Two dimensional KP systems and their solvability

Alexander Zheglov

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 42

MSC 2000

37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies

35Q53 KdV-like equations

35A27 Microlocal methods; methods of sheaf theory and homological algebra in PDE

Abstract
In this paper we introduce new various generalizations of the classical Kadomtsev-Petviashvili hierarchy in the case of operators in several variables. These generalizations are the candidates for systems that should play the role, analogous to the role of the KP hierarchy in the classical KP theory, in a generalized KP theory. In particular, they should describe flows of some generalized geometric datas, including those described by A.N.Parshin, for certain initial conditions. The unique solvability of the initial value problem for the generalized KP hierarchies is established. The connection of these systems with universal families of isospectral deformations of certain pairs of commuting differential operators is opened. To prove the solvability of the systems we generalize several results from the works of M.Mulase and A.N.Parshin.