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arXiv · 1608.06920

On the integrability conditions for a family of the Liénard-type equations

Nikolai A. Kudryashov·Dmitry I. Sinelshchikov
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Abstract

We study a family of Liénard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this equations family. To this end we use an approach, which is bases on application of nonlocal transformations. By studying connections between this family of Liénard--type equations and type III Painlevé--Gambier equations, we obtain four new integrability criteria. We illustrate our results by providing examples of some integrable Liénard--type equations. We also discuss relationships between linearizability via nonlocal transformations of this family of Liénard--type equations and other integrability conditions for this family of equations.

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Nikolai A. Kudryashov, Dmitry I. Sinelshchikov. 2016-08-24. On the integrability conditions for a family of the Liénard-type equations. https://doi.org/10.1134/s1560354716050063

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