Symmetry and singularity properties of the generalised Kummer–Schwarz and related equations

Abstract

We examine the generalised Kummer-Schwarz equation and some of its generalisations from the viewpoints of symmetry and singularity analyses. We determine the Complete Symmetry Group of the general equation and show that different forms of the fourthorder representative illustrate the three possible classes of Laurent series to be expected in the course of the singularity analysis.

Global Dynamics of the Kummer–Schwarz Differential Equation

“…Here the dot denotes derivative with respect to the independent variable t. When the right hand in equation (1) is taken at zero, the resulting equation is the KummerSchwarz equation which is given by (2) 2ẋ ... x − 3ẍ 2 = 0, and is of special interest due to its relationship to the Schwarzian derivative and its exceptional algebraic properties. This equation is also encountered in the study of geodesic curves in spaces of constant curvature, Lie lists the characteristic functions for its contact symmetries, see more results on this differential equation in [1], [4], [5] and [6]. But up to now nobody has described its global dynamics.…”

Ordinary differential equations described by their Lie symmetry algebra

“…Let us observe that in the case m = 1 the second set of equations of (18) is always satisfied, so that (19) reduces to (14) with α = 0, which turns out the most general scalar ODE (euclidean curvature equal to a constant) admitting the euclidean algebra of R 2 as a Lie symmetry subalgebra.…”

“…This is the situation discussed in [17], where the idea of complete symmetry group was proposed and exploited in order to characterize uniquely Kepler's equation. This idea was subsequently exploited by several authors in different ways for characterizing many differential equations [3,4,5,6,19,25,26]. Following the terminology introduced in [21,22,23,28,29], we call Lie remarkable a DE which is completely characterized by its Lie algebra of point symmetries.…”

“…Below we give the main theorems. Equation of item 3 is also known as generalized Kummer-Schwartz equation (see [19] for a discussion of this topic). Equation of item 4 can be realized as the vanishing of total derivative of the euclidean curvature u xx (1 + u 2 x ) − 3 2 of the curve u = u(x).…”

Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

“…The Kummer-Schwarz equation appears in various mathematical contexts like theory of functions, differential geometry, complex analysis, differential equations, integrable systems, mathematical physics, Sturm -Liouville equation, study of curves in a Lorentz space, the charge of density of dark energy [12,32,34]. In [26], Leach suggested several generalizations of the Kummer-Schwarz Equation (KS − 3):…”