1211.4374 (Raphael Boll)

On Bianchi permutability of Bäcklund transformations for asymmetric
quad-equations    [PDF]

Raphael Boll

Multidimensional Consistency becomes more and more important in the theory of discrete integrable systems. Recently, we gave a classification of all 3D consistent systems of quad-equations with the tetrahedron property, where several novel asymmetric systems have been found. This classification allows for derivation of B\"acklund transformations and zero-curvature representations for all participating 2D systems of quad-equations which is a justification for using 3D consistency as a synonym for integrability. In the present paper, we will consider the proof of the Bianchi permutability (existence of superposition principle) of these B\"acklund transformations. 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.

View original: http://arxiv.org/abs/1211.4374