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.

Figure 1

Bifurcation diagram showing ${\left|\right|A\left|\right|}_2$ vs $\mu$ for Eq. (4) with periodic boundary conditions, $\mathrm{\Lambda }=2\pi$ and $L=1$, where ${\left|\right|A\left|\right|}_2=\sqrt{\frac{1}{\mathrm{\Lambda }}{\int }_0^{\mathrm{\Lambda }}{\left|A\right|}^{2}dx\phantom{{}^1}}$ is the (normalized) $L^2$ norm. Solid lines represent unstable (thin) and stable (thick) stationary states. Filled circles denote primary bifurcations at $\mu =Q^2$ for each wave number $Q$ and open circles denote secondary bifurcations at $\mu =3Q^2-\frac{1}{2}{k}^2$ for each integer $k\in (0,2Q)$. In the insets (a), (b), and (c), we show sample profiles for the three key stationary states: solid, dashed, and dotted lines denote $\left|A\right|,\text{Re}A,$ and $\text{Im}A$, respectively, and the $y$ ticks denote zero. The homogeneous state (a) is given by $A=\sqrt{\mu }$. The phase-winding pattern states (b) have the form $A=\sqrt{\mu -Q^2\phantom{{}^1}}{e}^{iQx}$. The mixed modes (c) determine the basin of attraction of the pattern states. The corresponding locations of each inset profile in the bifurcation diagram are marked with diamonds.

Figure 2

Bifurcation diagram showing ${\left|\right|A\left|\right|}_2$ vs $L$ for $\mu =3$. Solution stability and bifurcation types are denoted as in Fig. 1. The primary bifurcations occur at $L=Q/\sqrt{\mu }, Q\in {\mathbb{Z}}^{+}$, and secondary bifurcations occur at $L=\sqrt{(3Q^2-\frac{1}{2}{k}^2)/\mu \phantom{{}^1}}$ for each $k\in (0,2Q)$ with $k\in {\mathbb{Z}}^{+}$. The trajectory for a $Q=2$ state on a growing domain $L\left(t\right)=e^{0.2t}$ is shown in blue with an upward arrow and the trajectory for a $Q=1$ state on a shrinking domain $L\left(t\right)=e^{-0.2t}$ is displayed in red with a downward arrow. The shaded region denotes the neighborhood around the trivial state for which delay is calculated; diamonds represent $L_{\text{exit}}$ (blue, right) and $L_{\text{enter}}$ (red, left) for the growing and shrinking domain, respectively. In the inset, the square denotes the turnaround point $L_{*}$. The additional domain growth $L_{\text{delay}}$ denotes bifurcation delay.

Figure 3

On an exponentially shrinking domain $L\left(t\right)=e^{\sigma t}$, where $\sigma <0$, as the domain shrinks faster, the delay time $t_{\text{delay}}$ decreases (top) but the length contraction $L_{\text{delay}}$ increases (bottom). The delay is calculated by comparing (18) to the quasistatic case. Here, $\mu =3,Q=1$, and $\epsilon =0.0001$, where $\epsilon$ is the size of the neighborhood around the trivial state used for the delay calculation [see (16) and Fig. 2].

Figure 4

Unstable $Q=2$ mode undergoes a phase slip leading to a stable $Q=1$ mode. The numerical simulation is run with $\mathrm{\Lambda }=2\pi , L=1, \mu =9$ and an initial perturbation $A^{\prime }=0.01e^{ix}$ of the base state $A=\sqrt{5}{e}^{2ix}$.

Figure 5

Squared norm of different $k=1$ perturbations with $arctan\left[\right|a_{k+}\left(0\right)/a_{k-}\left(0\right)\left|\right]=\{-\pi /4,0,\pi /4,\pi /2\}$ (solid, decreasing opacity) of a $Q=2$ pattern state from simulations with $L\left(t\right)=e^{-2t}$ and $\mu =14$. The norms are bounded from above by (29) (dashed).

Figure 6

A phase slip is arrested with $L\left(t\right)=1.95+0.95tanh\left[5\right(t-2.59\left)\right]$ (blue, top) but develops with $L\left(t\right)=1.85+0.85tanh\left[5\right(t-2.59\left)\right]$ (red, bottom). The phase slip time is indicated by a solid red point. Both states are initialized with a $Q=1$ pattern state $A_Q\left(\xi \right)$ defined by (20) and the same perturbation $A^{\prime }=0.25e^{i\xi }$.

Figure 7

Phase slip core width $\mathrm{\Delta }\xi$ for a $Q=1$ profile, defined to be the spatial width around a phase slip in which there is an accumulation of $\mathrm{\Delta }\varphi =\pi$ phase. Top: $\left|A\right|, \text{Re}A$, and $\text{Im}A$. Bottom: phase $\varphi$.

Figure 8

Critical core width $\mathrm{\Delta }{\xi }_{\text{cr}}$ of the $Q=1$ mixed mode as a function of $L$ when $\mu =2$, obtained by an interpolation of points obtained from numerical continuation in $L$.

Figure 9

The core width model classifies which regimes on a growing domain result in arrested phase slips between $Q=1$ and $Q=0$ by comparing the core width of the profile $\mathrm{\Delta }\xi$ at each time to the critical core width $\mathrm{\Delta }{\xi }_{\text{cr}}$ of the mixed mode. If $\mathrm{\Delta }\xi >\mathrm{\Delta }{\xi }_{\text{cr}}$ at some time, then the phase slip is arrested while $\mathrm{\Delta }\xi =0$ indicates the presence of a phase slip. Left: the core width $\mathrm{\Delta }\xi$ over time for various initial conditions (solid) and the critical core width $\mathrm{\Delta }{\xi }_{\text{cr}}$ (dashed). Right: space-time plots of $\text{Re}A$ for the selected profiles (colors), with the core width $\mathrm{\Delta }\xi$ overlaid (dotted). Top: slowly growing domain with $L\left(t\right)=e^{0.1t}$. The core width model works very well for 50 random finite-amplitude perturbations (gray lines). A phase slip (a) and arrested phase slip (b) are shown. Bottom: rapidly growing domain with $L\left(t\right)=1.93+0.93tanh\left[5\right(t-2.59\left)\right]$. Here the model is less successful. A phase slip (c) and arrested phase slip (e) are shown. The profile (d) reveals the limitations of this model, where $\mathrm{\Delta }\xi >\mathrm{\Delta }{\xi }_{\text{cr}}$ at some time but the phase slip occurs nonetheless.

Figure 10

Time evolution of $\left|A\right|$ (solid) and $\text{Re}A$ (dashed) of the different front types in the RGLE on a fixed domain $\mathrm{\Lambda }=20\pi$. Values of $\mu$ are varied to emphasize qualitative features on the same spatial scale. Homogeneous fronts (left) are generated by the supercritical bifurcation at $\mu =0$. Pattern-spreading fronts (middle) are generated by primary bifurcations with $Q\ne 1$. Eckhaus fronts (right) are generated by secondary Eckhaus instability between two different pattern states. For visual clarity, the imaginary part is not shown.

Figure 11

Space-time plots of $\text{Re}A$ for homogeneous (left), pattern-spreading (middle), and Eckhaus (right) fronts on a fixed domain (top) and exponentially growing domain $L\left(t\right)=e^{\sigma t}$ (bottom) with $\mathrm{\Lambda }=20\pi$. The values of $\mu$ and $\sigma$ are varied to emphasize qualitative features across the different types of fronts while retaining the same spatial and temporal scale. In all cases, the front slows down on the growing domain. For the homogeneous front, the initial condition is a narrow Gaussian. For the pattern-spreading front, the initial condition is a pattern modulated by a narrow Gaussian. The pattern is initially outside of the existence band at $t=0$ on the growing domain; therefore, front propagation is delayed. For the Eckhaus front, the initial condition is an Eckhaus-unstable pattern with a narrow Gaussian perturbation. On the growing domain the resulting phase slips occur less frequently and eventually stop, resulting in a phase-melting state.

Figure 12

Homogeneous front propagation in the RGLE with $\mu =1$ at times $t=5,15,25,35$ with various $L\left(t\right)$. Top: constant domain with $L\left(t\right)=1$. Middle: exponentially shrinking domain with $L\left(t\right)=e^{-0.02t}$. Bottom: exponentially growing domain with $L\left(t\right)=e^{0.06t}$. The black (solid, thick) lines are the nonlinear fronts computed with DNS. The blue (solid, medium) lines are the linear fronts calculated analytically from (37). The gray (dashed, thin) lines are the time-frozen asymptotic fronts. The red (dashed, medium) lines are the natural asymptotic fronts. In DNS, a delta function initial condition is approximated by a Gaussian $\frac{1}{\sqrt{2\pi s^2}}exp[-{(\xi -\mathrm{\Lambda }/2)}^2/2s^2]$ with standard deviation $s=0.1$. This is narrow enough to model a delta function but wide enough to avoid numerical issues with finite differences. To allow fronts to propagate for long times without reaching the boundary, we select $\mathrm{\Lambda }=100\pi$. This figure can be compared with Fig. 12 in [42].

Figure 13

Velocities of the homogeneous fronts in Fig. 12 with $C=0.5$. The black (solid, thick) lines denote the nonlinear position ${\xi }_C\left(t\right)$ and velocity ${\dot{\xi }}_C\left(t\right)$ obtained from DNS. The blue (solid, medium) lines denote the linear position (46) and velocity (47) calculated analytically. The gray (dashed, thin) lines denote the time-frozen asymptotic position and velocity (33). The red (dashed, medium) lines denote the natural asymptotic position and velocity (44). This figure can be compared with Fig. 5 in [76]. On the fixed domain, both the linear and nonlinear velocities approach the constant $v^{**}=2$. In the shrinking domain, the nonlinear velocity deviates from the asymptotic and linear velocities as time increases. In the growing domain, the linear and nonlinear velocities overshoot the asymptotic velocities.

Figure 14

Profiles of the homogeneous fronts depicted in Fig. 12, where $\mu =1$. In each time snapshot, the top subplot displays overlapping nonlinear (black, solid, thick), linear (blue, solid, medium), time-frozen asymptotic (gray, dashed, thin), and natural asymptotic (red, dashed, medium) profiles intersecting at $C=0.5$. The bottom subplot shows the phase plane in the variables $(u,w)$ for stationary states in the natural asymptotic frame (42) with some trajectories plotted as described by (49). The two fixed points (0,0) (stable, filled) and (1,0) (unstable, empty) and their eigenspaces are shown. The fixed domain has a degenerate eigenspace for the (0,0) fixed point which arises due to a double root. This is the well-known marginal stability property of the asymptotic nonlinear profile [42]. However, on a time-dependent domain, (0,0) loses its double root and becomes a generic stable node. In the growing domain, the time-frozen asymptotic profile, which tracks the degenerate state for all time, begins to deviate strongly from the other profiles. This is a further confirmation that $v^{**}\left(t\right)$ is the natural asymptotic velocity, not $v^{*}\left(t\right)$. Note that, for the growing domain, the scales of the plots are different. This figure can be compared with Fig. 4 in [76] and Fig. 9 in [42].

Figure 15

Eckhaus front velocities for $Q=1$ and $Q=2$ initial states. The solid lines are obtained from calculations in Appendix pp4 and solid points denote measured velocities from DNS. In (a), the velocity is obtained for various $\mu$ with the constant domain $L=1$. In (b), the velocity is obtained for various values of $L$ with fixed $\mu =3$. This is a quasistatic analysis and on these small domains does not reveal any time-dependent behavior arising from domain growth.

Figure 16

Dilution increases primary bifurcation delay. The system with dilution (dotted) has a larger turnaround size $L_{*}$ and exit size $L_{\text{exit}}$ than that without (solid). Both systems are initialized with the same initial condition and employ the same exponentially growing size, $L\left(t\right)=e^{0.02t}$, the same as the inset of Fig. 2.

Figure 17

Propagating homogeneous fronts in the RGLE with dilution (dotted) and without dilution (solid) obtained from DNS. In the exponentially shrinking domain (top), the velocity increases faster when dilution is included. The reverse applies to the exponentially growing domain (bottom). The final amplitude behind the front also changes.