Lie symmetries for Lie systems: Applications to systems of ODEs and PDEs

Abstract

A Lie system is a nonautonomous system of first-order differential equations admitting a superposition rule, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants. Using that a Lie system can be considered as a curve in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra, we associate every Lie system with a Lie algebra of Lie point symmetries induced by the Vessiot–Guldberg Lie algebra. This enables us to derive Lie symmetries of relevant physical systems described by first- and higher-order systems of differential equations by means of Lie systems in an easier way than by standard methods. A generalization of our results to partial differential equations is introduced. Among other applications, Lie symmetries for several new and known generalizations of the real Riccati equation are studied.