Partial integrability of the anharmonic oscillator

Details Partial integrability of the anharmonic oscillator Partial integrability of the anharmonic oscillator

2008-12-08

arxiv.org/abs/0812.1451v1

J. Nonlinear Mathematical Physics 2007 (14) 454-465

Nonlinear Sciences

Exactly Solvable and Integrable Systems

12 pages, no figure

Scientific paper

10.2991/jnmp.2007.14.3.11

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.

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