Contact transformations for difference schemes

TL;DR

This paper introduces a class of contact transformations for difference schemes that preserve solutions and connect to continuous contact transformations, demonstrating their existence through a simple example.

Contribution

It defines and proves the existence of contact transformations for discrete difference schemes, extending continuous symmetry concepts to discrete systems.

Findings

Contact transformations for difference schemes are formally defined.

Such transformations can be non-trivial and genuinely exist in discrete systems.

The class reduces to classical contact transformations in the continuous limit.

Abstract

We define a class of transformations of the dependent and independent variables in an ordinary difference scheme. The transformations leave the solution set of the system invariant and reduces to a group of contact transformations in the continuous limit. We use a simple example to show that the class is not empty and that such "contact transformations for discrete systems" genuinely exist.