Georgia Institute of Technology College of Sciences

I will discuss a two dimensional spatial pattern formation problem proposed by Doelman, Sandstede, Scheel, and Schneider in 2003 as a phenomenological model of convective fluid flow . In the same work the authors just mentioned use geometric singular perturbation theory to show that the coexistence of certain spatial patterns is equivalent to the existence of some heteroclinic orbits between equilibrium solutions in a four dimensional vector field. More recently Andrea Deschenes, Jean-Philippe Lessard, Jan Bouwe van den Berg and the speaker have shown, via a computer assisted argument, that these heteroclinic orbits exist. Taken together these arguments provide mathematical proof of the existence of some non-trivial patterns in the original planar PDE. I will present some of the ingredients of this computer assisted proof.