Eckhaus-Benjamin-Feir instability in systems with temporal modulation

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation with periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

Ising-Bloch transition for the parametric Ginzburg-Landau equation with rapidly varying perturbations

We study the effects of rapid periodic and stochastic modulations of parameters in systems described by the complex parametric Ginzburg-Landau equation. Amplitude equations, which govern the dynamics of the field averaged over the rapid modulations, are derived. For temporal modulations of the linear detuning the threshold for the transition from Ising to Bloch walls is shifted depending on the strength of the perturbation. In contrast to this, rapid perturbations of the linear gain lead only to a decrease of the amplitude of both wall types leaving the bifurcation point of the Ising-Bloch transition unchanged. Stochastic perturbations of the detuning lead to a Brownian motion of the Bloch wall beyond bifurcation where the velocity is given analytically. All theoretical predictions are confirmed by numerical simulations of the full stochastic Ginzburg-Landau equation.