A Gluing Problem for a Gauged Hyperbolic PDE

TL;DR

This paper develops a method to construct three-dimensional dynamical solutions of the gauged hyperbolic PDE by gluing two-dimensional vortex configurations, extending the understanding of vortex dynamics in higher dimensions.

Contribution

It proves the existence of 3D solutions close to 2D vortex configurations using a gluing technique, linking wave maps to the Abelian Higgs model in dimension 3.

Findings

Existence of 3D solutions approximating 2D vortex dynamics

Construction of solutions via gluing of vortex configurations

Small gauge field variables in the constructed solutions

Abstract

In this project, we study the hyperbolic Abelian Higgs model in dimension $3$ at the critical coupling. The stationary solutions to the two-dimensional version of this equation have been found by Jaffe and Taubes, the so called $N$-vortex configurations. One can consider the space of all $N$-vortex configurations $M_N$ as a smooth Riemannian manifold. Stuart has proved that near the critical coupling regime, the dynamic in dimension $2$ can be approximated by a finite dimensional Hamiltonian system on the moduli space $M_2$, for suitable initial data. In this thesis, we study how to glue the $N$-vortex configurations to construct dynamical solutions in dimension $3$. Namely, we prove that if $q:[0,T)\times \mathbb{R}\to M_N$ is a wave map, then for $\epsilon>0$ small enough, there exists a solution of the Abelian Higgs model in dimension $(1+3)$ on $[0,\frac{T_0}{\epsilon})\times \mathbb{R}^3$ for some $T_0>0$ which is close to $(\phi,\alpha)(.;q(\epsilon t,\epsilon z))$ in terms of $\epsilon$, where $(\phi,\alpha)(.)$ denotes the variables of the corresponding $N$-vortex configuration. Furthermore, the other gauge field variables are small in terms of $\epsilon$. This dissertation has been supervised by Prof. Robert Jerrard.