Dispersive partial differential equations are evolution equations whose solutions preserve the energy, but decay in large time due to the fact that various wave numbers propagate with distinct velocities. In some cases, there exist special solutions called solitons or lumps, which do not change their shape as time passes. The Soliton Resolution Conjecture predicts that solitons are the only obstruction to the decay of solutions. More precisely, every solution eventually decomposes into a superposition of asymptotically decoupled solitons and a decaying term called radiation.
 I will focus on the critical equivariant wave maps equation, which is the analog of the wave equation for maps from the Minkowski space of dimension 2+1 to the 2-dimensional unit sphere, under an additional symmetry assumption on the initial data (analogous to spherical symmetry). In this case, solitons are centered at the origin, but they can still decouple if their characteristic scales are very different. In a joint work with Andrew Lawrie, we prove that soliton resolution holds. In view of this result, it is natural to examine what can be the long-time behavior of the scales of several interacting solitons. In this direction, in a recent joint work with Joachim Krieger we construct solutions developing a singularity by a simultaneous concentration of two solitons at the origin.