Seminarium Gamma

This talk aims to present the theory of systems of first-order partial differential equations that admit a (nonlinear) superposition rule: the so-called PDE Lie systems. These systems are characterized by the property that the partial derivatives of their solutions depend only on the independent and dependent variables, and their general solution can be expressed as a function of a family of particular solutions together with several constants. In particular, we will discuss the Lie–Scheffers theorem and the Lie group approach to PDE Lie systems. We will present the main features of these systems as well as a range of classical and novel results concerning Bäcklund transformations, Lax pairs, Floquet theory, geometric phases, conditional symmetries, and further potential applications in hydrodynamical systems.