Partial integrability of the anharmonic oscillator

TL;DR

This paper provides a comprehensive overview of partial integrability conditions for a generalized anharmonic oscillator with time-dependent coefficients, unifying previous results and comparing them with Painleve' analysis.

Contribution

It introduces the most general two conditions on coefficients for the existence of a specific first integral, extending and generalizing prior findings.

Findings

Two general conditions for partial integrability are identified.

A natural interpretation of these conditions is provided.

Comparison with Painleve' analysis highlights differences and similarities.

Abstract

We consider the anharmonic oscillator with an arbitrary-degree anharmonicity, a damping term and a forcing term, all coefficients being time-dependent: u" + g_1(x) u' + g_2(x) u + g_3(x) u^n + g_4(x) = 0, n real. Its physical applications range from the atomic Thomas-Fermi model to Emden gas dynamics equilibria, the Duffing oscillator and numerous dynamical systems. The present work is an overview which includes and generalizes all previously known results of partial integrability of this oscillator. We give the most general two conditions on the coefficients under which a first integral of a particular type exists. A natural interpretation is given for the two conditions. We compare these two conditions with those provided by the Painleve' analysis.