On the mode stability of a self-similar wave map

TL;DR

This paper investigates the stability of a specific self-similar wave map, demonstrating that it is linearly stable except for a known gauge instability, through analysis of perturbation eigenvalues.

Contribution

It provides a rigorous proof that the self-similar wave map has no real unstable eigenvalues besides the gauge instability, confirming its linear stability.

Findings

No real unstable eigenvalues found apart from gauge instability

Supports the conjecture of linear stability for the wave map

Clarifies the spectral properties of perturbations

Abstract

We study linear perturbations of a self-similar wave map from Minkowski space to the three-sphere which is conjectured to be linearly stable. Considering analytic mode solutions of the evolution equation for the perturbations we prove that there are no real unstable eigenvalues apart from the well-known gauge instability.