On Bianchi permutability of B\"acklund transformations for asymmetric quad-equations

TL;DR

This paper proves the Bianchi permutability of Bäcklund transformations for asymmetric quad-equations, establishing a superposition principle within 3D consistent systems using 4D systems and geometric insights.

Contribution

It introduces a novel proof of Bianchi permutability for asymmetric quad-equations using 4D consistency and super-consistent cube configurations.

Findings

Bianchi permutability holds for asymmetric quad-equations.

Superposition principle is established via 4D consistent systems.

Structural insights through biquadratics patterns are key to the proof.

Abstract

We prove the Bianchi permutability (existence of superposition principle) of B\"acklund transformations for asymmetric quad-equations. Such equations and there B\"acklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations, the structural insights through biquadratics patterns and the consideration of super-consistent eight-tuples of quad-equations on decorated cubes.