Recent progress on the problem of soliton resolution

Abstract

Dispersive partial differential equations are evolution equations (that is, involving the time variable) whose solutions preserve the energy but can still 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, apart from exceptional cases, 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 critical wave maps equation, which is the analog of the wave equation for maps from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$. The solitons correspond to harmonic maps, which were classified by Eells and Wood in 1976. We consider equivariant solutions, which are solutions having a specific symmetry preserved by the flow. In a joint work with Andrew Lawrie, we prove that soliton resolution holds for these solutions. Our proof hinges on an analysis of collisions of solitons and an appropriate localized Lyapunov functional, which together allow us to prove a no-return lemma for multisoliton configurations.

Building on some of these ideas, we solve, in a joint work with Andrew Lawrie and Wilhelm Schlag, an analogous problem for the harmonic map heat flow of Eells and Sampson (1964) without assuming any symmetry of the initial data.