Superposition rules and second-order Riccati equations

Abstract

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the socalled Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose several generalisations of this notion to secondorder differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other secondorder differential equations of mathematical and physical interest.

Lie systems: theory, generalisations, and applications

“…For example, in the research on systems of second-order differential equations, which very frequently appear in Classical Mechanics, various relevant differential equations can be studied by means of Lie systems. Dissipative Milne-Pinney equations [45], Milne-Pinney equations [52], Caldirola-Kanai oscillators [54], t-dependent frequency harmonic oscillators [55], or second-order Riccati equations [48,225], are just some examples of such systems of second-order differential equations that have already been analysed successfully through Lie systems.…”

“…While studying second-order differential equations by means of Lie systems [52,53,202], a new type of 'superposition-like' expression describing the general solution of certain systems of second-order differential equations appeared. These essays led to the definition of a possible superposition rule notion for such systems whose main properties are still under analysis [48]. In addition, these works carried out different approaches to analyse second-order differential equations: by means of the SODE Lie system notion [52] and through regular Lagrangians [54].…”

“…In addition, these works carried out different approaches to analyse second-order differential equations: by means of the SODE Lie system notion [52] and through regular Lagrangians [54]. The relations between these approaches or even the existence of new approaches is still an open question that must be investigated in detail [48].…”

“…Finally, this work led, more or less indirectly, to the characterisation of families of systems of first-order differential equations admitting a t-dependent superposition rule [35] and the definition of the mixed and partial superposition rule notions [38,52]. Finally, we can mention the usefulness of the Lie scheme concept provided in [34], which enables us to generalise the Lie system notion and leads to the discovery of new properties for multiple systems of differential equations, including non-Lie systems, appearing in Physics and Mathematics [34,42,45,48,56].…”

“…Let us now turn to discuss some of the authors' contributions that gave rise to the redaction of this work. On one hand, Cariñena and his collaborators investigated the integrability of Lie systems [40,43,47,50,54,63], a generalisation of the Wei-Norman method devoted to the study of Lie systems [57], the application of Lie systems techniques to analyse systems of second-order differential equations [48,49,52,53], and other topics like the analysis of certain Schrödinger equations [46,51,59]. In this way, they provided a continuation of diverse previous articles dedicated to some of these themes [77,172,202,225] and they opened several research lines [59].…”