Yang-Mills instanton on a four dimensional wormhole: asymptotic stability in the energy space

TL;DR

This work analyzes SU(2) Yang-Mills dynamics on a 4+1 dimensional wormhole, reducing to a 1D semilinear wave equation with a kink as a distinguished stationary solution. It proves conditional asymptotic stability of the degree-one kink under small odd perturbations in the energy space, delivering precise decay on finite intervals and weighted global decay. The authors develop a robust virial framework with tailored weights to overcome the algebraic decay of the linearized operator, including a Darboux-type transformation to produce a repulsive potential and ensure coercivity. The results advance the understanding of soliton-like stability on wormhole geometries and showcase the versatility of virial methods for problems with slow dispersive decay.

Abstract

In this paper we consider an $SU(2)$ Yang-Mills field propagating in the $4+1$ dimensional wormhole spacetime. Assuming the spherically symmetric magnetic ansatz the problem reduces to a one dimensional non linear wave equation. This equation posses a degree one solution (instanton) which is odd in space. We consider small, odd perturbations of the instanton and show that it is conditionally asymptotically stable in the odd energy space.

Theorems & Definitions (17)

• Lemma 2.1
• Theorem 2.1
• Theorem 2.2
• Proposition 4.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 5.1: Lemma 4.7 KMMV
• Lemma 5.2