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 Nelin. Dinam., 2018, Volume 14, Number 1, Pages 53–61 (Mi nd596)

Original papers

Application of the hydrodynamic substitution for systems of equations with the same principal part

N. N. Fimina, V. M. Chechetkinab

a Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, pl. Miusskaya 4, Moscow, 125047, Russia
b Institute for Design Automation, Russian Academy of Sciences, ul. 2nd Brestskaya 19/18, Moscow, 123056, Russia

Abstract: The properties of quasi-linear differential equations with the same the principal part are considered. Their connection with the reduced system of Euler equations is established, which results from the hydrodynamic substitution in the kinetic Liouville and Vlasov equations. When considering the momentum equation of the Euler system, it turns out that it reduces to a special form such as Liouville – Jacobi equation. This equation can also be investigated using a hydrodynamic substitution, but of conjugate type. The application of this substitution (of the second order) makes it possible to symmetrize the technique of applying hydrodynamic substitution and to extend the class of equations of hydrodynamic type to which systems of (in the general case non-Hamiltonian) first-order autonomous differential equations. Examples are given of the use of the developed formalism for systems of gravitating particles in post-Newtonian approximation and the hydrodynamic systems described by Monge potentials, with the aim of constructing the Liouville – Jacobi equations and applying to them a modified hydrodynamic substitution.

Keywords: Liouville equation, quasi-linear equations, hydrodynamic substitution, Monge potentials, equations with the same principal part

 Funding Agency Grant Number Russian Science Foundation 16-11-10339

DOI: https://doi.org/10.20537/nd1801005

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UDC: 517.9
MSC: 34A25
Accepted:18.10.2017

Citation: N. N. Fimin, V. M. Chechetkin, “Application of the hydrodynamic substitution for systems of equations with the same principal part”, Nelin. Dinam., 14:1 (2018), 53–61

Citation in format AMSBIB
\Bibitem{FimChe18} \by N.~N.~Fimin, V.~M.~Chechetkin \paper Application of the hydrodynamic substitution for systems of equations with the same principal part \jour Nelin. Dinam. \yr 2018 \vol 14 \issue 1 \pages 53--61 \mathnet{http://mi.mathnet.ru/nd596} \crossref{https://doi.org/10.20537/nd1801005} \elib{https://elibrary.ru/item.asp?id=32773038}