Recent progress on the problem of soliton resolution

Abstract

Dispersive partial differential equations are evolution equations whose solutions decay in large time due to the fact that various frequencies propagate with distinct velocities. In some cases, there exist special solutions called solitons, 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 solitons and a decaying term called radiation. We discuss the conjecture in the context of the wave maps equation, which is the analog of the wave equation for sphere-valued maps.(1)

(1) This note is based on the talk given by the author at the 9th European Congress of Mathematics.