RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

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

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.

DOI: https://doi.org/10.4213/tmf954

Full text: PDF file (227 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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
\Bibitem{Fer97}
\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
\mathnet{http://mi.mathnet.ru/tmf954}
\crossref{https://doi.org/10.4213/tmf954}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1472017}
\zmath{https://zbmath.org/?q=an:0919.35132}
\transl
\jour Theoret. and Math. Phys.
\yr 1997
\vol 110
\issue 1
\pages 68--77
\crossref{https://doi.org/10.1007/BF02630370}

• http://mi.mathnet.ru/eng/tmf954
• https://doi.org/10.4213/tmf954
• http://mi.mathnet.ru/eng/tmf/v110/i1/p86

 SHARE:

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. 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
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
3. Grinevich, PG, “Conformal invariant functionals of immersions of tori into R3”, Journal of Geometry and Physics, 26:1–2 (1998), 51
4. C. Athorne, “Darboux maps and $\mathcal D$-modules”, Theoret. and Math. Phys., 122:2 (2000), 135–139
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
6. Yilmaz, H, “The geometrically invariant form of evolution equations”, Journal of Physics A-Mathematical and General, 35:11 (2002), 2619
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
8. A. M. Gurieva, A. V. Zhiber, “Laplace Invariants of Two-Dimensional Open Toda Lattices”, Theoret. and Math. Phys., 138:3 (2004), 338–355
9. S. Ya. Startsev, “On the variational integrating matrix for hyperbolic systems”, J. Math. Sci., 151:4 (2008), 3245–3253
10. Demskoi, DK, “On non-Abelian Toda A(2)((1)) model and related hierarchies”, Journal of Mathematical Physics, 50:12 (2009), 123516
11. Shemyakova E.S., “X- and Y-Invariants of Partial Differential Operators in the Plane”, Program Comput Softw, 37:4 (2011), 192–196
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
13. Athorne Ch., Yilmaz H., “Laplace Invariants for General Hyperbolic Systems”, J. Nonlinear Math. Phys., 19:3 (2012), 1250024
14. Vekslerchik V.E., “Explicit Solutions for a (2+1)-Dimensional Toda-Like Chain”, J. Phys. A-Math. Theor., 46:5 (2013), 055202
15. Athorne Ch. Yilmaz H., “Invariants of Hyperbolic Partial Differential Operators”, J. Phys. A-Math. Theor., 49:13 (2016), 135201
16. Ismagil Habibullin, Mariya Poptsova, “Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings”, SIGMA, 13 (2017), 073, 26 pp.
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
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
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
•  Number of views: This page: 421 Full text: 152 References: 27 First page: 1