Are there Contact Transformations for Discrete Equations?

TL;DR

This paper investigates the existence of contact transformations for discrete equations and proves that such transformations do not exist, extending the understanding of symmetry transformations in difference schemes.

Contribution

The paper introduces the concept of infinitesimal contact transformations for difference schemes and establishes their non-existence.

Findings

Contact transformations depend on multiple lattice points

Such transformations cannot be integrated into Lie groups

The non-existence result applies to all orders of difference schemes

Abstract

We define infinitesimal contact transformations for ordinary difference schemes as transformations that depend on $K+1$ lattice points $(K \geq 1)$ and can be integrated to form a local or global Lie group. We then prove that such contact transformations do not exist.