Lie-point symmetries of the discrete Liouville equation

TL;DR

This paper investigates the symmetry properties of the discrete Liouville equation, showing full symmetry preservation is impossible, but constructing an invariant scheme that better approximates solutions than standard methods.

Contribution

It demonstrates the impossibility of discretizing the Liouville equation while preserving its entire symmetry algebra and introduces an invariant difference scheme with improved approximation capabilities.

Findings

Full symmetry algebra cannot be preserved in discretization.

Constructed an explicit invariant difference scheme.

Invariant scheme outperforms standard non-invariant schemes in approximating solutions.

Abstract

The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subalgebra $ SL_x \lf 2 , \mathbb{R} \rg \otimes SL_y \lf 2 , \mathbb{R} \rg $. The invariant scheme is an explicit one and provides a much better approximation of exact solutions than comparable standard (non invariant) schemes.