Wave maps on a wormhole

TL;DR

This paper investigates wave maps on a wormhole spacetime, demonstrating the existence and stability of static solutions and providing numerical evidence for their role as attractors in the evolution of wave maps.

Contribution

It proves the existence and linear stability of harmonic maps for each topological degree and shows numerically that solutions tend to these harmonic maps over time.

Findings

Existence of unique harmonic maps for each topological degree.

Linear stability of these harmonic maps.

Numerical evidence of solutions evolving towards harmonic maps.

Abstract

We consider equivariant wave maps from a wormhole spacetime into the three-sphere. This toy-model is designed for gaining insight into the dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture. We first prove that for each topological degree of the map there exists a unique static solution (harmonic map) which is linearly stable. Then, using the hyperboloidal formulation of the initial value problem, we give numerical evidence that every solution starting from smooth initial data of any topological degree evolves asymptotically to the harmonic map of the same degree. The late-time asymptotics of this relaxation process is described in detail.