Planforms in two and three dimensions
نویسندگان
چکیده
Many systems of partial differential equations are posed on all of R" and have Euclidean symmetry. These include the Navier-Stokes equations, the Boussinesq equations, the Kuramoto-Sivashinsky equation and reaction-diffusion systems (with constant diffusion coefficients). In many applications, where these and related Euclidean equivariant equations are used, time independent, spatially periodic solutions are sought; and, typically, they are obtained by bifurcation from an invariant equilibrium. In this paper we attempt to classify, by symmetry, spatially periodic solutions that can be obtained through bifurcation. Our main result is a partial classification of such solutions in two and three spatial dimensions obtained using symmetry methods and equation independent genericity considerations. The remainder of this Introduction is devoted to making precise the kind of classification theorem we intend to prove. We show that a certain class of planforms may be found by solving an algebraic problem whose data is based on the irreducible representations of the symmetry groups of n-dimensional lattices. The planar planforms are classified in Section 2 (see Theorem 2.1, whose proof is given in Section 3). The main theoretical results (valid for all n) are also presented in Section 2. The classification of planforms in three dimensions is more complicated than in two. In Section 4 we describe our results for the standard cubic lattices; that is, for the standard spatially triply periodic planforms. Details for the other three-dimensional lattices may be found in Dionne [6].
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