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

• Published in: Applied mathematics and computation 2016-01, Vol.273, p.435-452
• Main Authors: Estévez, P.G., Herranz, F.J., de Lucas, J., Sardón, C.
• Format: Article
• Language: English
• Subjects: Lie algebra; Lie symmetry; Lie system; Ordinary differential equation; Partial differential equation; Riccati equation; Superposition rule
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2015.09.078

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.