Lie point symmetry analysis of a second order differential equation with singularity

TL;DR

This paper applies Lie symmetry analysis to a class of second order nonlinear differential equations with singularities, revealing invariance properties, superposition rules, and first integrals relevant to physical applications like nonlinear Schrödinger equations.

Contribution

It extends Lie symmetry methods to equations with singular terms, rediscovering superposition rules and invariants for equations related to the Ermakov-Pinney and Kummer-Schwarz types.

Findings

Identified symmetry groups for equations with singularities.

Rediscovered nonlinear superposition rules.

Derived first integrals such as Ermakov-Lewis invariants.

Abstract

By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear superposition rule for second order Kummer-Schwarz equation is rediscovered. Invariance under one-dimensional symmetry group is also used to obtain first integrals (Ermakov-Lewis invariants). Our motivation is a type of equations with singular term that arises in many applications, in particular in the study of general NLS (nonlinear Schr\"odinger) equations.