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TMF, 1997, Volume 110, Number 1, Pages 86–97 (Mi tmf954)  

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

Laplace transformations of hydrodynamic-type systems in Riemann invariants

E. V. Ferapontov

Institute for Mathematical Modelling, Russian Academy of Sciences

Abstract: The conserved densities of hydrodynamic-type systems in Riemann invariants satisfy a system of linear second-order partial differential equations. For linear systems of this type, Darboux introduced Laplace transformations, which generalize the classical transformations of a second-order scalar equation. It is demonstrated that the Laplace transformations can be pulled back to transformations of the corresponding hydrodynamic-type systems. We discuss finite families of hydrodynamic-type systems that are closed under the entire set of Laplace transformations. For $3\times3$ systems in Riemann invariants, a complete description of closed quadruples is proposed. These quadruples appear to be related to a special quadratic reduction of the $(2+1)$-dimensional 3-wave system.


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Theoretical and Mathematical Physics, 1997, 110:1, 68–77

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Received: 28.03.1996

Citation: E. V. Ferapontov, “Laplace transformations of hydrodynamic-type systems in Riemann invariants”, TMF, 110:1 (1997), 86–97; Theoret. and Math. Phys., 110:1 (1997), 68–77

Citation in format AMSBIB
\by E.~V.~Ferapontov
\paper Laplace transformations of hydrodynamic-type systems in Riemann invariants
\jour TMF
\yr 1997
\vol 110
\issue 1
\pages 86--97
\jour Theoret. and Math. Phys.
\yr 1997
\vol 110
\issue 1
\pages 68--77

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    This publication is cited in the following articles:
    1. S. I. Agafonov, E. V. Ferapontov, “Systems of conservation laws in the context of the projective theory of congruences”, Izv. Math., 60:6 (1996), 1097–1122  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Bogdanov, LV, “Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies”, Journal of Mathematical Physics, 39:9 (1998), 4701  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    3. Grinevich, PG, “Conformal invariant functionals of immersions of tori into R3”, Journal of Geometry and Physics, 26:1–2 (1998), 51  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    4. C. Athorne, “Darboux maps and $\mathcal D$-modules”, Theoret. and Math. Phys., 122:2 (2000), 135–139  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Kamran N., Tenenblat K., “Hydrodynamic systems and the higher-dimensional Laplace transformations of Cartan submanifolds”, Algebraic Methods in Physics - Symposium for the 60th Birthdays of Jiri Patera and Pavel Winternitz, CRM Series in Mathematical Physics, 2001, 105–120  mathscinet  zmath  isi
    6. Yilmaz, H, “The geometrically invariant form of evolution equations”, Journal of Physics A-Mathematical and General, 35:11 (2002), 2619  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    7. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. A. M. Gurieva, A. V. Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. S. Ya. Startsev, “On the variational integrating matrix for hyperbolic systems”, J. Math. Sci., 151:4 (2008), 3245–3253  mathnet  crossref  mathscinet  zmath  elib  elib
    10. 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  scopus
    11. 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  adsnasa  isi  elib  elib  scopus
    12. Pritula G.M., Vekslerchik V.E., “Toda-Heisenberg CHAIN: INTERACTING sigma-FIELDS IN TWO DIMENSIONS”, J Nonlinear Math Phys, 18:3 (2011), 443–459  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    13. Athorne Ch., Yilmaz H., “Laplace Invariants for General Hyperbolic Systems”, J. Nonlinear Math. Phys., 19:3 (2012), 1250024  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    14. Vekslerchik V.E., “Explicit Solutions for a (2+1)-Dimensional Toda-Like Chain”, J. Phys. A-Math. Theor., 46:5 (2013), 055202  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    15. Athorne Ch. Yilmaz H., “Invariants of Hyperbolic Partial Differential Operators”, J. Phys. A-Math. Theor., 49:13 (2016), 135201  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    16. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.  mathnet  crossref
    17. Athorne Ch., “Laplace Maps and Constraints For a Class of Third-Order Partial Differential Operators”, J. Phys. A-Math. Theor., 51:8 (2018), 085205  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    18. M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105  mathnet  crossref  isi
    19. M. N. Poptsova, “Simmetrii odnoi periodicheskoi tsepochki”, Kompleksnyi analiz. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 162, VINITI RAN, M., 2019, 80–84  mathnet
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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