Soliton resolution for critical co-rotational wave maps and radial cubic wave equation

TL;DR

This paper proves the soliton resolution conjecture for co-rotational wave maps and radial cubic wave equations, establishing a comprehensive understanding of their long-term behavior in the energy space.

Contribution

It is the first to verify the soliton resolution conjecture for all initial data in the energy space for wave-type equations.

Findings

Proved soliton resolution for all solutions of co-rotational wave maps.

Established soliton resolution for radial solutions of the energy-critical cubic wave equation.

Demonstrated boundedness of solutions in the energy norm.

Abstract

In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space for a wave-type equation. We also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.