Quasi-Lie families, schemes, invariants and their applications to Abel equations

Abstract

This work analyses types of group actions on families of t-dependent vector fields of a particular class, the hereby called quasi-Lie families. We devise methods to obtain the defined here quasi-Lie invariants, namely a kind of functions constant along the orbits of the above-mentioned actions. Our techniques lead to a deep geometrical understanding of quasi-Lie schemes and quasi-Lie systems giving rise to several new results. Our achievements are illustrated by studying Abel and Riccati equations. We retrieve the Liouville invariant and study other new quasi-Lie invariants of Abel equations. Several Abel equations with a superposition rule are described and we characterise Abel equations via quasi-Lie schemes.

Nonstandard Hamiltonian structures of the Liénard equation and contact geometry

“…This shows that a necessary and sufficient condition for the function M defined by (8) to be a JLM is (9). The compatibility condition between functions f and g, the so called Chiellini condition [15], is:…”

“…appear very often modelling phenomena in many different areas of applied science (see e.g. [2]) and have receiving a lot of attention the last years [2,3,4,5,6] These equations are very related with first and second-kind Abel equations [7,8,9]. Some generalisations of the Levinson and Smith type [10], where the function f (x) is replaced in (1) by another one f (x,ẋ), are also relevant and in particular the differential equationẍ + h(x)ẋ 2 + f (x)ẋ + g(x) = 0,…”

Quasi-Lie schemes for PDEs

“…The previous procedure is quite frequently applied in the literature on integrability conditions in a more or less clear way [1,24,37,45]. Obviously, it gives rise to finding an infinite number of integrability conditions for differential equations.…”

“…Quasi-Lie systems admit non-autonomous superposition rules, whose properties were analysed in the theory of Lie families in [13]. Moreover, quasi-Lie schemes have also found applications to the description of integrability conditions for Abel differential equations and other differential equations [19,21,24].…”

“…Finally, the theory of quasi-Lie invariants [24] is extended to systems of PDEs. Roughly speaking, every quasi-Lie scheme for systems of PDEs provides a naturally family of tdependent transformations mapping elements of a family of systems of PDEs into new members of the same family.…”