Ising and Bloch domain walls in a two-dimensional parametrically driven Ginzburg-Landau equation model with nonlinearity management

We study a parametrically driven Ginzburg-Landau equation model with nonlinear management. The system is made of laterally coupled long active waveguides placed along a circumference. Stationary solutions of three kinds are found: periodic Ising states and two types of Bloch states, staggered and unstaggered. The stability of these states is investigated analytically and numerically. The nonlinear dynamics of the Bloch states are described by a complex Ginzburg-Landau equation with linear and nonlinear parametric driving. The switching between the staggered and unstaggered Bloch states under the action of direct ac forces is shown.

Figure 1

(Color online) Schematic view of the system with nonlinearity management. The active region is concentrated in cylinders $N=24$, the blank part between cylinders represents the passive (idle) medium.

Figure 2

The Ising domain wall profile for different values of the radius of the nonlocal dispersion: the subcritical case $r\sqrt{{\gamma }_a+\mu }=0.1$ (solid line), the critical case $r\sqrt{{\gamma }_a+\mu }=1$ (dashed line), and the supercritical case $r\sqrt{{\gamma }_a+\mu }=1.5$ (dotted line).

Figure 3

(Color online) Wall profiles for periodic Ising states ($\gamma =1$, $\mu =1$, $j=2$).

Figure 4

(Color online) Wall profiles for staggered periodic Bloch states. The real (solid line) and imaginary (dashed line) parts of the complex amplitude are plotted for $\gamma =2$, $\mu =0.35$, $j=2$.

Figure 5

(Color online) Wall profiles for unstaggered periodic Bloch states. The real (solid line) and imaginary (dashed line) parts of the complex amplitude are plotted for $\gamma =2$, $\mu =0.35$, $j=2$.

Figure 6

The top panel shows the time evolution of the Ising state for the system with the number of sites $L=12$ for the time moments $t=0$ (dashed line), $t=270$ (dotted-dashed line), $t=300$ (solid line); the bottom shows the evolution for $L=24$ for the time moments $t=0$ (dashed line), $t=2000$ (dotted-dashed line), $t=50000$ (solid line).

Figure 7

(Color online) The evolution of the real part of the complex amplitude when the initial seed is given by Eq. (74).

Figure 8

(Color online) The evolution of the imaginary part of the complex amplitude when the initial seed is given by Eq. (73).

Figure 9

(Color online) The evolution of the imaginary part of the complex amplitude when the initial seed is given by Eq. (74).

Figure 10

(Color online) Effective energy profile for the amplitudes of staggered and unstaggered critical modes.

Figure 11

Phase portrait for a switching dynamics under the action of spatially homogeneous ac field for $\gamma =1$, $\mu =0.15$, $f_B=0.1$, and two different values of the driving frequency $\omega =0.1$ (top panel), $\omega =0.25$. Initially the system is in the in-phase Bloch state.

Figure 12

Phase portrait for a switching dynamics under the action of spatially homogeneous ac field for $\gamma =1$, $\mu =0.15$, $f_B=0.1$, and two different values of the driving frequency $\omega =0.1$ (top panel), $\omega =0.25$. Initially the system is in the staggered Bloch state.

Figure 13

Mean chirality switching from the in-phase Bloch state to the staggered Bloch state under the action of the external driving with the amplitude $f=0.1$ and the frequency $\omega =0.005$.

Figure 14

(Color online) The Bloch state evolution for the same parameters as in Fig. 13.