Dissipative switching waves and solitons in the systems with spontaneously broken symmetry

The paper addresses the bistability caused by spontaneous symmetry breaking bifurcation in a one-dimensional periodically corrugated nonlinear waveguide pumped by coherent light at normal incidence. The formation and the stability of the switching waves connecting the states of different symmetries are studied numerically. It is shown that the switching waves can form stable resting and moving bound states (dissipative solitons). The protocols of the creation of the discussed nonlinear localized waves are suggested and verified by numerical simulations.

Figure 1

Bifurcation diagram of the bright (blue line) and hybrid (green line) states, where $W=|U_b{|}^2+|U_d{|}^2=|U_{+}{|}^2+{\left|U_{-}\right|}^2$ is the intensity of states; dynamically unstable solutions are shown by the dashed line. Parameters are $\delta =0.05, \alpha =-1, \sigma =1, \gamma =0.001$, and $\mathrm{\Gamma }=0.299$. The inset shows a schematic view of the considered system.

Figure 2

(a) The part of the bifurcation diagram showing the dependencies of the intensities of the hybrid and bright (belonging to the lower branch) spatially uniform states on the pump amplitudes. Two vertical dashed lines mark the range of the existence of the stable hybrid states. The green lines and arrows show the range of the existence of resting $H_{+}B{H}_{-}$ and $H_{-}B{H}_{+}$ solitons. The range of existence of moving $H_{+}B{H}_{+}$ and $H_{-}B{H}_{-}$ solitons is shown by the red lines and arrows. The point where the bright states become unstable is marked as SSB. (b) Velocity dependencies of the domain walls on the pump. The bifurcation curves of the different domain walls are shown by different colors. The solid lines correspond to the stable domain walls and the dashed lines to the unstable domain walls. The inset shows the enlarged region near the Maxwell points of $H_{+}B$ and $B{H}_{+}$ domain walls. The points where the domain walls $H_{+}B$ and $B{H}_{+}$ (or $H_{-}B$ and $B{H}_{-}$) have equal velocities are marked as $P_{eq}$.

Figure 3

Field distributions of the domain walls. (a) $H_{+}B$ at $P=0.14$, (b) $H_{+}B$ at Maxwell point, (c) $H_{+}B$ at $P=0.27$, (d) $B{H}_{-}$ at $P=0.14$, (e) $B{H}_{-}$ at Maxwell point, (f) $B{H}_{-}$ at $P=0.27$, (g) $B{H}_{+}$ at $P=0.14$, (h) $B{H}_{+}$ at Maxwell point, and (i) $B{H}_{+}$ at $P=0.24$ of the unstable branch.

Figure 4

(a) The evolution of the unstable $H_{+}B$ wall at $P=0.31$ obtained by numerical simulations of Eq. (1). The colors shows the total intensity of the state $|U_{+}{|}^2+{\left|U_{-}\right|}^2$: the blue color corresponds to the lowest intensity and the yellow color corresponds to the highest intensity. (b)–(e) The distribution of the fields at different times.

Figure 5

(a) The evolution of the unstable $H_{-}B$ domain wall at $P=0.22$ obtained by numerical simulations of Eq. (1). The colors show the total intensity of the state, $|U_{+}{|}^2+{\left|U_{-}\right|}^2$: the blue color corresponds to the lowest intensity and the yellow color corresponds to the higher intensity. (b) Numerically found unstable domain wall taken as the initial conditions for the numerical simulations shown in (a). The domain wall can be seen as an unstable bound state of the $H_{+}B$ wall and a dark soliton ($H_{-}B{H}_{+}$ soliton) used as an initial condition in the numerical simulation. (c) Field intensity distributions at $t=10000$.

Figure 6

Bifurcation diagrams of $H_{+}B{H}_{-}$ (blue line) and $H_{-}B{H}_{+}$ (black line) solitons. The dashed lines correspond to the unstable states. It is seen that in the vicinity of the Maxwell points, the bifurcation curves form snaking patterns. Insets (i) and (ii) illustrate the field distributions of the solitons belonging to the snaking patterns of the bifurcation diagrams. These states correspond to the points (i) and (ii) of the bifurcation curves. The field distributions in the $H_{+}B{H}_{-}$ and $H_{-}B{H}_{+}$ solitons at pump $P=0.2$ are presented in insets (iii) and (iv) for $H_{+}B{H}_{-}$ and $H_{-}B{H}_{+}$ solitons, respectively. The relevant points on the bifurcation curves are marked as (iii) and (iv) in the main panel.

Figure 7

(a) Development of the instability of the $H_{+}B{H}_{-}$ soliton at $P=0.14$. (i) Dynamically unstable soliton is taken as an initial condition in the numerical simulation. (ii) Resulting stable soliton with the width smaller than the initial one. (b) Development of the instability of the $H_{+}B{H}_{-}$ soliton at $P=0.17$. (i) Initial conditions in the form of an unstable soliton. (ii) The final distributions of the fields where the separation of two domain walls is clearly seen. (c) Development of the instability of the $H_{+}B{H}_{+}$ soliton at $P=0.2$. (i) The initial condition in the form of the unstable soliton. (ii) Resulting stable soliton with the width smaller than the initial one. (d) Development of the instability of the $H_{+}B{H}_{+}$ soliton at $P=0.28$. (i) The initial condition in the form of the unstable soliton. (ii) The final spatially uniform field distribution corresponding to the hybrid state which appears after the collapse of the soliton.

Figure 8

(a) The velocities of $H_{+}B{H}_{+}$ (blue line) and $B{H}_{+}B$ (red line) moving solitons as a function of the pump $P$. $P_{eq}$ is the point where the domain walls $H_{+}B$ and $B{H}_{+}$ have the same velocities; see the inset in Fig. 2. The fields distributions of the (b) single-hump $H_{+}B{H}_{+}$ soliton at $P=0.22$, (c) multihump $H_{+}B{H}_{+}$ soliton at $P=0.22$, and (d) $B{H}_{+}B$ soliton at $P=0.24$. The snaking of the bifurcation curve of the moving $H_{+}B{H}_{+}$ soliton around the Maxwell point is shown in (b) in the $Z\text{−}P$ axis.

Figure 9

The formation of the $H_{-}B{H}_{+}$ dark soliton from the weak noise taken as the initial conditions. (a) The distributions of the amplitude of the spatially nonuniform pump $P$ shown by the blue curve. The distribution of the coupling strength coefficient $\sigma$ is shown in this panel by the red curve. (b) The numerically obtained evolution of the field. (c) The final distribution of the field, where the dark soliton is clearly seen.

Figure 10

Numerical simulation supporting the protocol of the formation of moving $H_{+}B{H}_{+}$ solitons suggested in the paper. (a) The temporal evolution of the intensity of the field. The intervals of different pumping regimes are marked by 1–4. The field distributions at the times marked by the vertical lines $S_{1-3}$ are presented in panels (e)–(g) correspondingly. (b) The profile of the pump acting on the system within the interval 1 ($t<5000$). (e) The quasistationary field formed by $t=5000$. (c) The distribution of the pump amplitude within the time interval 2 ($5000<t<20000$). The area of the low intensity of the pump is slowly shrinking and, finally, the pump becomes spatially uniform. Schematically, the shrinking of the low pump area is indicated by the red arrows. (f) The field distributions at $t=20000$. The pump distributions shown in (d) are for time intervals 3 ($20000<t<30000$) and 4 ($t>30000$), and they are marked by the numbers 3 and 4 correspondingly. (g) The final distribution of the field ($t=50000$). This field distribution perfectly coincides with the field distribution in the $H_{+}B{H}_{+}$ soliton existing at the pump $P=0.21$.