Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices

When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov--Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift--Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have eight-fold, ten-fold, twelve-fold, and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with six-fold rotational symmetry made up from the superposition of two equal amplitude hexagonal patterns rotated by almost any angle $\alpha$ with respect to each other. Here, we revisit the Swift--Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift--Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.