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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 5, Pages 548–555 (Mi rcd204)

On the Integrability Conditions for a Family of Liénard-type Equations

N. A. Kudryashov, D. I. Sinelshchikov

Department of Applied Mathematics, National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia

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 travelingwave reductions of some nonlinear partial differential equations. In this work we find new conditions for the integrability of this family of equations. To this end we use an approach which is based on the 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.

Keywords: Liénard-type equation, nonlocal transformations, closed-form solution, general solution, Painlevé–Gambier equations

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation MK-6624.2016.1 Russian Foundation for Basic Research 14-01-00493-a This research was partially supported by the grant for the state support of young Russian scientists MK-6624.2016.1, by RFBR grant 14–01–00493-a and by the Competitiveness Program of NRNU “MEPhI”.

DOI: https://doi.org/10.1134/S1560354716050063

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MSC: 34A05, 34A34, 34A25
Accepted:15.08.2016
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Citation: N. A. Kudryashov, D. I. Sinelshchikov, “On the Integrability Conditions for a Family of Liénard-type Equations”, Regul. Chaotic Dyn., 21:5 (2016), 548–555

Citation in format AMSBIB
\Bibitem{KudSin16} \by N. A. Kudryashov, D. I. Sinelshchikov \paper On the Integrability Conditions for a Family of Liénard-type Equations \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 5 \pages 548--555 \mathnet{http://mi.mathnet.ru/rcd204} \crossref{https://doi.org/10.1134/S1560354716050063} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000385167300006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84990932249} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. I. Sinelshchikov, N. A. Kudryashov, “On the Jacobi last multipliers and Lagrangians for a family of Lienard-type equations”, Appl. Math. Comput., 307 (2017), 257–264
2. D. I. Sinelshchikov, N. A. Kudryashov, “On the general traveling wave solutions of some nonlinear diffusion equations”, V International Conference on Problems of Mathematical and Theoretical Physics and Mathematical Modelling, Journal of Physics Conference Series, 788, IOP Publishing Ltd, 2017, UNSP 012033
3. A. D. Polyanin, I. K. Shingareva, “Nonlinear problems with blow-up solutions: numerical integration based on differential and nonlocal transformations, and differential constraints”, Appl. Math. Comput., 336 (2018), 107–137
4. A. D. Polyanin, I. K. Shingareva, “Non-linear problems with non-monotonic blow-up solutions: non-local transformations, test problems, exact solutions, and numerical integration”, Int. J. Non-Linear Mech., 99 (2018), 258–272
5. A. L. Kazakov, Sv. S. Orlov, S. S. Orlov, “Construction and study of exact solutions to a nonlinear heat equation”, Siberian Math. J., 59:3 (2018), 427–441
6. D. I. Sinelshchikov, N. A. Kudryashov, “On integrable non–autonomous Liénard–type equations”, Theoret. and Math. Phys., 196:2 (2018), 1230–1240
7. A. Ruiz, C. Muriel, “On the integrability of Liénard I-type equations via $\lambda$-symmetries and solvable structures”, Appl. Math. Comput., 339 (2018), 888–898
8. A. D. Polyanin, I. K. Shingareva, “Application of non-local transformations for numerical integration of singularly perturbed boundary-value problems with a small parameter”, Int. J. Non-Linear Mech., 103 (2018), 37–54