Opuscula Math. 41, no. 5 (2021), 685-699

Opuscula Mathematica

On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

Ivan Tsyfra

Abstract. We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

Keywords: ordinary differential equation, partial differential equation, integrability, symmetry, quadrature, Lie transformation group.

Mathematics Subject Classification: 34A30, 34C14.

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  1. Yu.V. Brezhnev, What does integrability of finite-gap or soliton potentials mean?, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1867, 923-945.
  2. F. Calogero, A. Degasperis, Spectral Transform and Solitons. Vol. I. Tools to Solve and Investigate Nonlinear Evolution Equations, Lecture Notes in Computer Science, 144. Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1982.
  3. J. Drash, Sur l'integration par quadratures de l' equation differentielle \(\tfrac{d^2y}{dx^2}=[\varphi(x)+h]y\), Compt. Rend. Acad. Sci 168 (1919), 337-340.
  4. B.A. Dubrovin, The inverse scattering problem for periodic short-range potentials, Funkcional. Anal. i Priložen 9 (1975), 65-66 [in Russian].
  5. S.P. Novikov, A periodic Korteweg-de Vries equation, I, Funkcional. Anal. i Priložen. 8 (1974), no. 3, 54–66 [in Russian].
  6. P.J. Olver, Application of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1986.
  7. L.V. Ovsiannikov, Group Analysis of Differential Equations, translated from the Russian by Y. Chapovsky, translation edited by William F. Ames, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.
  8. I.M. Tsyfra, T. Czyżycki, Symmetry and solution of neutron transport equations in nonhomogeneous media, Abstract Appl. Anal. 2014 (2014), Art. ID 724238, 9 pp.
  • Communicated by Piotr Oprocha.
  • Received: 2021-06-07.
  • Revised: 2021-07-31.
  • Accepted: 2021-09-15.
  • Published online: 2021-09-30.
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Cite this article as:
Ivan Tsyfra, On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations, Opuscula Math. 41, no. 5 (2021), 685-699, https://doi.org/10.7494/OpMath.2021.41.5.685

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