Geodesic
Conformally invariant elliptic Liouville equation and its symmetry-preserving discretization
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 419-433 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : Lie group
Keywords: partial differential equation, discretization procedure. 
@article{TMF_2018_196_3_a4,
     author = {D. Levi and L. Martina and P. Winternitz},
     title = {Conformally invariant elliptic {Liouville} equation and its symmetry-preserving discretization},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {419--433},
     year = {2018},
     volume = {196},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a4/}
}
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D. Levi; L. Martina; P. Winternitz. Conformally invariant elliptic Liouville equation and its symmetry-preserving discretization. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 419-433. http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a4/