##
New planforms in systems of
partial differential equations with Euclidean symmetry

* Arch. Rat. Mech. Anal. * **131** (1995) 199-224

** Ignacio Bosch Vivancos, Pascal Chossat and Ian Melbourne **

** Abstract **

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.

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