Continuous symmetries of discrete equations

doi:10.1016/0375-9601(91)90733-O

Physics Letters A

Volume 152, Issue 7, 28 January 1991, Pages 335-338

https://doi.org/10.1016/0375-9601(91)90733-O Get rights and content

Abstract

Lie group techniques for solving differential equations are extended to differential-difference equations. As an application, it is shown that the two-dimensional Toda lattice has an infinite dimensional symmetry group with a Kac-Moody-Virasoro Lie algebra.

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Citation Excerpt :
Here, by using the method presented in Refs. [17,18], we construct the optimal system of one-dimensional subalgebras of Eq. (1.1). Following the way to investigate optimal system and reductions of constant astigmatism Eq. (1.1) and some symbolic computation methods [26–55], one can obtain the following results. From the Theorem 4.2, one can obtain the reductions listed in Table 5 by using the general method.
In this paper, the constant astigmatism equation is investigated, which finds numerous applications in geometry and physics describing surfaces of constant astigmatism. Utilizing a set of non-singular local multipliers, we present a set of local conservation laws for the equation, based on which, we further find a tree of nonlocally related PDE systems. Moreover, by considering these nonlocally related PDE systems, we systematically present Lie symmetries, optimal systems and some interesting analytical solutions of the constant astigmatism equation. Here it is the first time to investigate the nonlocally related PDE systems for this model, which can be used to expand the solution space of the given PDE system. Finally, by using its local conservation laws and variational principle, we obtain four kinds of Lagrangians.
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Continuous symmetries of discrete equations

Abstract

Applications of Lie groups of differential equations

Math. Japonica

IMA J. Appl. Math.

Prog. Theor. Phys. Suppl.

Phys. Rev.

Prog. Theor. Phys.

J. Math. Phys.