Backlund-Transformation-Related Recursion Operators: Application to the Self-Dual Yang-Mills Equation

TL;DR

This paper explores the relationship between symmetries and recursion operators of the self-dual Yang-Mills equation and its potential form, establishing a Lie-algebra isomorphism and constructing recursion operators.

Contribution

It demonstrates a method to relate symmetries and recursion operators of connected PDEs via non-auto-Backlund transformations, specifically for the SDYM and PSDYM equations.

Findings

Proves Lie-algebra isomorphism between SDYM and PSDYM symmetries

Constructs recursion operators for both SDYM and PSDYM

Provides insights into the algebraic structure of potential symmetries

Abstract

By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relating symmetries and recursion operators of two partial differential equations connected to each other by a non-auto-Backlund transformation. We prove the Lie-algebra isomorphism between the symmetries of the SDYM equation and those of the potential SDYM (PSDYM) equation, and we describe the construction of the recursion operators for these two systems. Using certain known aspects of the PSDYM symmetry algebra, we draw conclusions regarding the Lie algebraic structure of the "potential symmetries" of the SDYM equation.