Lie-point symmetries of the discrete Liouville equation

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.

References in corpus (5)

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Cited by in corpus (3)

• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
• Invariant compact finite difference schemes
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization