Integrable Abelian vortex-like solitons

Abstract

We propose a modified version of the Ginzburg-Landau energy functional admitting static solitons and determine all the Painlevé-integrable cases of its Bogomolny equations of a given class of models. Explicit solutions are determined in terms of the third Painlevé transcendents, allowing us to calculate physical quantities such as the vortex number and the vortex strength. These solutions can be interpreted as the usual Abelian-Higgs vortices on surfaces of non-constant curvature with conical singularity.

Manton’s five vortex equations from self-duality

“…In each case the symmetry group is G C0 and the gauge group is G C . These integrable cases of (1.2) do not exhaust the list of all integrable vortices: there are other integrable cases related to the sinh-Gordon and the Tzitzeica equations [4,5,8], where the vortex is interpreted as a surface with conical singularities.…”

Vortices, Painlevé integrability and projective geometry

“…Chapters 2, 4 and 5 are based on (but are not the same as) my papers in collaboration with my supervisor [12,13,11], respectively, except for Section 4.4. Chapter 3 is essentially the same as my individual paper [9].…”

“…These integrable cases of (2.2) do not exhaust the list of all integrable vortices: there are other integrable cases related to the sinh-Gordon and the Tzitzeica equations [22,10,9].…”