Two-bubble dynamics for the equivariant wave maps equation

I will consider the energy-critical wave maps equation with values in the sphere in the equivariant case, that is for symmetric initial data. It is known that if the initial data has small energy, then the corresponding solution scatters. Moreover, the initial data of any scattering solution has topological degree 0. I try to answer the following question: what are the non-scattering solutions of topological degree 0 and the least possible energy? I will show how to construct such threshold solutions. Then I will describe the dynamical behavior of any threshold solution: in one time direction it is close to a superposition of two stationary states, in the other time direction it scatters. This is a joint work with Andrew Lawrie (MIT).