On the complete integrability of a nonlinear oscillator from group theoretical perspective

TL;DR

This paper demonstrates the complete integrability of a nonlinear oscillator using group theoretical methods, specifically through hidden symmetries and the $$-symmetry approach, despite limited Lie point symmetries.

Contribution

It introduces a novel application of the $$-symmetry method to establish integrability of a nonlinear oscillator lacking sufficient Lie symmetries.

Findings

Constructed a second integral for the oscillator

Derived the general solution using $$-symmetry approach

Established complete integrability from a group theoretical perspective

Abstract

In this paper, we investigate the integrability aspects of a physically important nonlinear oscillator which lacks sufficient number of Lie point symmetries but can be integrated by quadrature. We explore the hidden symmetry, construct a second integral and derive the general solution of this oscillator by employing the recently introduced $\lambda$-symmetry approach and thereby establish the complete integrability of this nonlinear oscillator equation from a group theoretical perspective.