Isospectral deformation of the reduced quasi-classical self-dual Yang--Mills equation

TL;DR

This paper introduces a new four-dimensional PDE derived from the reduced quasi-classical self-dual Yang--Mills equation, featuring an isospectral Lax representation, a recursion operator, and Bäcklund transformations linking related equations.

Contribution

It presents a novel four-dimensional PDE with an isospectral Lax pair, recursion operator, and Bäcklund transformations, expanding the understanding of integrable structures in gauge theories.

Findings

Derived a new four-dimensional PDE with isospectral Lax representation

Constructed a recursion operator for the new PDE

Established Bäcklund transformations connecting related equations

Abstract

We derive new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang--Mills equation. Then we find a recursion operator for the obtained equation and construct B{\"a}cklund transformations between this equation and the reduced quasi-classical self-dual Yang--Mills equation as well as the four-dimensional Mart{\'{\i}}nez Alonso--Shabat equation