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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 4, Pages 486–496 (Mi rcd28)

On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions

Nikolay A. Kudryashov, Dmitry I. Sinelshchikov

National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, Moscow, 115409, Russia

Abstract: The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear second-order differential equation has been discussed. Here we show that the whole family of quadratic Lienard equations can be transformed into an equation for the elliptic functions. We demonstrate that this connection can be useful for finding explicit forms of general solutions of the quadratic Lienard equation. We provide several examples of application of our approach.

Keywords: quadratic lienard equation, elliptic functions, nonlocal transformations, general solution

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 2296.2014.13694.2014.1 Russian Foundation for Basic Research 14-01-0049814-01-31078 This research was partially supported by the grant for Scientific Schools 2296.2014.1, by the grant for the state support of young Russian scientists 3694.2014.1 and by RFBR grants 14–01–00498 and 14–01–31078.

DOI: https://doi.org/10.1134/S1560354715040073

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Document Type: Article
MSC: 34A34, 34A05, 33E05
Language: English

Citation: Nikolay A. Kudryashov, Dmitry I. Sinelshchikov, “On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions”, Regul. Chaotic Dyn., 20:4 (2015), 486–496

Citation in format AMSBIB
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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. N. A. Kudryashov, D. I. Sinelshchikov, “Analytical solutions of a nonlinear convection-diffusion equation with polynomial sources”, Automatic Control and Computer Sciences, 51:7 (2017), 621–626
2. 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
3. M. B. Gavrikov, N. A. Kudryashov, B. A. Petrov, V. V. Savelyev, D. I. Sinelshchikov, “Solitary and periodic waves in two-fluid magnetohydrodynamics”, Commun. Nonlinear Sci. Numer. Simul., 38 (2016), 1–7
4. N. A. Kudryashov, D. I. Sinelshchikov, “On the criteria for integrability of the Lienard equation”, Appl. Math. Lett., 57 (2016), 114–120
5. R. Campoamor-Stursberg, “A functional realization of $\mathfrak{sl}(3,\Bbb{R})$ providing minimal Vessiot-Guldberg-Lie algebras of nonlinear second-order ordinary differential equations as proper subalgebras”, J. Math. Phys., 57:6 (2016), 063508
6. N. A. Kudryashov, D. I. Sinelshchikov, “Analytical solutions of the Rayleigh equation for arbitrary polytropic exponent”, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2015 (ICNAAM-2015), AIP Conf. Proc., 1738, eds. T. Simos, C. Tsitouras, Amer. Inst. Phys., 2016, 230010
7. D. I. Sinelshchikov, N. A. Kudryashov, “On the Jacobi last multipliers and Lagrangians for a family of Liénard-type equations”, Appl. Math. Comput., 307 (2017), 257–264
8. N. A. Kudryashov, D. I. Sinelshchikov, “On connections of the Lienard equation with some equations of Painlevé-Gambier type”, J. Math. Anal. Appl., 449:2 (2017), 1570–1580
9. C. Ozemir, “On some canonical classes of cubic-quintic nonlinear Schrodinger equations”, J. Math. Anal. Appl., 446:2 (2017), 1814–1832
10. D. I. Sinelshchikov, “On connections of the Lienard-type equations with type II Painlevé-Gambier equations”, Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2016 (ICNAAM-2016), AIP Conf. Proc., 1863, eds. T. Simos, C. Tsitouras, Amer. Inst. Phys., 2017, UNSP 380008-1
11. 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
12. N. A. Kudryashov, D. I. Sinelshchikov, “New non-standard Lagrangians for the Lienard-type equations”, Appl. Math. Lett., 63 (2017), 124–129
13. Ivan R. Garashchuk, Dmitry I. Sinelshchikov, Nikolay A. Kudryashov, “Nonlinear Dynamics of a Bubble Contrast Agent Oscillating near an Elastic Wall”, Regul. Chaotic Dyn., 23:3 (2018), 257–272
14. I. Garashchuk, D. Sinelshchikov, N. Kudryashov, “General solution of the Rayleigh equation for the description of bubble oscillations near a wall”, Mathematical Modeling and Computational Physics 2017 (MMCP 2017), EPJ Web Conf., 173, eds. G. Adam, J. Busa, M. Hnatic, D. Podgainy, EDP Sciences, 2018, UNSP 03008
15. A. A. Kosov, E. I. Semenov, “O tochnykh resheniyakh uravneniya nelineinoi diffuzii”, Sib. matem. zhurn., 60:1 (2019), 123–140