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Mat. Zametki, 2003, Volume 74, Issue 6, Pages 848–857 (Mi mz322)  

This article is cited in 19 scientific papers (total in 19 papers)

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

A. V. Zhibera, S. Ya. Startsevb

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

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


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English version:
Mathematical Notes, 2003, 74:6, 803–811

Bibliographic databases:

UDC: 517.956.32
Received: 01.08.2002

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
\by A.~V.~Zhiber, S.~Ya.~Startsev
\paper Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems
\jour Mat. Zametki
\yr 2003
\vol 74
\issue 6
\pages 848--857
\jour Math. Notes
\yr 2003
\vol 74
\issue 6
\pages 803--811

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    This publication is cited in the following articles:
    1. D. K. Demskoi, “One Class of Liouville-Type Systems”, Theoret. and Math. Phys., 141:2 (2004), 1509–1527  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. D. K. Demskoi, S. Ya. Startsev, “On construction of symmetries from integrals of hyperbolic partial differential systems”, J. Math. Sci., 136:6 (2006), 4378–4384  mathnet  crossref  mathscinet  zmath  elib
    3. 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  mathnet  crossref  elib
    4. S. Ya. Startsev, “Cascade Method of Laplace Integration for Linear Hyperbolic Systems of Equations”, Math. Notes, 83:1 (2008), 97–106  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. S. P. Tsarev, E. S. Shemyakova, “Differential Transformations of Parabolic Second-Order Operators in the Plane”, Proc. Steklov Inst. Math., 266 (2009), 219–227  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. 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  mathnet  zmath  elib
    7. Demskoi, DK, “On non-Abelian Toda A(2)((1)) model and related hierarchies”, Journal of Mathematical Physics, 50:12 (2009), 123516  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. 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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. 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  crossref  zmath  isi  scopus  scopus
    10. Yu. G. Voronova, “O zadache Koshi dlya lineinykh giperbolicheskikh sistem uravnenii s nulevymi obobschennymi invariantami Laplasa”, Ufimsk. matem. zhurn., 2:2 (2010), 20–26  mathnet  zmath  elib
    11. E. I. Ganzha, “Euler Integrals and Multi-Integrals of Linear Partial Differential Equations”, Math. Notes, 89:1 (2011), 37–50  mathnet  crossref  crossref  mathscinet  isi
    12. Shemyakova E.S., “X- and Y-Invariants of Partial Differential Operators in the Plane”, Program. Comput. Softw., 37:4 (2011), 192–196  crossref  mathscinet  zmath  isi  elib  elib  scopus
    13. 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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    14. Ganzha E.I., “On Laplace and Dini Transformations for Multidimensional Equations with a Decomposable Principal Symbol”, Program. Comput. Softw., 38:3 (2012), 150–155  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    15. Shemyakova E., “Laplace Transformations as the Only Degenerate Darboux Transformations of First Order”, Program. Comput. Softw., 38:2 (2012), 105–108  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. S. Ya. Startsev, “On differential substitutions for evolution systems”, Ufa Math. J., 9:4 (2017), 108–113  mathnet  crossref  isi  elib
    17. Sara Froehlich, “The Variational Bi-Complex for Systems of Semi-Linear Hyperbolic PDEs in Three Variables”, SIGMA, 14 (2018), 096, 49 pp.  mathnet  crossref
    18. 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  mathnet  crossref  isi
    19. S. Ya. Startsev, “Draivery simmetrii i formalnye integraly giperbolicheskikh sistem uravnenii”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 110–119  mathnet  mathscinet
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