Chern—Simons Systems: Non-Abelian Case

Abstract

In this chapter we present static multisoliton solutions of the non-Abelian Chern—Simons equations. In §6.1 we review some basic facts about a complex semi-simple Lie algebra such as the Cartan—Weyl bases and Cartan matrices to be used in the development to follow. In §6.2 we consider the solution of the non-Abelian gauged Schrödinger equations coupled with a Chern—Simons dynamics via the Toda equations, which is a non-relativistic Chern—Simons theory. In §6.3 we introduce the relativistic Chern—Simons equations and state a general existence theorem. In §6.4 we reduce the governing equations into a nonlinear elliptic system and formulate a variational principle. In §6.5 we present an analysis of the elliptic system and prove all the results stated. In §6.6 we apply our existence theory to some concrete examples.