Real Ginzburg-Landau equation on a time-dependent domain

Abstract

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation (RGLE) with periodic boundary conditions on monotonically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and find that the additional domain growth before pattern onset 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 alters the amount of additional compression which occurs before pattern decay. 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 profile which deviate from the values obtained by the marginal stability criterion. Incorporating the cubic nonlinearity, this mismatch reveals a broken degeneracy of spatial eigenvalues and presents 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, and these more complex fronts can also experience delayed onset. Lastly, we show that dilution—a phenomenon realized by prescribing a conservation law—increases bifurcation delay time and amplifies changes in homogeneous front velocity under a time-dependent domain. These analyses provide general insight into the delay of pattern onset and pattern transitions as well as front propagation in systems across many scientific fields.