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

This article is cited in 15 scientific papers (total in 15 papers)

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 2296.2014.1
Russian Foundation for Basic Research 14-01-00498
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 140100498 and 140131078.


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Document Type: Article
MSC: 34A34, 34A05, 33E05
Received: 29.05.2015
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
\by Nikolay A. Kudryashov, Dmitry I. Sinelshchikov
\paper On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions
\jour Regul. Chaotic Dyn.
\yr 2015
\vol 20
\issue 4
\pages 486--496

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    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  mathnet  crossref  crossref  mathscinet  elib
    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  mathnet  crossref
    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  crossref  mathscinet  isi  scopus
    4. N. A. Kudryashov, D. I. Sinelshchikov, “On the criteria for integrability of the Lienard equation”, Appl. Math. Lett., 57 (2016), 114–120  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi  scopus
    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  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    9. C. Ozemir, “On some canonical classes of cubic-quintic nonlinear Schrodinger equations”, J. Math. Anal. Appl., 446:2 (2017), 1814–1832  crossref  mathscinet  zmath  isi  scopus
    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  crossref  isi  scopus
    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  crossref  isi  scopus
    12. N. A. Kudryashov, D. I. Sinelshchikov, “New non-standard Lagrangians for the Lienard-type equations”, Appl. Math. Lett., 63 (2017), 124–129  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  mathscinet  adsnasa
    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  crossref  isi
    15. A. A. Kosov, E. I. Semenov, “O tochnykh resheniyakh uravneniya nelineinoi diffuzii”, Sib. matem. zhurn., 60:1 (2019), 123–140  mathnet  crossref
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