Solvability of the 2 dimensional KP systems

Alexander Zheglov

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 18 pages

MSC 2000

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

37K20 Relations with algebraic geometry, complex analysis, special functions

Abstract
In this paper we define a series of modified Parshin's KP-hierarchies for the case of the ring of pseudodifferential operators in 2 variables, which are parametrized by a real positive parameter. All these systems satisfy the same properties as the original system (the conversations laws, Zhaharov-Shabat equations and so on). We prove their unique solvability for some initial values. Also we prove that for all values except one the systems have nontrivial solutions.