On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators

TL;DR

This paper extends the study of the KP hierarchy to a broader algebra of formal pseudo-differential operators, establishing existence, uniqueness, and Hamiltonian structures in this extended setting.

Contribution

It introduces an extended algebra of formal pseudo-differential operators and proves the existence and uniqueness of KP hierarchy solutions within this new framework.

Findings

Established algebraic properties of the extended operator class.

Proved existence and uniqueness of KP solutions in the extended algebra.

Extended KP hierarchy to complex order operators with Hamiltonian structures.

Abstract

We study the existence and uniqueness of the Kadomtsev-Petviashvili (KP) hierarchy solutions in the algebra of $\F Cl(S^1,\K^n)$ of formal classical pseudo-differential operators. The classical algebra $\Psi DO(S^1,\K^n)$ where the KP hierarchy is well-known appears as a subalgebra of $\F Cl(S^1,\K^n).$ We investigate algebraic properties of $\F Cl(S^1,\K^n)$ such as splittings, r-matrices, extension of the Gelfand-Dickii bracket, almost complex structures. Then, we prove the existence and uniqueness of the KP hierarchy solutions in $\F Cl(S^1,\K^n)$ with respect to extended classes of initial values. Finally, we extend this KP hierarchy to complex order formal pseudo-differential operators and we describe their Hamiltonian structures similarly to previously known formal case.