Analytic Aspects of the Toda System: Ii. Bubbling Behavior and Existence of Solutions

system (1.1) is completely integrable, which is well-known in integrable systems theory. The Liouville equation and the Toda system arise in many physical models. In Chern-Simons theories, the Liouville equation is closely related to Abelian models, while the Toda system is related to non-Abelian models. See for instance the books [10] and [24] and the references therein. Though the Liouville equation has been extensively studied after Liouville [16], the precise behavior of convergence of its solutions has been rather well understood only in the last two decades, see for example [3, 4, 5, 6, 7, 8, 14, 15, 17]. Such a delicate study leads to many applications in the Abelian Chern-Simons theories and mean field equations. To well understand the non-Abelian Chern-Simons model, we have to study the analytic aspects of the Toda system. It is natural to ask if we