<正>Bogomol'nyi-type vortices and self-dual solutions appear in a variety of gauge theories scenarios in 2 + 1 dimension. Based on the U(1) gauge potential decomposition theory andφ-mapping theory,the topological inner structure of self-duality (Bogomol'nyi-type) equations are studied. We start with a Abelian CSH Lagrangian density.The self-dual vortices that arise