  Bifurcation problems in which periodic boundary conditions (PBC) or Neumann
boundary conditions (NBC) are imposed often involve partial differential
equations that have Euclidean symmetry. In this case posing the bifurcation
problem with either PBC or NBC on a finite domain can lead to a symmetric
bifurcation problem for which the manifest symmetries of the domain do not
completely characterize the constraints due to symmetry on the bifurcation
equations. Additional constraints due to the Euclidean symmetry of the
equations can have a crucial influence on the structure of the bifurcation
equations. An example is the bifurcation of standing waves on the surface of
fluid layer. The Euclidean symmetry of an infinite fluid layer constrains the
bifurcation of surface waves in a finite container with square cross section
because the waves satisfying PBC or NBC can be shown to lie in certain
finite-dimensional fixed point subspaces of the infinite-dimensional problem.
These constraints are studied by analyzing the finite-dimensional vector fields
obtained on these subspaces by restricting the bifurcation equations for the
infinite layer. Particular emphasis is given to determining which bifurcations
might reveal observable effects of the rotational symmetry of the infinite
layer. It turns out that a necessary condition for this possibility to arise is
that the subspace for PBC must carry a reducible representation of the
normalizer subgroup acting on the subspace. This condition can be met in
different ways in both codimension-one and codimension-two bifurcations.

Pattern Formation and Solitons;Pattern Formation and Solitons