2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/53209Based on the U(1) gauge potential decomposition theory and $ϕ$-mapping theory, the topological inner structure of the self-duality (Bogomol'nyi-type) equations are studied. The special form of the gauge potential decomposition is obtained directly from the first of the self-duality equations. Using this decomposition, the topological inner structure of the Chern-Simons-Higgs (CSH) vortex is discussed. Furthermore, we obtain a rigorous self-dual equation with topological term for the first time, in which the topological term has been ignored by other physicists.LaTex, 8 pages, no figuresHigh Energy Physics - Theorytext