Chern—Simons Systems: Non-Abelian Case
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.
KeywordsCartan Subalgebra Vortex Point Cartan Matrix Root Vector Toda Equation
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