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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 10D, 249-276.
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