General information
Latest issue
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS


Personal entry:
Save password
Forgotten password?

TMF, 1998, Volume 114, Number 1, Pages 3–55 (Mi tmf827)  

This article is cited in 38 scientific papers (total in 38 papers)

Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems

V. G. Danilova, V. P. Maslovb, V. M. Shelkovichc

a Moscow State Institute of Electronics and Mathematics (Technical University)
b M. V. Lomonosov Moscow State University
c St. Petersburg State University of Architecture and Civil Engineering

Abstract: An associative commutative algebra of distributions that contains homogeneous and associated homogeneous distributions is constructed. This algebra is used to analyze generalized solutions to strictly hyperbolic partial differential equations. Possible types of singularities are studied and the necessary (analogues of Hugoniуt conditions for shock waves) and sufficient conditions for the existence of such solutions are obtained.


Full text: PDF file (525 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 1998, 114:1, 1–42

Bibliographic databases:

Received: 29.07.1997

Citation: V. G. Danilov, V. P. Maslov, V. M. Shelkovich, “Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems”, TMF, 114:1 (1998), 3–55; Theoret. and Math. Phys., 114:1 (1998), 1–42

Citation in format AMSBIB
\by V.~G.~Danilov, V.~P.~Maslov, V.~M.~Shelkovich
\paper Algebras of the singularities of singular solutions to first-order quasi-linear strictly hyperbolic systems
\jour TMF
\yr 1998
\vol 114
\issue 1
\pages 3--55
\jour Theoret. and Math. Phys.
\yr 1998
\vol 114
\issue 1
\pages 1--42

Linking options:

    SHARE: FaceBook Twitter Livejournal

    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. Dobrokhotov, SY, “Hugoniot-Maslov chains for solitary vortices of the shallow water equations, I. - Derivation of the chains for the case of variable Coriolis forces and reduction to the Hill equation”, Russian Journal of Mathematical Physics, 6:2 (1999), 137  mathscinet  zmath  isi
    2. S. Yu. Dobrokhotov, “Integrability of truncated Hugoniot–Maslov chains for trajectories of mesoscale vortices on shallow water”, Theoret. and Math. Phys., 125:3 (2000), 1724–1741  mathnet  crossref  crossref  mathscinet  zmath
    3. Dobrokhotov, SY, “On Maslov's conjecture about the structure of weak point singularities of shallow-water equations”, Doklady Mathematics, 64:1 (2001), 127  mathscinet  zmath  isi
    4. Dobrokhotov, SY, “Proof of Maslov's conjecture about the structure of weak point singular solutions of the shallow water equations”, Russian Journal of Mathematical Physics, 8:1 (2001), 25  mathscinet  zmath  isi
    5. Danilov V.G., Shelkovich V.M., “Propagation and interaction of nonlinear waves to quasilinear equations”, Hyperbolic Problems: Theory, Numerics, Applications, International Series of Numerical Mathematics, 140, 2001, 267–276  mathscinet  isi
    6. Khrennikov, AY, “Locally convex spaces of vector-valued distributions with multiplicative structures”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 5:4 (2002), 483  crossref  mathscinet  zmath  isi  scopus
    7. Danilov, VG, “Dynamics of the interface between two immiscible liquids with nearly equal densities under gravity”, European Journal of Applied Mathematics, 13 (2002), 497  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Smolyanov, OG, “Multiplicative structures in the linear space of vector-valued distributions”, Doklady Mathematics, 65:2 (2002), 169  mathscinet  zmath  isi
    9. S. Yu. Dobrokhotov, E. S. Semenov, B. Tirozzi, “Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory”, Journal of Mathematical Sciences, 124:5 (2004), 5209–5249  mathnet  crossref  mathscinet  zmath
    10. Albeverio, S, “Associated homogeneous p-adic distributions”, Doklady Mathematics, 68:3 (2003), 354  isi
    11. Danilov, AD, “Long-term trends of foF2 independent of geomagnetic activity”, Geomagnetism and Aeronomy, 43:2 (2003), 196  isi
    12. G. A. Omel'yanov, D. A. Kulagin, “Asymptotics of Kink–Kink Interaction for Sine-Gordon Type Equations”, Math. Notes, 75:4 (2004), 563–567  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. S. A. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, “Associative Algebras of $p$-Adic Distributions”, Proc. Steklov Inst. Math., 245 (2004), 22–33  mathnet  mathscinet  zmath
    14. S. Yu. Dobrokhotov, E. S. Semenov, B. Tirozzi, “Calculation of Integrals of the Hugoniot–Maslov Chain for Singular Vortical Solutions of the Shallow-Water Equation”, Theoret. and Math. Phys., 139:1 (2004), 500–512  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Nedeljkov, M, “Delta and singular delta locus for one-dimensional systems of conservation laws”, Mathematical Methods in the Applied Sciences, 27:8 (2004), 931  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    16. Shvedov, OY, “Approximations for strongly singular evolution equations”, Journal of Functional Analysis, 210:2 (2004), 259  crossref  mathscinet  zmath  isi  scopus  scopus
    17. Danilov, VG, “Propagation and interaction of delta-shock waves for hyperbolic systems of conservation laws”, Doklady Mathematics, 69:1 (2004), 4  mathscinet  zmath  isi
    18. Shelkovich V.M., “Delta-shocks, the Rankine-Hugoniot conditions and singular superposition of distributions”, Days on Diffraction 2004, Proceedings, 2004, 175–196  crossref  isi
    19. Danilov, VG, “Delta-shock wave type solution of hyperbolic systems of conservation laws”, Quarterly of Applied Mathematics, 63:3 (2005), 401  crossref  mathscinet  isi  scopus  scopus
    20. Shelkovich, VM, “New versions of the Colombeau algebras”, Mathematische Nachrichten, 278:11 (2005), 1318  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    21. Danilov, VG, “Dynamics of propagation and interaction of delta-shock waves in conservation law systems”, Journal of Differential Equations, 211:2 (2005), 333  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    22. Albeverio, S, “p-adic Colombeau-Egorov type theory of generalized functions”, Mathematische Nachrichten, 278:1–2 (2005), 3  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    23. Dobrokhotov S., Tirozzi B., “A perturbative theory of the evolution of the center of typhoons”, Zeta Functions, Topology and Quantum Physics, Developments in Mathematics, 14, 2005, 31–50  crossref  mathscinet  zmath  isi
    24. V. M. Shelkovich, “The Rankine–Hugoniot conditions and balance laws for $\delta$-shocks”, J. Math. Sci., 151:1 (2008), 2781–2792  mathnet  crossref  mathscinet  zmath  elib  elib
    25. K. A. Volosov, “Eigenfunctions of structures described by the “shallow water” model in a plane”, J. Math. Sci., 151:1 (2008), 2639–2650  mathnet  crossref  mathscinet  zmath  elib  elib
    26. Kulagin, DA, “Interaction of kinks for semilinear wave equations with a small parameter”, Nonlinear Analysis-Theory Methods & Applications, 65:2 (2006), 347  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    27. Albeverio, S, “Associated homogeneous p-adic distributions”, Journal of Mathematical Analysis and Applications, 313:1 (2006), 64  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    28. Reutskiy, S, “Forecast of the trajectory of the center of typhoons and the Maslov decomposition”, Russian Journal of Mathematical Physics, 14:2 (2007), 232  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    29. Bermudez, PR, “Hugoniot-Maslov chains of a shock wave in conservation law with polynomial flow”, Mathematische Nachrichten, 280:8 (2007), 907  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    30. Sarrico C.O.R., “Products of Distributions and Singular Travelling Waves as Solutions of Advection-Reaction Equations”, Russ. J. Math. Phys., 19:2 (2012), 244–255  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    31. Sarrico C.O.R., “A Distributional Product Approach To Delta-Shock Wave Solutions For a Generalized Pressureless Gas Dynamics System”, Int. J. Math., 25:1 (2014), 1450007  crossref  mathscinet  zmath  isi  scopus  scopus
    32. Sarrico C.O.R., “the Riemann Problem For the Brio System: a Solution Containing a Dirac Mass Obtained Via a Distributional Product”, Russ. J. Math. Phys., 22:4 (2015), 518–527  crossref  mathscinet  zmath  isi  scopus  scopus
    33. Sarrico C.O.R., Paiva A., “Products of distributions and collision of a ? -wave with a ?? -wave in a turbulent model”, J. Nonlinear Math. Phys., 22:3 (2015), 381–394  crossref  mathscinet  isi  scopus  scopus
    34. Sarrico C.O.R., Paiva A., “New Distributional Travelling Waves For the Nonlinear Klein-Gordon Equation”, Differ. Integral Equ., 30:11-12 (2017), 853–878  mathscinet  zmath  isi
    35. Sarrico C.O.R., Paiva A., “The Multiplication of Distributions in the Study of a Riemann Problem in Fluid Dynamics”, J. Nonlinear Math. Phys., 24:3 (2017), 328–345  crossref  mathscinet  isi  scopus  scopus  scopus
    36. Sarrico C.O.R., Paiva A., “Newton'S Second Law and the Multiplication of Distributions”, J. Math. Phys., 59:1 (2018), 013505  crossref  mathscinet  zmath  isi  scopus  scopus
    37. Rodriguez-Bermudez P., Valino-Alonso B., “Asymptotic Maslov'S Method For Shocks of Conservation Laws Systems With Quadratic Flux”, Appl. Anal., 97:6 (2018), 888–901  crossref  mathscinet  zmath  isi  scopus  scopus
    38. Sarrico Carlos Orlando R., “Multiplication of Distributions and Travelling Wave Solutions For the Keyfitz-Kranzer System”, Taiwan. J. Math., 22:3 (2018), 677–693  crossref  isi  scopus  scopus  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
    Number of views:
    This page:541
    Full text:126
    First page:5

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019