,t=800)$ is plotted.

Figure 8

(a) Time evolution of the magnitude $u\left(x_0\right)$ at the center of the domain ($x_0=L/2$) for a train of DSs. $h=0.08, c=0.12$, and $L=209$. (b) Snapshots of the spatial profile $u(x,t)$ at different times [as indicated in individual panels and corresponding to the blue diamonds in panel (a)], illustrating the growth and depinning of a DS from the spatial defect. The defect is shown in a red dashed line.

Figure 9

Contour plots of $u(x,t)$ showing the spatiotemporal evolution of a train of DSs for $h=0.136$ (a) and a small amplitude oscillation for $h=0.164$ (b). $c=0.12$ and $L=209$. Above the contour plots the spatial profile $u(x,t=0.25)$ is plotted. (c) Canard explosion illustrated by plotting the maxima and minima of an oscillating DS, evaluated at $u(x=x_m)$ as function of $h$, for $x_m\approx 123$ and $c=0.4$.

Figure 10

Scaling of the oscillation period $T$ in function of $h$ for Type I (a) and Type II [(b) and (c)] excitability. (a) $c=0.12$; (b) $c=0.6$; (c) $c=0.4$.

Figure 11

Type I excitability (SNIC): Region D. An excitable excursion of the DS is shown close to the SNIC for $c=0.12,h=0.085$ and $\mathrm{\Delta }h=-0.035,\mathrm{\Delta }t=10$. Panel (a) shows the contour plot of the real field $u$, while several spatial profiles $u\left(x\right)$ for fixed values of $t$ are shown in (b).

Figure 12

Type II excitability (${\mathrm{H}}^{+}$): Region A. An excitable excursion of the DS is shown close to the FC for $c=0.6,h=0.092$ and $\mathrm{\Delta }h=0.035\mathrm{\Delta }t=10$. Panel (a) shows the contour plot of the real field $u$, while several spatial profiles $u\left(x\right)$ for fixed values of $t$ are shown in (b).

Figure 13

Type II excitability (${\mathrm{H}}^{-}$): Region C. An excitable excursion of the DS is shown close to the ${\mathrm{H}}^{-}$ for $c=0.4,h=0.175$ and $\mathrm{\Delta }h=-0.045,\mathrm{\Delta }t=10$. Panel (a) shows the contour plot of the real field $u$, while several spatial profiles $u\left(x\right)$ for fixed values of $t$ are shown in (b).

Figure 14

Unfolding of the Takens-Bodganov ${\mathrm{TB}}_1$ bifurcation, completing the phase diagram in Fig. 5 for $a=1.2$ and $r=-0.2$. The black dashed lines refer to Fig. 16 where the corresponding bifurcation diagrams and eigenvalues are shown. The dot-dashed blue line corresponds to ${\mathrm{H}}^{+}$.

Figure 15

Unfolding of the Takens-Bodganov ${\mathrm{TB}}_2$ bifurcation, completing the phase diagram in Fig. 5 for $a=1.2$ and $r=-0.2$. The black dashed lines refer to Fig. 17 where the corresponding bifurcation diagrams and eigenvalues are shown. The dot-dashed blue line corresponds to ${\mathrm{H}}^{-}$ and the red dashed line to a SL bifurcation.

Figure 16

Bifurcation diagrams as a function of $h$ (top), and the real part of the leading eigenvalues (bottom) for fixed increasing values of $c$ crossing the ${\mathrm{TB}}_1$ point: (a) $c=0.1171$, (b) $c=0.1173$, (c) $c=0.1175$, (d) $c=0.1177$. These values of $c$ are also indicated as horizontal dashed lines in Fig. 14.

Figure 17

Bifurcation diagrams as a function of $h$ (top), and the real part of the leading eigenvalues (bottom) for fixed increasing values of $c$ crossing the ${\mathrm{TB}}_2$ point: (a) $c=0.154$, (b) $c=0.158$, (c) $c=0.162$, and (d) $c=0.164$. These values of $c$ are also indicated as horizontal dashed lines in Fig. 15.

Figure 18

Two-parameter ($c$ vs $h$) phase diagram of the system for $a=1$ and $r=-0.09$. The bifurcation curves and the labels of the regions corresponds to the ones in Fig. 5. The points labeled with (a), (b), and (c) correspond to the solutions shown in Fig. 19 for $c=0.08$.

Figure 19

Contour plots of $u(x,t)$ showing the spatiotemporal evolution of oscillatory solutions for $c=0.08$ (see dashed line in Fig. 18). The defect strength is varied: $h=0.04$ (a), $0.05$ (b), $0.085$ (c).

Figure 20

Bifurcation diagram for Eq. (18) with the defect in the gain term at $c=0.0001$. Branches corresponding to DSs with two peaks are shown in blue, DS with a single peak in black, and the asymmetric rung states in red. Solid and dashed lines represent stable and unstable states, respectively.

Figure 21

Spatial profiles $u\left(x\right)$ of the main attractors of the system for $c=0$. The red dashed line shows the defect profile.

Figure 22

Bifurcation diagrams for Eq. (18) with the defect in the gain term for increasing values of the drift $c$: $c=0.05$ (a), $c=0.15$ (b), and $c=0.3$ (c). Only the branches for $h>0$ relevant to the discussion are shown. The main dynamical regimes are indicated in red and explained in the main text. Panel (d) shows the spatial profiles of the main attractors for $c\ne 0$.

Figure 23

Contour plots of $u(x,t)$ showing the spatiotemporal evolution of oscillatory solutions of the high amplitude DS (train of solitons) for Eq. (18) with the defect in the gain term for $c=0.15$ and $h=0.268$. Above the contour plots the spatial profile $u(x,t=800)$ is plotted.

Figure 24

Excitable excursion of a DS for Eq. (18) with the defect in the gain term. The parameters are $c=0.15, h=0.281, \mathrm{\Delta }h=-0.03$, and $\mathrm{\Delta }t=10$. Panel (a) shows the contour plot of $u(x,t)$ illustrating the spatiotemporal evolution of the excited DS. In (b) several snapshots of spatial profiles are shown for fixed values of $t$.

Figure 25

Two-parameter ($c$ vs $h$) phase diagram for Eq. (18) with the defect in the gain term. The bifurcations and regions A–D are explained in the main text.

Figure 26

Detail of the diagram in Fig. 20 showing the imperfect transcritical bifurcations around $h=0$. Branches corresponding to DSs with two peaks are shown in blue, DS with a single peak in black, and the asymmetric rung states in red. Solid and dashed lines represent stable and unstable states, respectively. ${\mathrm{T}}_1,\cdots ,{\mathrm{T}}_6$ represent transcritical bifurcations.

Figure 27

Transcritical bifurcations ${\mathrm{T}}_2$ and ${\mathrm{T}}_3$ and their imperfections corresponding to Fig. 26 are shown, together with examples of the spatial profiles of the L states. The two red (dotted and dashed) lines correspond to the bifurcation for $c\ne 0$, while the black dashed lines correspond to the overlapping solution branches for $c=0$.

Figure 28

Bifurcation diagram showing the transcritical bifurcation ${\mathrm{T}}_4$ and its imperfection for $c=0.0001$. In the bottom panel the profiles of the states corresponding to those branches are shown. The thin dashed lines represent the solution branches at $c=0$.

Figure 29

Pitchfork bifurcation (P) for $c=0$ and its imperfection for $c=0.0001$ are shown.

Figure 30

Reconnection of solution branches 13 and 19. In (a) we plot a zoom of the diagram in Fig. 20, showing branches 13 and 19 for $c=0.0001$. In panel (b) a zoom of panel (a) illustrates how branches 13 and 19 approach for increasing $c$ until they reconnect for $c=0.0003$.