Classification of String Solutions for the Self-Dual Einstein–Maxwell–Higgs Model

Abstract

In this paper, we are concerned with an elliptic system arising from the Einstein–Maxwell–Higgs model which describes electromagnetic dynamics coupled with gravitational fields in spacetime. Reducing this system to a single equation and setting up the radial ansatz, we classify solutions into three cases: topological solutions, nontopological solutions of type I, and nontopological solutions of type II. There are two important constants: $$a>0$$ representing the gravitational constant and $$N\ge 0$$ representing the total string number. When $$0\le aN<2$$ , we give a complete classification of all possible solutions and prove the uniqueness of solutions for a given decay rate. In particular, we obtain a new class of topological solitons, with nonstandard asymptotic value $$\sigma <0$$ at infinity, when the total string number is sufficiently large such that $$1<aN<2$$ . We also prove the multiple existence of solutions for a given decay rate in the case $$aN \ge 2$$ . Our classification improves previous results which are known only for the case $$0<aN<1$$ .