Dynamics of Domain Walls in Pattern Formation with Traveling-Wave Forcing

We study dynamics of domain walls in pattern forming systems that are externally forced by a moving space-periodic modulation close to $2:1$ spatial resonance. The motion of the forcing induces nongradient dynamics, while the wave number mismatch breaks explicitly the chiral symmetry of the domain walls. The combination of both effects yields an imperfect nonequilibrium Ising-Bloch bifurcation, where all kinks (including the Ising-like one) drift. Kink velocities and interactions are studied within the generic amplitude equation. For nonzero mismatch, a transition to traveling bound kink-antikink pairs and chaotic wave trains occurs.

Figure 1

Projections of domain walls into the complex plane of $A$. The kinks approach fixed points (gray dots) in the limit of infinite $|x|$. Dotted curves correspond to unstable solutions. The dashed and dotted curves in (e), (f) are kinks created in the saddle node bifurcation of the imperfect Ising-Bloch transition.

Figure 2

Plot of the NIB transition ($q=0$, dashed lines) and the imperfect transition ($q=0.1$, solid lines) for $\omega =0$. The graphs show the imaginary component of $A$ at the center of each kink that connects $A_h^{+}$ with $A_h^{-}$. Thick lines correspond to stable kinks.

Figure 3

Velocity of all three types of kinks for $\tilde{\mu }=8$ and $q=0.1$. The two stable branches $B1$ and $B2$ end in a saddle node at $\omega =2$, where $|v|=1.652$ and $|v|=1.592$, respectively.

Figure 4

Complex asymptotic eigenvalues exist above the bold line for $\omega =0$. For $\omega =0.1$ the thin solid and the dotted lines show the location of zero discriminant (numerical convergence was not sufficient for the dotted line close to $\tilde{\mu }=-1$). The dashed line represents the kink saddle node for $\omega =0$.

Figure 5

The stable distance of kinks (stars) and the critical distance for repulsion (diamonds) for $\mu =1$ and $\omega =0$. The transition to oscillating wall interaction occurs at $q\approx 0.28$.