Solutions of the Yang-Mills-Higgs equations in the plane

Question

Is there any reference describing the analytical properties of the critical points of the functional $$ E(\phi,A)=\int_{\mathbb{R}^2}\frac{1}{2}|(\partial_j-iA_j)\phi|^2+\frac{1}{4}F_{jk}F_{jk}+\frac{\lambda}{8}(1-|\phi|^2)^2 $$ with $F_{jk}=\partial_jA_k-\partial_kA_j$ (i.e. solutions of the Yang-Mills-Higgs equations)?

All that I found in the literature is related to (much) more general cases. Any help is appreciated.

Answer 1

In this 2D model, the system is often called the Ginzburg-Landau system, as it was introduced independently as a model of superconductors by Ginzburg and Landau. There is a large literature devoted to this problem, so I will just mention a couple books you might find useful. The first three consider the more general problem but have sections related to the 2D model. The latter three specialize to the 2D case.

1. Manton and Sutcliffe, Topological Solitons
2. Jaffe and Taubes, Vortices and Monopoles
3. Yang, Solitons in Field Theory and Nonlinear Analysis
4. Bethuel, Brezis, and Helein, Ginzburg-Landau Vortices
5. Pacard and Riviere, Linear and Nonlinear Aspects of Vortices
6. Sandier and Serfaty, Vortices in the Magnetic Ginzburg-Landau Model