Explicit Flow Equations and Recursion Operator of the ncKP hierarchy

TL;DR

This paper derives explicit flow equations and a recursion operator for the noncommutative KP hierarchy, revealing new commutator terms and solving an open problem related to nonlocal recursion operators.

Contribution

It provides explicit expressions for flow equations and a recursion operator for the ncKP hierarchy, including the nonlocal recursion operator for the ncKdV hierarchy, addressing an open problem.

Findings

Derived explicit flow equations for ncKP hierarchy.

Identified commutator terms as additional components.

Constructed a nonlocal recursion operator for ncKdV.

Abstract

The explicit expression of the flow equations of the noncommutative Kadomtsev-Petviashvili(ncKP) hierarchy is derived. Compared with the flow equations of the KP hierarchy, our result shows that the additional terms in the flow equations of the ncKP hierarchy indeed consist of commutators of dynamical coordinates \{$u_i$\}. The recursion operator for the flow equations under $n$-reduction is presented. Further, under 2-reduction, we calculate a nonlocal recursion operator $\Phi(2)$ of the noncommutative Korteweg-de Vries(ncKdV) hierarchy, which generates a hierarchy of local, higher-order flows. Thus we solve the open problem proposed by P.J. Olver and V.V. Sokolov(Commun.Math.Phys. 193 (1998), 245-268).