A Gluing Problem for a Gauged Hyperbolic PDE

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.