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 E2. 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.
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.
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
This publication is cited in the following 3 articles:
L. Xia, M. Wu, X. Ge, “Symmetry preserving discretization of the Hamiltonian systems with holonomic constraints”, Mathematics, 9:22 (2021), 2959
D. Levi, M. A. Rodriguez, Z. Thomova, “The discretized Boussinesq equation and its conditional symmetry reduction”, J. Phys. A-Math. Theor., 53:4 (2020), 045201
D. Levi, M. A. Rodriguez, Z. Thomova, “Differential equations invariant under conditional symmetries”, J. Nonlinear Math. Phys., 26:2 (2019), 281–293