Ordinary differential equations described by their Lie symmetry algebra

Abstract

The theory of Lie remarkable equations, i.e., differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on R k and characterize Lie remarkable equations admitted by the considered Lie algebras.

Consistent approximate Q-conditional symmetries of PDEs: application to a hyperbolic reaction-diffusion-convection equation

“…As a last remark, we observe that approximate Lie symmetries of differential equations can be considered also in the framework of the inverse Lie problem, that is the problem of characterizing the form of differential equations by requiring the invariance with respect to a given Lie algebra of point symmetries. As far as the inverse Lie problem is concerned, in recent years, Lie remarkable equations [65,66,67,68], i.e., differential equations uniquely determined by the algebra of their point symmetries, have been defined and characterized. Therefore, a possible perspective could be that of considering suitable approximate Lie algebras and determine the differential equations containing small terms uniquely characterized by them at a fixed order of approximation.…”

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

“…In [119], within the framework of inverse Lie problem, strongly and weakly Lie remarkable differential equations have been defined; relevant examples of such equations have been studied in [120][121][122]. Their analysis involves the study of the rank of the distribution of prolongations of a Lie algebra of generators.…”

“…By using the program, it is possible to compute almost automatically: Lie point symmetries, conditional symmetries, contact symmetries, variational symmetries (all these symmetries may be either exact or approximate) of differential equations, and equivalence transformations for classes of differential equations containing arbitrary elements. Moreover, the program implements functions for computing Lie brackets, the commutator table of a list of Lie generators, and the distribution of an algebra of Lie symmetries (useful in the context of inverse Lie problem [119][120][121][122]). Remarkably, the program can be used interactively in all the cases where the determining equations are not automatically solved (for instance, when one looks for conditional symmetries or in some group classification problems).…”

“…Here we show with a simple example how ReLie can be used for investigating the Lie remarkability of differential equations [119][120][121][122]. The following code shows how one can easily verify that Monge-Ampère equation describing a surface in R 3 with zero Gaussian curvature,…”

Lie remarkable partial differential equations characterized by Lie algebras of point symmetries

“…Moreover, in [22,23], differential equations uniquely determined by some relevant Lie algebras of vector fields in R 3 have been characterized. Finally, in [24], the case of ordinary differential equations uniquely characterized by their Lie point symmetries has been investigated.…”