B. Dionne and M. Golubitsky
Planforms in two and three dimensions
ZAMP. 43 (1992) 36-62.
When solving systems of PDE with two space dimensions it is often assumed that the solution is spatially doubly periodic. This assumption is usually made in systems such as the Boussinesq equation or reaction-diffusion equations where the equations have Euclidean invariance. In this article we use group theoretic techniques to determine a large class of spatially doubly periodic solutions that are forced to exist near a steady-state bifurcation from a translation-invariant equilibrium.
This type of bifurcation problem has been considered by many authors when studying a number of different systems of PDE. Typically, these studies focus at the beginning on equilibria that are spatially periodic with respect to a fixed planar lattice type - such as square or hexagonal. our focus is different in that we attempt to find all spatially periodic equilibria that bifurcate on all lattices. This point of view leads to some technical simplifications such as being able to restrict to translation free irreducible representations.
Of course, many of the types of solutions that we find are well-known - such as hexagon and roll solutions on a hexagonal lattice. This coordinated group theoretic approach does lead, however, to solutions which seem not to have been discussed previously (antisquare solutions on a square lattice) as well as to a more complete classification of the symmetry types of possible solutions. Moreover, our methods extend to triply periodic solutions of PDE with three spatial variables. Some of these results, namely those concerned with primitive cubic lattices, are presented here. The complete results on triply periodic solutions may be found in [6,7].