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

Abstract

This paper presents a resolution of the gaugedO(3) sigma model proposed by B.J. Schroers in which the matter field ø mapsR 2 intoS 2 while the vector gauge potential gives rise to a magnetic field. It is shown that for each natural numberN there are solutions to saturate the classical energy lower boundE≧4πN for the field configurations in the topological family deg(ø)=N if and only ifN≠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 byN prescribed lumps of the magnetic field, simulatingN identical particles in equilibrium, and are governed by a nonlinear elliptic equation with both vortex and anti-vortex source terms.