Amirmasoud Geevechi | Searcharxiv

arXiv2025

Adiabatic approximation of Abelian Higgs models

We construct novel solutions in $d\ge 3$ space dimensions of a family of nonlinear evolutions equations that includes the critical hyperbolic Abelian Higgs model (AHM). For the AHM, these solutions exhibit an ensemble of $N\ge 1$ slowly-moving, nearly parallel vortex filaments, whose leading-order dynamics are described by a wave map from ${\mathbb R}^{d-2}$ into the Abelian Higgs moduli space, a manifold carrying a natural Riemannian structure that parametrizes stationary 2d solutions of the AHM. We also prove extremely similar results that relate the critical Abelian Higgs heat flow, modeling certain superconductors, to the harmonic map heat flow into the Moduli space, as well as some parallel results for near-critical equations. When $d=3$, these results allow for the study of the poorly-understood phenomenon of vortex reconnection in this setting.

math.AP↗

arXiv2024

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 $M_N$ 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 $ε>0$ small enough, there exists a solution of the Abelian Higgs model in dimension $(1+3)$ on $[0,\frac{T_0}ε)\times \mathbb{R}^3$ for some $T_0>0$ which is close to $(ϕ,α)(.;q(εt,εz))$ 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.

math.AP↗