A scattering theory construction of dynamical solitons in 3d

TL;DR

This paper develops a scattering theory framework for energy critical wave equations in 3D, constructing solutions near solitons with polynomial decay, and extends methods to supercritical cases.

Contribution

It introduces a novel scattering theory construction for 3D solitons, including energy boundedness and conormal solutions near blow-up, extending to supercritical equations.

Findings

Energy boundedness for linearized problem around soliton

Construction of scattering solutions near timelike infinity

Extension of methods to supercritical equations

Abstract

We study the energy critical wave equation in 3 dimensions around a single soliton. We obtain energy boundedness (modulo unstable modes) for the linearised problem. We use this to construct scattering solutions in a neighbourhood of timelike infinity ($i_+$), provided the data on null infinity ($\scri$) decay polynomially. Moreover, the solutions we construct are conormal on a blow-up of Minkowski space. The methods of proof also extend to some energy supercritical modifications of the equation.