Development of Singularities of the Skyrme Model

TL;DR

This paper investigates singularity formation in the equivariant Skyrme Model, demonstrating the existence of smooth self-similar solutions in 5+1 dimensions that lead to finite-time blow-up from smooth initial data.

Contribution

It proves the existence of self-similar solutions in the strong-field limit of the Skyrme Model, establishing finite-time blow-up for certain smooth initial conditions.

Findings

Existence of smooth self-similar solutions in 5+1 dimensions.

Finite-time blow-up from smooth initial data.

Extension of singularity formation results from Wave Maps to the Skyrme Model.

Abstract

The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what's known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the $5+1$-dimensional Skyrme Model in the strong-field limit, and use that to conclude that the solution to the corresponding Cauchy problem blows up in finite time, starting from a particular class of everywhere smooth initial data.