Lie-point symmetries of the discrete Liouville equation
For researchers in numerical analysis and integrable systems, this work provides a method to preserve partial symmetry in discretizations, though the result is incremental as it only retains a finite subalgebra.
The authors show that the infinite-dimensional Lie point symmetry algebra of the Liouville equation cannot be preserved under discretization, but construct a difference scheme invariant under a maximal finite subalgebra, which yields better approximations of exact solutions than standard non-invariant schemes.
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.