A necessary and sufficient condition for the existence of multisolitons in a self-dual gauged sigma model

This paper presents a resolution of the gauged O(3) sigma model proposed by B.J. Schroers in which the matter field φ maps R² into S² while the vector gauge potential gives rise to a magnetic field. It is shown that for each natural number N there are solutions to saturate the classical energy lower bound E ≧ 4πN for the field configurations in the topological family deg(φ) = N if and only if N ≠ 1. Furthermore the solutions obtained depend on at least 4N - 3 continuous parameters, the associated magnetic flux can assume its value in an open interval, and the decay rates of the field strengths may be specified in a suitable range. These solutions are multisolitons represented by N prescribed lumps of the magnetic field, simulating N identical particles in equilibrium, and are governed by a nonlinear elliptic equation with both vortex and anti-vortex source terms.