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

The non-linear superposition principle and the Wei-Norman method

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei--Norman method is applied for obtaining the associated differential equation in the group SL(2; R ). The superposition principle for first order differential equation systems and Lie-Scheffers theorem are also analysed from this group theoretical perspective. Finally, the theory is applied in the solution of second order differential equations like time independent Schrodinger equation. 1 Introduction Nonlinear phenomena are having everyday more and more importance in almost all branches of science, and in particular in Physics. One of the most important nonlinear differential equation is the Riccati equation (see e.g. [1] and references therein). This differential equation has recently been shown to be related with the factorization method (see e.g.[2, 3, 4, 5]). The recent interest of this equation is stea..