Spherically symmetric 't Hooft–Polyakov monopoles

In 1974 G. 't Hooft and A. Polyakov independently proved that the equations of motion of the SU(2)-gauge model with a triplet of scalar fields have static spherically symmetric solutions with finite energy and magnetic charge. These solutions represent an important example of topological solitons and have attracted a lot of theoretical attention. Soon the exact analytical solution was found by E. Bogomolny, and also by M. Prasad and S. Sommerfeld, and then a simpler solution by D. Singleton. These are all spherically symmetric solutions that have been known to date. All monopoles are divided into classes of homotopically nonequivalent solutions, which are characterized by the index of the map (topological charge) of two-dimensional spheres. E. Bogomolny proved that in the case of massless scalar fields in each class of solutions there are solutions with minimal energy that satisfy a nonlinear system of partial differential equations of the first order. In his work M. Katanaev found a general spherically symmetric solution of the Bogomolny equations [1]. It depends on two constants and one arbitrary function of the radius. In particular cases, the well-known solutions of Bogomolny-Prasad-Sommerfeld and Singleton are obtained. Thus, all spherically symmetric 't Hooft-Polyakov monopoles with massless scalar fields and minimal energy are found.