Luigi Martina

Decio Levi, Luigi Martina, Pavel Winternitz

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.

Decio Levi, Luigi Martina, Pavel Winternitz

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra $o(3,1)$ as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane $E_2$. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a discretisation procedure developed earlier, we present a difference scheme that is invariant under the group $O(3,1)$ and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under $O(3,1)$ and is itself invariant under a subgroup of $O(3,1)$, namely the $O(2)$ rotations of the Euclidean plane.