Recursion operators for dispersionless integrable systems in any dimension

TL;DR

This paper introduces a novel method for constructing recursion operators for multidimensional integrable systems with Lax representations, demonstrated on several complex systems including the Manakov--Santini system and higher-dimensional heavenly equations.

Contribution

A new approach to derive recursion operators for multidimensional integrable systems with Lax pairs, applicable to various complex equations in multiple dimensions.

Findings

Successfully constructed recursion operators for several multidimensional integrable systems.

Extended the method to systems with arbitrary dimensions and complex structures.

Provided explicit examples demonstrating the effectiveness of the approach.

Abstract

We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the Manakov--Santini system which is a hyperbolic system in N dependent and N + 4 independent variables, where N is an arbitrary natural number, the six-dimensional generalization of the first heavenly equation, the modified heavenly equation, and the dispersionless Hirota equation.