Front propagation in weakly subcritical pattern-forming systems

The speed and stability of fronts near a weakly subcritical steady-state bifurcation are studied, focusing on the transition between pushed and pulled fronts in the bistable Ginzburg-Landau equation. Exact nonlinear front solutions are constructed and their stability properties investigated. In some cases, the exact solutions are stable but are not selected from arbitrary small amplitude initial conditions. In other cases, the exact solution is unstable to modulational instabilities which select a distinct front. Chaotic front dynamics may result and is studied using numerical techniques.

Figure 1

The nonlinear front with parameters $(a_1,a_2)=(2,3)$ and $\mu =0$ shown at a fixed time. (a) The real (green) and imaginary (red) parts of the solution $A(\xi ,·)$ along with its amplitude $\left|A\right|$ in blue. (b) A three-dimensional representation of the solution in (a).

Figure 2

The length scale $\lambda \equiv {\left(e_1{a}_N^2\right)}^{-1}$ at the Maxwell point $\mu ={\mu }_M(a_1,a_2)$ for fronts on (a) the $a_{N+}$ branch and (b) the $a_{N-}$ branch. Regions colored dark red represent values $\ge 8$. The solution that lies below the fold on the branch of front solutions has a smaller amplitude and decay rate as compared to that above the fold.

Figure 3

The existence regions for $a_{N\pm }$ when (a) $\mu =-0.1$, (b) $\mu =0$, (c) $\mu =0.1$. The dark gray indicates existence of both solutions, light gray indicates existence of only $a_{N+}$, black indicates existence of only $a_{N-}$, while white implies nonexistence of both solutions. The lines $\mathrm{\Gamma }=0$ (red), $\mathrm{\Lambda }=6$ (blue), $5\mathrm{\Lambda }=6$ (magenta), $\mathrm{{\rm Y}}=0$ (green), and $\mathrm{\Delta }=0$ (orange) are shown restricted to the region $\mathrm{\Gamma }>0$, required for the validity of the ansatz (10). The dots indicate the locations $(a_1,a_2)=(\pm 3,\pm 1),\left(\pm \sqrt{5},\mp \frac{1}{\sqrt{5}}\right)$ on the curve $\mathrm{\Gamma }=0$.

Figure 4

The front amplitudes $a_{N+}$ (blue) and $a_{N-}$ (red) for (a) $\mathrm{{\rm Y}}>0,\mathrm{\Delta }>0$, (b) several values of $\mathrm{{\rm Y}}\approx 0$ while $\mathrm{\Delta }>0$, and (c) for several values of $\mathrm{\Delta }\approx 0$ while $\mathrm{{\rm Y}}<0$. The parameter values in each of these plots are $a_2=0$ and (a) $a_1=(0,0.7,1)$, (b) $a_1=\left(1.6,\sqrt{\frac{8}{3}},1.65\right)$, and (c) $a_1=(1.92,2,2.1)$ as indicated in the panels. The special cases $\mathrm{{\rm Y}}=0$ and $\mathrm{\Delta }=0$ are shown in black.

Figure 5

The front amplitudes $a_{N+}$ (blue) and $a_{N-}$ (red) as functions of $\mu$ for $a_2=1$ and $a_1=-2.3,-1.9,-1.85,-\frac{9}{5}$; the latter branch, corresponding to $5\mathrm{\Lambda }=6$, is shown in black.

Figure 6

Stability of the rotating wave selected in the wake of an $a_{N+}$ front (a) and an $a_{N-}$ front (b) in the $(a_1,a_2)$ plane at $\mu =-0.1$. Stable regimes are indicated in dark gray, unstable regimes in light gray, and regions with no front solutions in white. The lines $\mathrm{\Gamma }=0$ (red) and the lines $\mathrm{\Lambda }=6$ (blue) and $\mathrm{\Delta }=0$ (orange) in the region $\mathrm{\Gamma }\ge 0$ are also shown.

Figure 7

Stability of the rotating wave selected in the wake of an $a_{N+}$ front (a) and an $a_{N-}$ front (b) in the $(a_1,a_2)$ plane at $\mu =0.1$. Stable regimes are indicated in dark gray, unstable regimes in light gray, and regions with no front solutions in white. The lines $\mathrm{\Gamma }=0$ (red), $\mathrm{\Lambda }=6$ (blue), and $\mathrm{\Delta }=0$ (orange) in the region $\mathrm{\Gamma }\ge 0$ are also shown.

Figure 8

Stability of the rotating wave selected in the wake of the stationary front at the Maxwell point $\mu =-\frac{3}{\mathrm{\Gamma }}$ in the $(a_1,a_2)$ plane. The relevant solution branch is determined by the sign of the quantity $\mathrm{\Lambda }-6$. Stable regimes are indicated in dark gray, unstable regimes in light gray, and regions with no front solutions in white. The lines $\mathrm{\Gamma }=0$ (red), $\mathrm{\Lambda }=6$ (blue), and $\mathrm{\Delta }=0$ (orange) in the region $\mathrm{\Gamma }\ge 0$ are also shown.

Figure 9

The velocity $v_{N\pm }$ of the exact front solution is shown in blue and red, respectively. The parameters $(a_1,a_2)$ are indicated next to each curve and dashed lines represent instability of the essential spectrum. The line $v=0$, shown as a black dotted line, is included for reference.

Figure 10

Front speed $v_N$ (blue) relative to the linear spreading speed $v^{*}$ (red) for parameters $(a_1,a_2)$: (a) $(0,0)$, (b) $\left(-\frac{8}{5},\frac{1}{2}\right)$, (c) $\left(\sqrt{\frac{8}{3}},0\right)$. The selected (not selected) speed is indicated by a solid (dashed) line according to Eq. (27). The black dots represent speeds calculated by time-stepping Heaviside initial conditions with FD in the stationary frame and tracking the motion of the front. The space and time discretizations are $\mathrm{\Delta }x=0.05$ and $\mathrm{\Delta }t={\left(\mathrm{\Delta }x\right)}^2$; further details are contained in Appendix pp3. The deterministic corrections as a result of the FD approximation (computed in Appendix pp3) are shown using error bars on the data points.

Figure 11

Front speed $v_N$ (blue) relative to the linear spreading speed $v^{*}$ (red) for parameters $(a_1,a_2,\pm )$, where the symbol $\pm$ specifies the front: (a) $(-15,16,+)$, (b) $(4,1.86,-)$, (c) $(-2,1,+)$. The selected (not selected) speed is indicated by a solid (dashed) line according to Eq. (27). The black squares represent speeds calculated in a domain of length $L=400\lambda$ by time-stepping Gaussian initial conditions in the stationary frame using a spectral method with parameters $\mathrm{\Delta }t=0.01,N_x=4096$, and $\epsilon ={10}^{-8}$ (Appendix pp3). The black dots represent speeds calculated by time-stepping Heaviside initial conditions in the stationary frame using a finite difference code with space and time discretizations $\mathrm{\Delta }x=0.05,\mathrm{\Delta }t={\left(\mathrm{\Delta }x\right)}^2$, and tracking the motion of the front (Appendix pp3). The deterministic corrections as a result of the FD approximation (computed in Appendix pp3) are shown using error bars on the data points.

Figure 12

Speeds $v^{*}$ (red), $v_N$ (green), and $v_{\text{BF}}$ (blue) for pulled fronts as a function of the variable $h$ defined in Appendix pp1-s3 for $(a_1,a_2)=\left(\frac{5}{2},1\right)$. Increasing $h$ corresponds to increasing $\mu$. The parameters corresponding to $\mu =0,\infty$ are indicated with dotted lines and the region in which pulled fronts are selected and $v_{\text{BF}}>v^{*}$ is delimited by blue shading. The transition from pushed to BF-unstable pulled fronts, and subsequently from BF-unstable to BF-stable pulled fronts is marked by black dots.

Figure 13

Space-time plot of the evolution of (a) $\left|A\right(x,t\left)\right|$ and (b) $\text{Re}\left[A\right(x,t\left)\right]$ from Heaviside initial conditions in the stationary frame for $(a_1,a_2)=\left(\frac{5}{2},1\right)$ and $\mu =0.4$ computed using FD. The calculation is done on a domain $[-100,100]$ with Dirichlet boundary conditions on both $\text{Re}\left(A\right)$ and $\text{Im}\left(A\right)$ and only half of the simulation window is shown. The space and time discretizations are $\mathrm{\Delta }x=0.2$ and $\mathrm{\Delta }t=0.0025$. In this regime after an initial transient the front is pulled, traveling at speed $v^{*}$ to a good approximation. An offset line representing propagation at speed $v^{*}$ is shown in red.

Figure 14

Space-time plot of the evolution of (a) $\left|A\right(x,t\left)\right|$ and (b) $\text{Re}\left[A\right(x,t\left)\right]$ from Heaviside initial conditions in the stationary frame for $(a_1,a_2)=\left(\frac{5}{2},1\right)$ and $\mu =1$ computed using FD. The calculation is done on a domain $[-100,100]$ with Dirichlet boundary conditions on both $\text{Re}\left(A\right)$ and $\text{Im}\left(A\right)$ and only half of the simulation window is shown. The space and time discretizations are $\mathrm{\Delta }x=0.1$ and $\mathrm{\Delta }t=0.005$. In this regime after an initial transient the front is pulled, traveling at speed $v^{*}$ and a secondary front separates from the leading edge traveling at a speed $v_{\text{BF}}$. Offset lines representing propagation at speeds $v^{*}$ (red) and $v_{\text{BF}}$ (blue) are also pictured.

Figure 15

Solution $A(x,t=35)$ computed from Heaviside initial conditions in the stationary frame using FD for $(a_1,a_2)=\left(\frac{5}{2},1\right)$ and $\mu =1$, showing $\text{Re}\left(A\right)$ (green), $\text{Im}\left(A\right)$ (red), and $\left|A\right|$ (blue) on half of the computation domain. The discretization parameters are $\mathrm{\Delta }x=0.1$ and $\mathrm{\Delta }t=0.005$.

Figure 16

Space-time plot of the evolution of (a) $\left|A\right(\xi ,t\left)\right|$ and (b) $\text{Re}\left[A\right(\xi ,t\left)\right]$ from Heaviside initial conditions in the moving frame for $(a_1,a_2)=\left(-2,\frac{5}{2}\right)$ and $\mu =0$. The speed of the moving frame is $v_N$ and the front is pushed. In this simulation, we use time step $\mathrm{\Delta }t=0.01$, number of Fourier modes $N_x=6144$, and cutoff $\epsilon ={10}^{-10}$.

Figure 17

Space-time plot of the evolution of (a) $|\tilde{A}(\xi ,t)|$ and (b) $\text{Re}\left[\tilde{A}(\xi ,t)\right]$ from Heaviside initial conditions in a frame moving at speed $v_{\text{BF}}$ for $(a_1,a_2)=\left(-2,\frac{5}{2}\right)$ and $\mu =0$. In this simulation, we use time step $\mathrm{\Delta }t=0.005$, number of Fourier modes $N_x=6144$, and cutoff $\epsilon ={10}^{-10}$.

Figure 18

Space-time plot of the evolution of $A(\xi ,t)$ from Heaviside initial conditions in the moving frame for $\left(a_1,a_2\right)=\left(\frac{9}{2},5\right)$ and $\mu =0.025$. The speed of the moving frame is $v_N$ and the front is pushed. In this simulation, we use time step $\mathrm{\Delta }t=0.01$, number of Fourier modes $N_x=3072$, and cutoff $\epsilon ={10}^{-5}$.

Figure 19

Continuation of the space-time plot in Fig. 18 over a longer time interval with an initial condition of different amplitude. Offset lines representing propagation at speed $v_{\text{BF}}$ (blue) are also pictured.

Figure 20

Time series of $\left|A\right({\xi }_0,t\left)\right|$ (red) and $ln\left|A\right({\xi }_0,t\left)\right|$ (blue) representing a vertical slice of the evolution shown in Fig. 19 at ${\xi }_0=100$. The amplitude $A=a_N$ is plotted with a thick dashed line (black) and $A=0$ is also shown for reference.

Figure 21

Four cases of the $\text{Re}\left[q\right]\ne 0$ generic solutions to the Benjamin-Feir stability equations for the positive root: (a) $g>0,d>0$, (b) $g>0,d<0$, (c) $g<0,d>0$, (d) $g<0,d<0$. Here, the zero level set of Eq. (A10) is plotted in red and the region $u,v>0$ and $\text{Im}\left[\sigma \right]>0$ is plotted in blue. The solutions correspond to the intersections of the red curve with the blue regions.

Figure 22

Four cases of the $\text{Re}\left[q\right]\ne 0$ generic solutions to the Benjamin-Feir stability equations for the negative root: (a) $g>0,d>0$, (b) $g>0,d<0$, (c) $g<0,d>0$, (d) $g<0,d<0$. Here, the zero level set of Eq. (A10) is plotted in red and the region $u,v>0$ and $\text{Im}\left[\sigma \right]>0$ is plotted in blue. The solutions correspond to the intersections of the red curve with the blue regions.

Figure 23

Front speed $v_N$ (blue) relative to the linear spreading speed $v^{*}$ (red) for parameters $(a_1,a_2)=(0,0)$. The selected (not selected) speed is indicated by a solid (dashed) line according to Eq. (27). The navy ($\mathrm{\Delta }x=0.1$) and orange ($\mathrm{\Delta }x=0.05$) circles represent speeds calculated by time-stepping Heaviside initial conditions using FD in the stationary frame and tracking the motion of the front with $\mathrm{\Delta }t={\left(\mathrm{\Delta }x\right)}^2$. The squares represent the values of the speeds after subtracting the corrections identified in Eq. (C4).