A Gluing Problem for a Gauged Hyperbolic PDE

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 MN 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 M₂, 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)× R→ MN is a wave map, then for ε>0 small enough, there exists a solution of the Abelian Higgs model in dimension $(1+3)$ on [0,T₀/ε)× R³ for some T₀>0 which is close to (φ,α)(.;q(ε t,ε z)) in terms of ε, where (φ,α)(.) denotes the variables of the corresponding N-vortex configuration. Furthermore, the other gauge field variables are small in terms of ε. This dissertation has been supervised by Prof. Robert Jerrard.