Tristability between stripes, up-hexagons, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons

Third-order amplitude equations on hexagonal lattices can be used for predicting the existence and stability of stripes, up-hexagons, and down-hexagons in pattern-forming systems. These amplitude equations predict the nonexistence of bistable ranges between up- and down-hexagons and tristable ranges between stripes, up-, and down-hexagons. In the present work we use fifth-order amplitude equations for finding such bistable and tristable ranges for a generalized Swift-Hohenberg equation and discuss stationary front connections between up- and down-hexagons.

Figure 1

The system (4) for the parameter set (5) has one trivial homogeneous state $(n,w)=(0,p)$, which is stable for $p<p_0$ and unstable for $p>p_0$ with $p_0\approx 0.157$. At this transition of stability a nontrivial homogeneous state bifurcates, which is stable at the beginning, becomes Turing unstable a bit later (in $p\approx 0.169$), and becomes stable again later on (in $p\approx 0.413$). The black line in (a) represents the nontrivial homogeneous states. Here and in the following, thick and thin lines represent stable and unstable states, respectively. We use the continuation and bifurcation software tool pde2path [33, 34] to follow the solution branches, which bifurcate in the second Turing point. Here we use the domain $\mathrm{\Omega }=(-l_x,l_x)\times (-l_y,l_y)$ with $l_x=20\pi /k_c$, $l_y=20\pi /\left(\sqrt{3}{k}_c\right)$ and Neumann boundary conditions, while $k_c=0.206$ is the critical wave number corresponding to the second Turing point. The red, green, and blue branches in (a) and (b) correspond to $H^{+}$, $S$, and $H^{-}$, respectively. All three states are stable for $p=0.301$ and their density plots for $H^{+}$, $S$, and $H^{-}$ are shown over the domain $\mathrm{\Omega }=(-l_x/2,l_x/2)\times (-l_y/2,l_y/2)$ in (c), (d), and (e), respectively. We set $\beta =3.5$ for developing (b).

Figure 2

We set $c_3=-1$, $c_4=5$, $c_5=-2$ for creating the plots in the present figure. $H^{+}$, $S$, and $H^{-}$ are monostable in the red, yellow, and blue regions, respectively. We have bistable ranges between $H^{-}$ and $S$, $H^{-}$ and $H^{+}$, $H^{+}$ and $S$ in the green, violet, and orange regions, respectively. In the black region all three states are stable. We set $c_2=-2$ for creating (b) and (c). Here we use the amplitude equations (7) for creating the solid lines. $•$ and + represent stable and unstable states, respectively, which we found numerically via the finite element and Newton's method on the domain $\mathrm{\Omega }=(-l_x,l_x)\times (-l_y,l_y)$ with $l_x=10\pi$, $l_y=10\pi /\sqrt{3}$ and Neumann boundary conditions. Here we use $\varepsilon =0.5$ and $\varepsilon =1$ for (b) and (c), respectively. The vertical axis shows the maximum of $u$ for stripes and up-hexagons and the minimum of $u$ for down-hexagons.

Figure 3

(a) Numerically calculated solution branches of (6) for $\varepsilon =0.5$ and $c_2=-2$, $c_3=-1$, $c_4=5$, $c_5=-2$ [as for Fig. 2]. Here and in the following it holds ${\parallel u\parallel }_2=\sqrt{\frac{1}{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}{\left|u\right|}^{2}dz}$, where $\mathrm{\Omega }$ is the considered domain and $z$ represents the spatial coordinates. Here $\mathrm{\Omega }=(-l_x,l_x)\times (-l_y,l_y)$ with $l_x=16\pi$, $l_y=2\pi /\sqrt{3}$ and we use Neumann boundary conditions. The orange branch $B^{+}$ is a branch of mixed mode patterns. (b) Density plot of a solution, which lies more or less in the middle of the bean branch. (c) Density plot of a solution, which lies on the gray branch and which is labeled with 235. (d) Left and right part of solution 235 plotted separately.

Figure 4

(a) Numerically calculated solution branches of (6) for $\varepsilon =0.5$ and $c_2=-2$, $c_3=-1$, $c_4=5$, $c_5=-2$ [as for Fig. 2 and Fig. 3]. Here $\mathrm{\Omega }=(-l_x,l_x)\times (-l_y,l_y)$ with $l_x=30.5\pi$, $l_y=2\pi /\sqrt{3}$ and we use Neumann boundary conditions. (b) Density plot of stripes. (c)–(h) Density plots of solution 53, 80, 120, 410, 458, and 810, respectively. (i) Left and right part of solution 458 plotted separately.

Figure 5

Potential energy $E_{\text{p}}$ (9) and Hamiltonian for up-hexagons (red), stripes (green), and down-hexagons (blue) are shown in (a) and (b) respectively.