Resumo

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non linear evolution equations (flows) which shares a common affine Lie algebraic structure. A generalized framework to accommodate higher graded (generalized) integrable hierarchies is proposed, primarily extending the conventional algebraic formalism. This approach utilizes a Generalized Riemann-Hilbert-Birkhoff (g-RHB) decomposition to systematically generate and classify multi-component nonlinear integrable models. Explicit examples of the positive and negative flows for the mKdV, Chen-Lee-Liu (CLL) hierarchies and its various reductions, including Burgers hierarchy are considered. Two classes of vacua, namely zero and non-zero constant vacuum solutions are shown to be admissible. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A particular set of solitons solutions constructed by a judicious choice of vertices are shown to yield in a closed form, the multi soliton solutions for the Burgers hierarchy.