Non-symmetric discrete Toda systems from quad-graphs

Abstract

For all non-symmetric discrete relativistic Toda-type equations, we establish a relation to 3D consistent systems of quad-equations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of ℤ³ carry different equations. Our construction allows for an algorithmic derivation of the zero curvature representations and yields analogous results also for the continuous time case.

Keywords:

• quad-graph
• quad-equations
• integrable lattice equations
• integrable discretization
• relativistic Toda lattice
• discrete Toda-type equations
• multidimensional consistency
• zero curvature representation
• Bäcklund transformation

AMS Subject Classifications:

• 35Q58
• 37J35
• 37K05
• 37K10
• 39A12