The Cauchy problem of the Kadomtsev-Petviashvili hierarchy with arbitrary coefficient algebra

Abstract

Mulase solved the Cauchy problem of the Kadomtsev-Petviashvili (KP) hierarchy in an algebraic category in “Solvability of the super KP equation and a generalization of the Birkhoff decomposition” (Inventiones Mathematicae, 1988), making use of a delicate factorization of an infinite-dimensional group of formal pseudodifferential operators of infinite order. We prove Mulase’s factorization theorem in a smooth category in the setting of formal pseudo-differential operators with coefficients in a (non-commutative) algebra equipped with a valuation. As an application, we solve the initial value problem for the KP hierarchy using r-matrix theory.