Mixed superposition rules and the Riccati hierarchy

Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, characterizing systems admitting a mixed superposition rule. This somehow unexpected result says that such systems are exactly Lie systems, i.e., they admit a standard superposition rule. This provides a new and powerful tool for finding Lie systems, which is applied here to studying the Riccati hierarchy and to retrieving some known results in a more efficient and simpler way.Comment: 20 page

Lie systems: theory, generalisations, and applications

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure

Sistemas de Lie, simetrías de Lie y transformaciones recíprocas

[ES]En esta tesis, estamos interesados en sistemas de interés físico y matemático, descritos por medio de ecuaciones diferenciales ordinarias y en derivadas parciales. Como es bien sabido, gran parte de los fenómenos naturales pueden modelizarse a través de estas ecuaciones. Por ejemplo, las cuatro ecuaciones de la Electrodinámica de Maxwell, o las ecuaciones de Einstein son ecuaciones diferenciales. Vamos a centrar nuestra investigación en dos tipos de sistemas: los llamados sistemas de Lie, muy recurrentes en la literatura, dadas sus múltiples propiedades geométricas y las ecuaciones diferenciales en derivadas parciales que aparecen en modelos físicos como los pertenecientes a la Mecánica de Fluidos, Física del Plasma o la Neurociencia, entre otros. Dada la importancia de los métodos geométricos en el tratamiento de ecuaciones diferenciales, vamos a formular nuestra investigación desde el punto de vista de la geometría diferencial

Poisson-Hopf algebra deformations of Lie-Hamilton systems

Hopf algebra deformations are merged with a class of Lie systems of Hamiltonian type, the so-called Lie-Hamilton systems, to devise a novel formalism: the Poisson-Hopf algebra deformations of Lie-Hamilton systems. This approach applies to any Hopf algebra deformation of any Lie-Hamilton system. Remarkably, a Hopf algebra deformation transforms a Lie-Hamilton system, whose dynamic is governed by a finite-dimensional Lie algebra of functions, into a non-Lie-Hamilton system associated with a Poisson-Hopf algebra of functions that allows for the explicit description of its t-independent constants of the motion from deformed Casimir functions. We illustrate our approach by considering the Poisson-Hopf algebra analogue of the non-standard quantum deformation of sl(2) and its applications to deform well-known Lie-Hamilton systems describing oscillator systems, Milne-Pinney equations, and several types of Riccati equations. In particular, we obtain a new position-dependent mass oscillator system with a time-dependent frequency