Quasi-Lie schemes for PDEs

Abstract

The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining $t$-dependent superposition rules and integrability conditions are analysed. We develop a procedure of constructing quasi-Lie systems through a generalisation to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess-Zumino-Novikov-Witten models, generalised Abel differential equations, Baecklund transformations, as well as other differential equations of physical and mathematical relevance.

32 pages

References in corpus (14)

• Superposition rules, Lie theorem and partial differential equations
• A nonlinear superposition rule for solutions of the Milne--Pinney equation
• Recent Applications of the Theory of Lie Systems in Ermakov Systems
• Applications of Lie systems in dissipative Milne--Pinney equations
• Lie-Hamilton systems on the plane: Applications and superposition rules
• k-symplectic Lie systems: theory and applications
• Solvability of a Lie algebra of vector fields implies their integrability by quadratures
• Quasi-Lie schemes and Emden--Fowler equations
• Integrable Abel equations and Vein's Abel equation
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• A geometric approach to integrability conditions for Riccati equations
• Geometry of Riccati equations over normed division algebras
• On the surfaces associated with $\mathbb{C}{P}^{N-1}$ models
• Quasi-Lie families, schemes, invariants and their applications to Abel equations

Cited by in corpus (2)

• On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications
• Stratified Lie systems: Theory and applications