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 subgroup SLx(2, double-struck R) ⊗ SLy(2, double-struck R). The invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard (noninvariant) scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries.
Levi, D., Martina, L., Winternitz, P. (2015). Lie-point symmetries of the discrete Liouville equation. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL, 48(2), 025204 [10.1088/1751-8113/48/2/025204].
Lie-point symmetries of the discrete Liouville equation
LEVI, Decio;
2015-01-01
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 subgroup SLx(2, double-struck R) ⊗ SLy(2, double-struck R). The invariant scheme is an explicit one and provides a much better approximation of exact solutions than a comparable standard (noninvariant) scheme and also than a scheme invariant under an infinite dimensional group of generalized symmetries.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.