Theory for the Spatiotemporal Dynamics of Domain Walls close to a Nonequilibrium Ising-Bloch Transition

We derive a generic model for the interaction of domain walls close to a nonequilibrium-Bloch transition. The universal scenario predicted by the model includes stationary Ising and Bloch localized structures (dissipative solitons), as well as drifting and oscillating Bloch structures. Our theory also explains the behavior of Bloch walls during a collision. The results are confirmed by numerical simulations of the Ginzburg-Landau equation forced at twice its natural frequency and are in agreement with previous observations in several physical systems.

Figure 1

(a) Ising LS of Eq. (1) for $\gamma =0.4$, (b) resting Bloch LS for $\gamma =0.35$, and (c) drifting Bloch LS for $\gamma =0.3$, above the IB transition. Other parameters: $\mu =1$, $\nu =0.2$, $\beta =-0.1$, and $\alpha =1$. Panels (d)–(f) show the homoclinic trajectory in the plane ($\Re [A]$,$\Im [A]$).

Figure 2

Oscillatory Bloch LS. (a) Cycle in the phase space ($\chi$,$d$), where the chirality $\chi$ is measured as $\Im [A]$ at the spatial location where $\Re [A]=0$. (b) Spatiotemporal representation of $\Re [A]$. Bright (dark) colors indicate high (low) values. Here $\mu =1$, $\nu =-0.2$, $\gamma =0.35$, $\beta =-0.5$, and $\alpha =2$.

Figure 3

Bouncing of two Bloch walls above the IB transition. Here $\gamma =0.3$. Other parameters as in Fig. 2.

Figure 4

Phase diagram of Eq. (7) for $\kappa =0.3$ (see text). We show only the bifurcation lines of the fixed points $d_n^I$ and $d_n^B$ with $n=1,2$. The phase diagram is qualitatively the same for other values of $n$.

Figure 5

Phase space of Eq. (7) for (a) $\eta =-1.5$ and $\beta =-3$ (region III), (b) $\eta =-1.5$ and $\beta =-2.8$ (region II), and (c) $\eta =-1.0$ and $\beta =-2.4$ (region III). $\kappa =0.3$. Symbols indicate fixed points and solid lines stable and unstable manifolds. The bold line in (a) corresponds to the limit cycle. The undulating dotted line displays $f(d)$ for reference.

Figure 6

As Fig. 5 for (a) $\eta =-0.15$ and $\beta =-3.17$ (region III), (b) $\eta =-0.05$ and $\beta =-3.15$ (region V), and (c) $\eta =0.2$ and $\beta =-3.17$ (region IV). The blue bold dashed line in panel (c) corresponds to a bouncing trajectory of two Bloch walls.

Figure 7

Oscillatory Bloch LS in the PCGLE after the cycle has destroyed by changing a parameter. It expands until a large width Ising LS is formed. Here $\gamma =0.32$. Other parameters as in Fig. 2.