Pattern formation on a sphere

Paul Matthews

School of Mathematical Sciences,

University of Nottingham, UK

Pattern formation on the surface of a sphere is described by equations involving interactions of spherical harmonics of degree $l$. When $l$ is even, the leading-order equations are determined uniquely by the symmetry, regardless of the physical context. Existence and stability results are found for even $l$ up to $l=12$. Using either a variational or eigenvalue criterion, the preferred solution has icosahedral symmetry for $l=6$, $l=10$ and $l=12$. Numerical simulations of a model pattern-forming equation are in agreement with these theoretical predictions near onset and show more complex patterns further from onset.