Arch. Rat. Mech. Anal. 131 (1995) 199-224
Ignacio Bosch Vivancos, Pascal Chossat and Ian Melbourne
Dionne and Golubitsky consider the classification of planforms bifurcating (simultaneously) in scalar PDEs that are equivariant with respect to the Euclidean group in the plane. In particular, those planforms corresponding to isotropy subgroups with one-dimensional fixed-point space are classified.
Many important Euclidean-equivariant systems of PDEs essentially reduce to a scalar PDE, but this is not always true for general systems. We extend the classification of Dionne and Golubitsky obtaining precisely three planforms that can arise for general systems and do not exist for scalar PDEs. In particular, there is a class of one-dimensional `pseudoscalar' PDEs for which the new planforms bifurcate in place of three of the standard planforms from scalar PDEs. For example the usual rolls solutions are replaced by a nonstandard planform called anti-rolls. Scalar and pseudoscalar PDEs are distinguished by the representation of the Euclidean group.