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

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Teoreticheskaya i Matematicheskaya Fizika, 2001, Volume 127, Number 1, Pages 63–74
DOI: https://doi.org/10.4213/tmf1926 (Mi tmf1926)

This article is cited in 12 scientific papers (total in 12 papers)

Hyperbolic Equations Admitting Differential Substitutions

S. Ya. Startsev

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

Full-text PDF (232 kB) Citations (12)

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

Abstract: We show that the first

n - 1

$n-1$ Laplace invariants of a scalar hyperbolic equation obtained from an equation of the same form under a differential substitution of the

n

$n$ 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

$x$ and an arbitrary function of

y

$y$ .

Received: 24.10.2000

English version:
Theoretical and Mathematical Physics, 2001, Volume 127, Issue 1, Pages 460–470
DOI: https://doi.org/10.1023/A:1010359808044

Bibliographic databases:

Language: Russian

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

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Linking options:

https://www.mathnet.ru/eng/tmf1926

https://doi.org/10.4213/tmf1926

https://www.mathnet.ru/eng/tmf/v127/i1/p63

This publication is cited in the following 12 articles:

Sergei Igonin, “Simplifications of Lax pairs for differential–difference equations by gauge transformations and (doubly) modified integrable equations”, Partial Differential Equations in Applied Mathematics, 11 (2024), 100821
M. N. Kuznetsova, “On nonlinear hyperbolic systems related by Bäcklund transforms”, Ufa Math. J., 15:3 (2023), 80–87
Rustem N. Garifullin, Ravil I. Yamilov, “Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation”, SIGMA, 15 (2019), 062, 15 pp.
V. M. Zhuravlev, “Mnogofunktsionalnye podstanovki i solitonnye resheniya integriruemykh nelineinykh uravnenii”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2019, no. 3, 93–119
Sergey ((((((() Grishin ((((((()“. . 140 . 1 2 (History of the Volga Railway is the First Part of the Second)”, SSRN Journal, 2015
M. N. Kuznetsova, “O nelineinykh giperbolicheskikh uravneniyakh, svyazannykh differentsialnymi podstanovkami s uravneniem Kleina–Gordona”, Ufimsk. matem. zhurn., 4:3 (2012), 86–103
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.
Uenal G., Turkeri H., Khalique Ch.M., “Explicit Solution Processes for Nonlinear Jump-Diffusion Equations”, J Nonlinear Math Phys, 17:3 (2010), 281–310
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
S. Ya. Startsev, “On the variational integrating matrix for hyperbolic systems”, J. Math. Sci., 151:4 (2008), 3245–3253
A. M. Gurieva, A. V. Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355
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

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

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