Abstract:
Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.
Keywords:Lax representation, Liénard type equations, Painlevé – Gambier equations, first integrals.
Citation:
P. Guha, S. Garai, A. G. Choudhury, “Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List”, Rus. J. Nonlin. Dyn., 16:4 (2020), 637–650
\Bibitem{GuhGarCho20}
\by P.~Guha, S.~Garai, A. G. Choudhury
\paper Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List
\jour Rus. J. Nonlin. Dyn.
\yr 2020
\vol 16
\issue 4
\pages 637--650
\mathnet{http://mi.mathnet.ru/nd734}
\crossref{https://doi.org/10.20537/nd200408}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4198785}
Linking options:
https://www.mathnet.ru/eng/nd734
https://www.mathnet.ru/eng/nd/v16/i4/p637
This publication is cited in the following 1 articles:
Dmitry I. Sinelshchikov, Partha Guha, A. Ghose Choudhury, “Lax representation and a quadratic rational first integral for second-order differential equations with cubic nonlinearity”, Communications in Nonlinear Science and Numerical Simulation, 112 (2022), 106553