Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere

TL;DR

This paper constructs a family of self-similar wave maps from Minkowski spacetime into the 3-sphere, analyzes their stability, and discusses potential generalizations to higher dimensions.

Contribution

It introduces a countable family of spherically symmetric self-similar wave maps and examines their stability properties, highlighting the significance of the first excitation as a critical solution.

Findings

Existence of a countable family of wave maps into the 3-sphere.

The number of unstable modes equals the excitation index.

Potential for extending results to higher dimensions.

Abstract

We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state wave map found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold of singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its excitation index. Finally, we formulate a condition under which these results can be generalized to higher dimensions.