Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study

TL;DR

This paper reviews the use of Lie symmetry analysis in solving nonlinear ordinary differential equations, illustrating various symmetry methods and their interconnections through the modified Emden equation as a case study.

Contribution

It provides a comprehensive overview of multiple symmetry techniques and their relationships, demonstrated on a nonlinear oscillator example.

Findings

Derivation of various symmetries for a second-order ODE

Connections between symmetries and integrating factors

Order reduction leading to general solutions

Abstract

Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, $\lambda$-symmetries, adjoint symmetries and telescopic vector fields of a second-order ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between (i) symmetries and integrating factors and (ii) symmetries and integrals are also discussed and illustrated through the same example. The interconnections between some of the above symmetries, that is (i) Lie point symmetries and $\lambda$-symmetries and (ii) exponential nonlocal symmetries and $\lambda$-symmetries are also discussed. The order reduction procedure is invoked to derive the general solution of the second-order equation.