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 TMF, 2001, Volume 127, Number 1, Pages 63–74 (Mi tmf1926)

S. Ya. Startsev

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: We show that the first $n-1$ Laplace invariants of a scalar hyperbolic equation obtained from an equation of the same form under a differential substitution of the $n$th order have a zeroth order with respect to one of the characteristics. It follows that all Laplace invariants of an equation admitting substitutions of an arbitrarily high order must have a zeroth order. Three special cases of such equations are considered: those admitting autosubstitutions, those obtained from a linear equation by a differential substitution, and those with solutions depending simultaneously on both an arbitrary function of $x$ and an arbitrary function of $y$.

DOI: https://doi.org/10.4213/tmf1926

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English version:
Theoretical and Mathematical Physics, 2001, 127:1, 460–470

Bibliographic databases:

Citation: S. Ya. Startsev, “Hyperbolic Equations Admitting Differential Substitutions”, TMF, 127:1 (2001), 63–74; Theoret. and Math. Phys., 127:1 (2001), 460–470

Citation in format AMSBIB
\Bibitem{Sta01} \by S.~Ya.~Startsev \paper Hyperbolic Equations Admitting Differential Substitutions \jour TMF \yr 2001 \vol 127 \issue 1 \pages 63--74 \mathnet{http://mi.mathnet.ru/tmf1926} \crossref{https://doi.org/10.4213/tmf1926} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1863519} \zmath{https://zbmath.org/?q=an:0990.35106} \elib{http://elibrary.ru/item.asp?id=13371184} \transl \jour Theoret. and Math. Phys. \yr 2001 \vol 127 \issue 1 \pages 460--470 \crossref{https://doi.org/10.1023/A:1010359808044} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000170446000005} 

• http://mi.mathnet.ru/eng/tmf1926
• https://doi.org/10.4213/tmf1926
• http://mi.mathnet.ru/eng/tmf/v127/i1/p63

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Zhiber, S. Ya. Startsev, “Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems”, Math. Notes, 74:6 (2003), 803–811
2. A. M. Gurieva, A. V. Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355
3. S. Ya. Startsev, “On the variational integrating matrix for hyperbolic systems”, J. Math. Sci., 151:4 (2008), 3245–3253
4. V. M. Zhuravlev, “The method of generalized Cole–Hopf substitutions and new examples of linearizable nonlinear evolution equations”, Theoret. and Math. Phys., 158:1 (2009), 48–60
5. Uenal G., Turkeri H., Khalique Ch.M., “Explicit Solution Processes for Nonlinear Jump-Diffusion Equations”, J Nonlinear Math Phys, 17:3 (2010), 281–310
6. M. N. Kuznetsova, “O nelineinykh giperbolicheskikh uravneniyakh, svyazannykh differentsialnymi podstanovkami s uravneniem Kleina–Gordona”, Ufimsk. matem. zhurn., 4:3 (2012), 86–103
7. Mariya N. Kuznetsova, Asli Pekcan, Anatoliy V. Zhiber, “The Klein–Gordon Equation and Differential Substitutions of the Form $v=\varphi(u,u_x,u_y)$”, SIGMA, 8 (2012), 090, 37 pp.
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