Algebro-geometric solutions of the Dirac hierarchy

Abstract

We introduce a Lenard equation and present two special solutions of it. We use one solution to derive an extended Dirac hierarchy and the other to construct the generating function. The generating function yields conserved integrals of the Dirac Hamiltonian system and defines an algebraic curve. Based on the theory of algebraic curves, we prove that the Dirac Hamiltonian system is integrable and obtain algebro-geometric solutions of the Dirac hierarchy.