2024/05/25 by Amirmasoud Geevechi, Geevechi, Amirmasoud
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2405.16092
openalex publication_date 2024/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,\fracT0ε)× ℝ3 for some T0>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.