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Talk Abstract

Sliced planforms: a representation of three-dimensional symmetries in thin domains

Sliced planforms: a representation of three-dimensional symmetries in thin domains

Institute for Mathematics and Its Applications

**Gabriela M. Gomes**, University of Porto

Problems described by PDEs in thin domains are often formulated as
planar problems (see for example Gunaratne, Ouyang & Swinney
[1] for a reaction-diffusion problem and Golubitsky, Swift & Knobloch
[2] for the Rayleigh-Béenard convection problem).
In particular for problems with Euclidean symmetry (which reaction-diffusion
equations satisfy),
the expected solutions are often described as planforms which are doubly
periodic with respect to a certain two-dimensional lattice.
Here we show that some nontrivial symmetries may
be missed by this assumption. We consider the
thin domain as a slice of a fully three-dimensional problem
whose symmetry is described by a lattice in three dimensions. The corresponding
*sliced planforms* have now a three-dimensional characterization
and different planforms may have different structure along the thin
direction. As a result we find
symmetries that are not expect in planar systems. In particular,
we find that two planar planforms with different wavelength may
extend to planforms with the same wavelenght in the three dimensional space.

REFERENCES
[1] Gunaratne, G.H., Ouyang, Q. & Swinney, H.L. (1994)
Pattern formation in the presence of symmetries. *Phys. Rev.
E*,** 50**(4), 2802-2820.
[2] Golubitsky, M., Swift, J.W. & Knobloch, E. (1984)
Symmetries and Pattern Selection in Rayleigh-Bénard Convection.
*Physica* ** 10**D, 249-276.