Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

TL;DR

This paper explores a specific reduction of the lattice potential KdV equation, revealing its connection to the discrete Painlevé equation with E6 symmetry, and finds explicit rational and hypergeometric solutions.

Contribution

It identifies a new reduction linking the lattice potential KdV to a discrete Painlevé equation and analyzes its symmetries and solutions.

Findings

Connection between lattice potential KdV reduction and discrete Painlevé E6 symmetry

Description of symmetry relations of the reduced equations

Explicit rational and hypergeometric solutions derived

Abstract

We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlev\'e equation with $E^{(1)}_6$ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlev\'e equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlev\'e equation.