A. V. Zhiber, S. Ya. Startsev, “Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems”, Mat. Zametki, 74:6 (2003), 848–857; Math. Notes, 74:6 (2003), 803

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Matematicheskie Zametki, 2003, Volume 74, Issue 6, Pages 848–857
DOI: https://doi.org/10.4213/mzm322 (Mi mzm322)

This article is cited in 22 scientific papers (total in 22 papers)

Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems

A. V. Zhiber^a, S. Ya. Startsev^b

^a Institute of Mechanics, Ufa Centre of the Russian Academy of Sciences
^b Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Full-text PDF (210 kB) Citations (22)

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DOI: https://doi.org/10.4213/mzm322

Abstract: We generalize the notions of Laplace transformations and Laplace invariants for systems of hyperbolic equations and study conditions for their existence. We prove that a hyperbolic system admits the Laplace transformation if and only if there exists a matrix of rank

k

$k$ mapping any vector whose components are functions of one of the independent variables into a solution of this system, where

k

$k$ is the defect of the corresponding Laplace invariant. We show that a chain of Laplace invariants exists only if the hyperbolic system has a entire collection of integrals and the dual system has a entire collection of solutions depending on arbitrary functions. An example is given showing that these conditions are not sufficient for the existence of a Laplace transformation.

Received: 01.08.2002

English version:
Mathematical Notes, 2003, Volume 74, Issue 6, Pages 803–811
DOI: https://doi.org/10.1023/B:MATN.0000009016.91968.ed

Bibliographic databases:

UDC: 517.956.32

Language: Russian

Citation: A. V. Zhiber, S. Ya. Startsev, “Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems”, Mat. Zametki, 74:6 (2003), 848–857; Math. Notes, 74:6 (2003), 803–811

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm322

https://doi.org/10.4213/mzm322

https://www.mathnet.ru/eng/mzm/v74/i6/p848

This publication is cited in the following 22 articles:

E. A. Sozontova, “Usloviya suschestvovaniya i edinstvennosti resheniya zadachi Gursa dlya sistemy uravnenii s dominiruyuschimi chastnymi proizvodnymi”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 27:2 (2023), 375–383
Zhang H., Su Ch., Bai J., Yuan R., Ma Yu., Wang W., “The Rheological Analytical Solution and Parameter Inversion of Soft Soil Foundation”, Symmetry-Basel, 13:7 (2021), 1228
Sara Froehlich, “The Variational Bi-Complex for Systems of Semi-Linear Hyperbolic PDEs in Three Variables”, SIGMA, 14 (2018), 096, 49 pp.
S. Ya. Startsev, “Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants”, Ufa Math. J., 10:4 (2018), 103–110
S. Ya. Startsev, “Symmetry Drivers and Formal Integrals of Hyperbolic Systems of Equations”, J. Math. Sci. (N. Y.), 252:2 (2021), 232–241
S. Ya. Startsev, “On differential substitutions for evolution systems”, Ufa Math. J., 9:4 (2017), 108–113
Lin Ch.-Sh., Wei J., Ye D., “Classification and Nondegeneracy of Su(N+1) Toda System with Singular Sources”, Invent. Math., 190:1 (2012), 169–207
Ganzha E.I., “On Laplace and Dini Transformations for Multidimensional Equations with a Decomposable Principal Symbol”, Program. Comput. Softw., 38:3 (2012), 150–155
Shemyakova E., “Laplace Transformations as the Only Degenerate Darboux Transformations of First Order”, Program. Comput. Softw., 38:2 (2012), 105–108
E. I. Ganzha, “Euler Integrals and Multi-Integrals of Linear Partial Differential Equations”, Math. Notes, 89:1 (2011), 37–50
Shemyakova E.S., “X- and Y-Invariants of Partial Differential Operators in the Plane”, Program. Comput. Softw., 37:4 (2011), 192–196
Yu. G. Voronova, “O zadache Koshi dlya lineinykh giperbolicheskikh sistem uravnenii s nulevymi obobschennymi invariantami Laplasa”, Ufimsk. matem. zhurn., 2:2 (2010), 20–26
S. P. Tsarev, E. S. Shemyakova, “Differential Transformations of Parabolic Second-Order Operators in the Plane”, Proc. Steklov Inst. Math., 266 (2009), 219–227
A. V. Zhiber, Yu. G. Mikhailova, “Algoritm postroeniya obschego resheniya $n$ -komponentnoi giperbolicheskoi sistemy uravnenii s nulevymi invariantami Laplasa i kraevye zadachi”, Ufimsk. matem. zhurn., 1:3 (2009), 28–45
Demskoi, DK, “On non-Abelian Toda A(2)((1)) model and related hierarchies”, Journal of Mathematical Physics, 50:12 (2009), 123516
Shemyakova E., “Multiple Factorizations of Bivariate Linear Partial Differential Operators”, Computer Algebra in Scientific Computing, Proceedings, Lecture Notes in Computer Science, 5743, ed. Gerdt V. Mayr E. Vorozhtsov E., Springer-Verlag Berlin, 2009, 299–309
Shemyakova E., “Invariant Properties of Third-Order Non-Hyperbolic Linear Partial Differential Operators”, Intelligent Computer Mathematics, Proceedings, Lecture Notes in Computer Science, 5625, ed. Carette J. Dixon L. Coen C. Watt S., Springer-Verlag Berlin, 2009, 154–169
S. Ya. Startsev, “Cascade Method of Laplace Integration for Linear Hyperbolic Systems of Equations”, Math. Notes, 83:1 (2008), 97–106
A. V. Zhiber, Yu. G. Mikhailova, “On hyperbolic systems of equations with zero generalized Laplace invariants”, Proc. Steklov Inst. Math. (Suppl.), 261, suppl. 1 (2008), S154–S164
Sergey P. Tsarev, Proceedings of the 2005 international symposium on Symbolic and algebraic computation, 2005, 325

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

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