A Simple Construction of Recursion Operators for Multidimensional Dispersionless Integrable Systems

TL;DR

This paper introduces a straightforward method to construct recursion operators for multidimensional dispersionless integrable systems with Lax pairs linear in the spectral parameter, providing new examples including equations relevant to Einstein's vacuum solutions.

Contribution

The paper presents a novel, simple technique for deriving recursion operators for a class of multidimensional dispersionless integrable systems, expanding the toolkit for analyzing such equations.

Findings

Constructed recursion operators for the general heavenly equation.

Derived recursion operators for a six-dimensional equation from Ferapontov and Khusnutdinova.

Provided new examples of recursion operators for integrable systems.

Abstract

We present a simple novel construction of recursion operators for integrable multidimensional dispersionless systems that admit a Lax pair whose operators are linear in the spectral parameter and do not involve the derivatives with respect to the latter. New examples of recursion operators obtained using our technique include {\em inter alia} those for the general heavenly equation, which describes a class of anti-self-dual solutions of the vacuum Einstein equations, and a six-dimensional equation resulting from a system of Ferapontov and Khusnutdinova.