Vortex dynamics for the Gross-Pitaevskii equation

: (Alternative Format Thesis)

Abstract

This thesis presents a rigorous construction of multi-vortex solutions to the Gross-Pitaevskii equation in the plane, verifying formal asymptotics derived by Neu in 1990.

In precise terms we consider the Schrödinger evolution equation associated to the Ginzburg-Landau model in two dimensions, and construct solutions to this problem resembling a product of canonical profiles in a certain asymptotic regime. These solutions have n≥2 zeroes of degree ±1, whose dynamics is governed at leading order by a Hamiltonian ODE known as the Helmholtz-Kirchhoff system.

Compared with previous measure-theoretic works on the same subject, our approach is constructive and provides a detailed description of the solutions under consideration. In particular, we detect lower order corrections to the dynamics depicting the interaction of vortices with induced radiation, justifying a formal expansion obtained by Ovchinnikov and Sigal.

Date of Award	25 Nov 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Monica Musso (Supervisor) & Manuel Del Pino (Supervisor)