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.
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”.
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
\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
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\crossref{https://doi.org/10.1134/S1560354716050063}
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https://www.mathnet.ru/eng/rcd204
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A. D. Polyanin, I. K. Shingareva, “Non-linear blow-up problems for systems of ODEs and PDEs: non-local transformations, numerical and exact solutions”, Int. J. Non-Linear Mech., 111 (2019), 28–41
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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
D. I. Sinelshchikov, N. A. Kudryashov, “On integrable non–autonomous Liénard–type equations”, Theoret. and Math. Phys., 196:2 (2018), 1230–1240
A. Ruiz, C. Muriel, “On the integrability of Liénard I-type equations via λ-symmetries and solvable structures”, Appl. Math. Comput., 339 (2018), 888–898
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
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
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