Universal wave-number selection laws in apical growth

We study pattern-forming dissipative systems in growing domains. We characterize classes of boundary conditions that allow for defect-free growth and derive universal scaling laws for the wave number in the bulk of the domain. Scalings are based on a description of striped patterns in semibounded domains via strain-displacement relations. We compare predictions with direct simulations in the Swift-Hohenberg, the complex Ginzburg-Landau, the Cahn-Hilliard, and reaction-diffusion equations.

Figure 1

Apical growth in SH equation (1) with $\mu =1.5$: (a) $c=0.01,k\sim 0.928$ and (b) $c=1$ (right, $k\sim 0.981$).

Figure 2

(a) Schematic illustration of the shooting problem, connecting boundary conditions to periodic orbits; (b) strain-displacement curve of Eq. (1) with free boundary conditions and $\mu =1.5$; specific boundary layer profiles as insets in (b).

Figure 3

Equilibria of Eq. (1) continued in the domain size $L$, with select saddle nodes marked as orange stars; saddle nodes approach extrema of $K$ [Eq. (5)]. (a) $L^2$ norm and (b) wave number.

Figure 4

(a) Asymptotically linear relation of $k-k_{\min }$ vs $\sqrt{c_{\mathrm{pe}}}$ with slope $\sqrt{2}\zeta (1/2)$; (b) plot exhibiting 1/4 exponent in corrections; (c) sample plots of $\theta (t,0)$ for range of $c$ values; $K\left(\theta \right)=1+0.3sin\left(\theta \right),d_{\mathrm{eff}}=1.$

Figure 5

Measured (blue crosses) vs predicted (orange line) wave number with associated strain-displacement relations for (a) CGL Eq. (16), (${\mu }_k=0.4,{\mu }_0=0.1,d_{\mathrm{eff}}\approx 0.830$); (b) SH Eq. (1) ($\mu =1.5,d_{\mathrm{eff}}\approx 3.531$); (c) RD Eq. (17) (${\mu }_0=0.95,d_u=0.1,d_v=2,\gamma =0.2,d_{\mathrm{eff}}\approx 0.872$); and (d) CH Eq. (18) (${\mu }_k=0.6,{\mu }_{\theta }=0.15,{\mu }_2=0.2,d_{\mathrm{eff}}\approx 0.8275$). Errors in the leading-order coefficient $k_{1/2}=-\sqrt{2}\zeta (1/2)$ are 0.0061, $-0.0033$, 0.0914, and $-0.1970$, respectively.

Figure 6

Selected wave numbers plotted on a logarithmic scale for small and large speeds: (a) phase-diffusion Eq. (6), $K\left(\theta \right)=1+0.3sin\left(\theta \right)$; (b) CGL Eq. (16). Blue curves show direct simulations; orange and gold curves show small and large speed predictions.