Generalized Abelian Higgs Equations

Abstract

In this chapter we present a thorough study of a most natural generalization of the Abelian Higgs theory in (2 + 1) dimensions containing m Higgs scalar fields. We are led to an m × m system of nonlinear elliptic equations which is not integrable. The main tool here is to use the Cholesky decomposition theorem to reveal a variational structure of the problem. In §4.1 we formulate our problem and state an existence and uniqueness theorem. In §4.2 we treat the problem as a pure differential equation problem and state a series of general results for the system defined on a compact surface and the full plane. In §4.3–§4.5, we establish the existence part of our results stated in §4.1 and §4.2. In §4.6 we establish the nonexistence results stated in §4.2. In §4.7 we extend our study to the situation when the coefficient matrix of the nonlinear elliptic system is arbitrary.