Discrete relativistic Toda equation from the perspective of shifted [formula omitted] transformation

Abstract

We investigate the discrete relativistic Toda (drToda) equation, which is a Poincaré-invariant and integrability-preserving generalization of the discrete Toda equation, from the perspective of shifted LR transformation. This enables us to discuss several important properties of the equation in an elementary and unified manner. In particular, we derive a Lax representation of the drToda equation in a purely matrix-theoretic way, without assuming knowledge on the Lax matrices of the continuous relativistic Toda equation. A condition for positivity of the variables is derived on the basis of this representation. We also consider the relationship between the drToda equation and the discrete hungry Lotka–Volterra equation and construct a Bäcklund transformation between them. Finally, we obtain conserved quantities of the drToda equation based on this relationship.