The Schwarzian variable associated with discrete KdV-type equations

TL;DR

This paper introduces a new Schwarzian variable linked to discrete KdV-type equations, revealing connections to the lattice Schwarzian KP and KdV equations, thereby advancing understanding of their integrability properties.

Contribution

It constructs a Schwarzian variable for systems of consistent polynomials related to discrete KdV-type equations, unifying their description with known lattice Schwarzian equations.

Findings

The Schwarzian variable satisfies the lattice Schwarzian KP equation in the generic case.

In degenerate cases, it transforms to the lattice Schwarzian KdV equation.

The construction applies to the primary model Q4 and its sub-cases.

Abstract

We present a new construction related to systems of polynomials which are consistent on a cube. The consistent polynomials underlie the integrability of discrete counterparts of integrable partial differential equations of Korteweg- de Vries-type (KdV-type). The construction reported here associates a Schwarzian variable to such systems. In the generic case, including the primary model Q4, the new variable satisfies the lattice Schwarzian Kadomtsev-Petviashvili (KP) equat ion in three dimensions. For the degenerate sub-cases of Q4 the same construction reveals an invertible transformation to the lattice Schwarzian KdV equation.