Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order

TL;DR

This paper presents a systematic method to find nonlocal symmetries of scalar and coupled nonlinear ODEs of any order by leveraging the symmetries of related linear ODEs, enabling reduction and explicit solutions.

Contribution

It introduces a novel procedure to derive nonlocal symmetries from linear ODE symmetries, applicable to scalar and coupled nonlinear ODEs of arbitrary order.

Findings

Derived nonlocal symmetries for classes of nonlinear ODEs

Obtained reduction transformations and explicit solutions

Extended method to coupled higher order nonlinear ODEs

Abstract

In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use of the Lie point symmetries of the linear ODEs and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear ODEs. Using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples. We find the reduced equations can be explicitly integrated to deduce the general solutions for these cases. We also extend this procedure to coupled higher order nonlinear ODEs with specific reference to second order nonlinear ODEs.