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Science China Mathematics

, Volume 61, Issue 11, pp 1947–1962 | Cite as

Normal forms of linear second order partial differential equations on the plane

  • Alexey Davydov
Open Access
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Abstract

The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the results obtained in the last decades and some problems, which are important both for the development of this theory and the applications. The reduction theorem, which was used to obtain many of recent results in the theory, is included in the paper in the parametric form together with proof. There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.

Keywords

normal form mixed type partial differential equation main symbol 

MSC(2010)

34A09 34C20 

Notes

Acknowledgements

This work was supported by the Ministry of Education and Science of the Russian Federation (Grant No. 1.638.2016/FPM).

Open access funding provided by International Institute for Applied Systems Analysis (IIASA).

References

  1. 1.
    Arnold V I. Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer-Verlag, 1983CrossRefGoogle Scholar
  2. 2.
    Arnold V I, Ilyashenko Y S. Ordinary differential equations. Encyclopaedia Math Sci, 1988, 1: 1–148Google Scholar
  3. 3.
    Arnold V I, Varchenko A N, Gusein-Sade S M. Singularities of Differentiable Mapping, Volume 1. Monographs in Mathematics, vol. 82. Boston: Birkhäuser, 1985Google Scholar
  4. 4.
    Bogaevsky I A. Implicit ordinary differential equations: Bifurcations and sharpening of equivalence. Izv Math, 2014, 78: 1063–1078MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bruce J W, Tari F. Generic 1–parameter families of binary differential equations of Morse type. Discrete Contin Dyn Syst, 1997, 3: 79–90MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bruce J W, Tari F, Fletcher G J. Bifurcations of binary differential equations. Proc Roy Soc Edinburgh Sect A, 2000, 130: 485–506MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cibrario M. Sulla reduzione a forma canonica delle equazioni lineari alle derivative parzialy di secondo ordine di tipo misto. Rend Lombardo, 1932, 65: 889–906zbMATHGoogle Scholar
  8. 8.
    Dara L. Singularities generiques des equations differentielles multiformes. Bol Soc Bras Mat, 1975, 6: 95–128MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davydov A A. The normal form of a differential equation that is not solved with respect to derivative, in the neighbourhood of its singular point. Funct Anal Appl, 1985, 19: 81–89CrossRefGoogle Scholar
  10. 10.
    Davydov A A. Structural stability of control systems on orientable surfaces. Mat Sb, 1992, 72: 1–28MathSciNetCrossRefGoogle Scholar
  11. 11.
    Davydov A A. Qualitative Theory of Control Systems. Translations of Mathematical Monographs, vol. 141. Providence: Amer Math Soc, 1994Google Scholar
  12. 12.
    Davydov A A, Diep L T T. Normal forms for families of linear equations of mixed type near non-resonant folded singular points. Russian Math Surveys, 2010, 65: 984–986MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davydov A A, Diep L T T. Reduction theorem and normal forms of linear second order mixed type PDE families in the plane. TWMS J Pure Appl Math, 2011, 2: 44–53MathSciNetzbMATHGoogle Scholar
  14. 14.
    Davydov A A, Ishikawa G, Izumiya S, et al. Generic singularities of implicit systems of first order differential equations on the plane. Jpn J Math, 2008, 3: 93–119MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Davydov A A, Ortiz-Bobadilla L. Smooth normal forms of folded elementary singular points. J Dyn Control Syst, 1995, 1: 463–483MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Davydov A A, Ortiz-Bobadilla L. Normal forms of folded elementary singular points. Russian Math Surveys, 1995, 50: 1260–1261MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Davydov A A, Rosales-Gonzales E. Complete classification of generic linear second-order partial differential equations in the plane. Dokl Math, 1996, 350: 151–154MathSciNetGoogle Scholar
  18. 18.
    Davydov A A, Rosales-Gonzales E. Smooth normal forms of folded resonance saddles and nodes and complete classi fication of generic linear second order PDE’s on the plane. In: International Conference on Differential Equation. Singapore: World Scientific, 1998, 59–68Google Scholar
  19. 19.
    Grishina Y A, Davydov A A. Structural stability of simplest dynamical inequalities. Proc Steklov Inst Math, 2007, 256: 80–91MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hormander L. On the theory of general partial differential operators. Acta Math, 1955, 94: 161–248MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kasten J A. Solvability of the boundary value problem for a Tricomi type equation in the exterior of a disk. J Math Sci, 2013, 188: 268–272MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kondratiev V A, Landis E M. Qualitative theory of second order linear partial differential equations. Itogi Nauki i Tekhniki Ser Sovrem Probl Mat Fund Napr, 1988, 32: 99–215MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kuzmin A G. On the behavior of the characteristics of equations of mixed type near the line of degeneracy. Differ Uravn, 1981, 17: 2052–2063MathSciNetGoogle Scholar
  24. 24.
    Kuzmin A G. Non-Classical Equations of Mixed Type and Their Applications in Gas Dynamics. International Series of Numerical Mathematics, vol. 109. Basel: Birkhäuser, 1992Google Scholar
  25. 25.
    Pilija A D, Fedorov V I. Singularities of electromagnetic wave field in cold anisotropic plasma with two-dimensional non-homogeneity. J Exp Theor Phys, 1971, 60: 389–400Google Scholar
  26. 26.
    Pkhakadze A V, Shestakov A A. On the classification of the singular points of a first order differential equation not solved for the derivative. Mat Sb, 1959, 49: 3–12MathSciNetzbMATHGoogle Scholar
  27. 27.
    Rassias J M. Lecture Notes on Mixed Type Partial Differential Equations. Singapore: World Scientific, 1990CrossRefzbMATHGoogle Scholar
  28. 28.
    Smirnov M M. Equations of Mixed Type. Translations of Mathematical Monographs, vol. 51. Providence: Amer Math Soc, 1978Google Scholar
  29. 29.
    Sokolov P V. On the paper of A. V. Phadadze and A. A. Šhestakov “on the classification of the singular point of a first order differential equation not solved for the derivative” (in Russian). Mat Sb, 1961, 53: 541–543MathSciNetGoogle Scholar
  30. 30.
    Takens F. Constrained equations: A study of implicit differential equations and their discontinuous solutions. In: Structural Stability, the Theory of Catastrophes, and Applications in the Sciences. Berlin-Heidelberg: Springer, 1976, 143–234CrossRefGoogle Scholar
  31. 31.
    Thom R. Sur les equations differentielles multiformes et leurs integrales singulieres. Bol Soc Bras Mat, 1972, 3: 1–11CrossRefzbMATHGoogle Scholar
  32. 32.
    Tricomi F. Sulle equazioni lineari alle derivate partziali di secondo ordine di tipo misto. Rend Reale Accad Lincei, 1923, 14: 134–247Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe National University of Science and Technology MISiSMoscowRussia
  2. 2.Department of Theory of Dynamical SystemsLomonosov Moscow State UniversityMoscowRussia
  3. 3.International Institute for Applied Systems AnalysisLaxenburgAustria

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