Recursion operators for KP, mKP and Harry-Dym Hierarchies

TL;DR

This paper presents a unified method to construct recursion operators for KP, mKP, and Harry-Dym hierarchies from their Lax representations, revealing new relationships among these integrable systems.

Contribution

It introduces a unified construction approach for recursion operators across multiple hierarchies, demonstrating explicit calculations for specific reductions.

Findings

Unified recursion operator construction for KP, mKP, and Harry-Dym.

Explicit recursion operators for 2- and 3-reduction cases.

Revealed inherent relationships among the hierarchies.

Abstract

In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and Harry-Dym under $n$-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for $2$-reduction and $3$-reduction.