On the criteria for integrability of the Liénard equation

Abstract

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of previously known criteria for integrability of the Liénard equation as a straightforward consequences of our approach's application. We illustrate our results by several new examples of integrable Liénard equations.

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• Cheillini integrability and quadratically damped oscillators
• On connections of the Liénard equation with some equations of Painlevé--Gambier type
• Sundman transformation and alternative tangent structuresSundman transformation and alternative tangent structures
• On the integrability of the Abel and of the extended Liénard equations
• Isochronous waveforms of Liénard equations via commutative factorization