Analytic-bilinear approach to integrable hierarchies. I. Generalized KP hierarchy

Abstract

An analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is proposed. It starts with the generalized Hirota identity for the Cauchy–Baker–Akhiezer (CBA) function and leads to a generalized KP hierarchy in the form of compact functional equations containing a special shift operator. A generalized KP hierarchy incorporates the basic KP hierarchy, modified KP hierarchy, KP singularity manifold equation hierarchy, and corresponding hierarchies of linear problems. Different “vertical” levels of the generalized KP hierarchy are connected via invariants of the Combescure symmetry group. The resolution of functional equations also gives rise to the τ function and the addition formulas for it.