Spatial Hidden Symmetries in Pattern Formation

Partial differential equations that are invariant under Euclidean transformations are traditionally used as models in pattern formation. These models are often posed on finite domains (in particular, multidimensional rectangles). Defining the correct boundary conditions is often a very subtle problem. On the other hand, there is pressure to choose boundary conditions which are attractive to mathematical treatment. Geometrical shapes and mathematically friendly boundary conditions usually imply spatial symmetry. The presence of symmetry effects that are “hidden” in the boundary conditions was noticed and carefully investigated by several researchers during the past 15–20 years. Here we review developments in this subject and introduce a unifying formalism to uncover spatial hidden symmetries (hidden translations and hidden rotations) in multidimensional rectangular domains with Neumann boundary conditions.