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 TMF, 2018, Volume 196, Number 3, Pages 419–433 (Mi tmf9523)

Conformally invariant elliptic Liouville equation and its symmetry-preserving discretization

D. Leviab, L. Martinacd, P. Winternitzef

a Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Roma, Italy
b Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Roma, Italy
c Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, Lecce, Italy
d Dipartimento di Matematica e Fisica, Università del Salento, Lecce, Italy
e Département de Mathématiques et de Statistique, Université de Montréal, Montréal (QC), Canada
f Centre de Recherches Mathématiques, Université de Montréal, Montréal (QC), Canada

Abstract: 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 previously developed discretization procedure, 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.

Keywords: Lie group, partial differential equation, discretization procedure.

 Funding Agency Grant Number Italian Ministry of Education, University and Research 2010 PRIN Instituto Nazionale di Fisica Nucleare IS-CSN4 Natural Sciences and Engineering Research Council of Canada (NSERC) The research of D. Levi and L. Martina was supported in part by the Italian Ministry of Education and Research, 2010 PRIN “Continuous and discrete nonlinear integrable evolutions: From water waves to symplectic maps" and by INFN IS-CSN4 "Mathematical Methods of Nonlinear Physics.” The research of P. Winternitz is supported in part by an NSERC discovery grant.

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DOI: https://doi.org/10.4213/tmf9523

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English version:
Theoretical and Mathematical Physics, 2018, 196:3, 1307–1319

Bibliographic databases:

MSC: 22E60, 35J15, 39A20

Citation: D. Levi, L. Martina, P. Winternitz, “Conformally invariant elliptic Liouville equation and its symmetry-preserving discretization”, TMF, 196:3 (2018), 419–433; Theoret. and Math. Phys., 196:3 (2018), 1307–1319

Citation in format AMSBIB
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