A Quasi-Lie Schemes Approach to Second-Order Gambier Equations

Abstract

Abstract. A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geometric way. This allows us to study the transformation of these equations into simpler canonical forms, which solves a gap in the previous literature, and other relevant differential equations, which leads to derive new constants of motion for families of second-order Gambier equations. Additionally, we describe general solutions of certain second-order Gambier equations in terms of particular solutions of Riccati equations, linear systems, and t-dependent frequency harmonic oscillators.

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].…”

Lie point symmetries classification of the mixed Liénard-type equation

“…We find that the cases (ii), (iii) and (iv) are either subcases of case (6) or they belong to linearizable cases. Hence we need to study only the case (i) separately.…”

“…In this paper, we intend to classify/identify linearizable and integrable nonlinear ODEs of a general mixed quadratic-linear (inẋ) Liénard type equation [6][7][8][9][10][11][12][13] A(ẍ,ẋ, x) ≡ẍ + f (x)ẋ 2 + g(x)ẋ + h(x) = 0, (2) where f (x), g(x) and h(x) are arbitrary functions of x, which is much more challenging than the study of (1). One can observe that (1) is a subcase of (2).…”

Lie–Scheffers Systems

“…One can proceed in a similar way with the Gambier equation [CGL13], which can be described as the coupling of two Riccati equations:…”