Alternative integrable discretisation of Korteweg de Vries equation

TL;DR

This paper introduces a new integrable discretization of the differential-difference KdV equation using Hirota's bilinear formalism, providing insights into its relation with classical forms and integrability properties.

Contribution

It proposes an alternative discretization method based on two tau functions, expanding the understanding of integrable discretizations of the KdV equation.

Findings

Discretization using two tau functions yields the known discrete KdV equation.

The approach clarifies the relation between different bilinear forms.

Discussion on the integrability of the new discretization.

Abstract

We present an alternative integrable discretization of differential-difference KdV equation based on Hirota bilinear formalism. It is shown that using two tau functions the direct discretisation of the bilinear equations gives immediately the well known discrete KdV equation. We comment also on integrability and relation with the classical bilinear form involving only one tau function.