Hexagon patch wavenumber-selection conjecture for the planar Swift–Hohenberg equation

Let $(\mu,\nu)$ be fixed with $\mu>0$. A stable hexagon 2D patch solution is a stationary or invading stable hexagon patch solution of the planar Swift–Hohenberg equation. For perpendicular hexagon $\langle10\rangle$- and $\langle11\rangle$-fronts, denote their selected far-field wavenumbers by $(k_x^{\langle10\rangle},k_y^{\langle10\rangle})$ and $(k_x^{\langle11\rangle},k_y^{\langle11\rangle})$, respectively.

Hexagon patch wavenumber-selection conjecture. Every stationary and invading stable hexagon 2D patch solution selects a unique hexagon cellular pattern. Its wavenumbers are determined by compatibility of the selected far-field wavenumbers of the perpendicular fronts, namely

$$(k_x^{\langle10\rangle},k_y^{\langle10\rangle})=(k_x^{\langle11\rangle},k_y^{\langle11\rangle}).$$

The conjecture proposes a criterion for pattern selection in the bistable region of the planar Swift–Hohenberg equation: a localized hexagon patch is viewed as being composed of perpendicular $\langle10\rangle$- and $\langle11\rangle$-hexagon-like fronts. The source presents this as a heuristic criterion, and no resolution is supplied here.