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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.: Year: Volume: Issue: Page: Find

 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2019, Volume 162, Pages 85–92 (Mi into444)

Conservation laws for hyperbolic equations: search algorithm for local preimage with respect to the total derivative

S. Ya. Startsev

Institute of Mathematics with Computing Centre — Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa

Abstract: We propose an algorithm, which allows one to eliminate flows from conservation laws for hyperbolic equations by expressing partial derivatives of these flows in terms of the corresponding densities. In particular, the application of this algorithm allows one to prove that the decreasing of order of at least one of Laplace $y$-invariants of the equation $u_{xy}=F(x,y,u,u_x,u_y)$ is a necessary condition for the function $F_{u_y}$ belonged to the image of the total derivative $D_x$ by virtue of this equation. Thus, we obtain constructive necessary conditions for the existence of differential substitutions that transform a hyperbolic equation into a linear equation or into the Klein–Gordon equation.

Keywords: nonlinear hyperbolic equation, integrability, higher symmetry, conservation law, Laplace invariant, differential substitution

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Bibliographic databases:
UDC: 517.957, 517.956.3, 514.763.8
MSC: 35L70, 37K05, 37K10, 37K35

Citation: S. Ya. Startsev, “Conservation laws for hyperbolic equations: search algorithm for local preimage with respect to the total derivative”, Complex Analysis. Mathematical Physics, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 162, VINITI, Moscow, 2019, 85–92

Citation in format AMSBIB
\Bibitem{Sta19} \by S.~Ya.~Startsev \paper Conservation laws for hyperbolic equations: search algorithm for local preimage with respect to the total derivative \inbook Complex Analysis. Mathematical Physics \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2019 \vol 162 \pages 85--92 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into444} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3981820}