Advertisement

Theory of Multidimensional Delsarte–Lions Transmutation Operators. I

  • A. M. Samoilenko
  • Ya. A. Prykarpatsky
  • D. Blackmore
  • A. K. Prykarpatsky
Article
  • 3 Downloads

We present a brief survey of the original results obtained by the authors in the theory of Delsarte–Lions transmutations of multidimensional spectral differential operators based on the classical works by Yu. M. Berezansky, V. A. Marchenko, B. M. Levitan, and R. G. Newton, on the well-known L. D. Faddeev’s survey, the book by L. P. Nyzhnyk, and the generalized de Rham–Hodge theory suggested by I. V. Skrypnik and developed by the authors for the differential-operator complexes. The operator structure of the Delsarte–Lions transformations and the properties of their Volterra factorizations are analyzed in detail. In particular, we study the differential-ga generalized de Rham–Hodge theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform, Society of Industrial and Applied Mathematics, Philadelphia (1981).CrossRefzbMATHGoogle Scholar
  2. 2.
    Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).Google Scholar
  3. 3.
    F. A. Berezin and M. A. Shubin, Schrödinger Equation [in Russian], Moscow University, Moscow (1983).zbMATHGoogle Scholar
  4. 4.
    D. Blackmore, A. K. Prykarpatsky, and V. Hr. Samoylenko, Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Differential-Geometrical Integrability Analysis, World Scientific, Singapore (2011).CrossRefzbMATHGoogle Scholar
  5. 5.
    D. Blackmore, A. K. Prykarpatsky, and J. Zagrodzinski, “Lax-type flows on Grassmann manifolds and dual momentum mappings,” Rep. Math. Phys., 40, No. 3, 539–549 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. L. Bukhgeim, Volterra Equations and Inverse Problems [in Russian], Nauka, Moscow (1983).Google Scholar
  7. 7.
    F. Calogero and A. Degasperis, Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, Vol. 1, North-Holland Publ. Co., Amsterdam (1982).zbMATHGoogle Scholar
  8. 8.
    S. S. Chern, Complex Manifolds, University of Chicago, Chicago (1956).zbMATHGoogle Scholar
  9. 9.
    R. W. Carroll, Topics in Soliton Theory, North-Holland Publ. Co., Amsterdam (1991).zbMATHGoogle Scholar
  10. 10.
    R. W. Carroll, Transmutation and Operator Differential Equations, North-Holland Publ. Co., Amsterdam (1979).zbMATHGoogle Scholar
  11. 11.
    R. Carroll, Transmutation, Scattering Theory and Special Functions, North-Holland Publ. Co., Amsterdam (1982).zbMATHGoogle Scholar
  12. 12.
    R. Carroll, Transmutation Theory and Applications, North Holland Publ. Co., Amsterdam (1986).zbMATHGoogle Scholar
  13. 13.
    K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer, Berlin (1989).CrossRefzbMATHGoogle Scholar
  14. 14.
    N. Dunford and J. T. Schwartz, Linear Operators, Part 2. Spectral Theory. Self-Adjoint Operators in Hilbert Space, Interscience Publ., New York (1963).zbMATHGoogle Scholar
  15. 15.
    B. N. Datta and D. R. Sarkissian, “Feedback control in distributed parameter gyroscopic systems: a solution of the partial eigenvalue assignment problem,” Mech. Syst. Signal Proc., 16, No. 1, 3–17 (2002).CrossRefGoogle Scholar
  16. 16.
    J. Delsarte, “Sur certaines transformations fonctionelles relatives aux équations linéaires aux dérivées partielles du second ordre,” C. R. Acad. Sci. Paris, 206, 178–182 (1938).zbMATHGoogle Scholar
  17. 17.
    J. Delsarte and J. L. Lions, “Transmutations d’opérateurs differentielles dans le domain complexe,” Comment. Math. Helv., 52, 113–128 (1957).zbMATHGoogle Scholar
  18. 18.
    J. Delsarte and J. L. Lions, “Moyennes généralisées,” Comment. Math. Helv., No. 34, 59–69 (1959).Google Scholar
  19. 19.
    G. de Rham, Variétés Différentielles, Hermann, Paris (1955).zbMATHGoogle Scholar
  20. 20.
    G. de Rham, “Sur la théorie des formes différentielles harmoniques,” Ann. Univ. Grenoble, 22, 135–152 (1946).MathSciNetzbMATHGoogle Scholar
  21. 21.
    P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford Univ. Press, Oxford (1935).zbMATHGoogle Scholar
  22. 22.
    N. Dunford and T. Schwartz, Linear Operators. Spectral Operators, Wiley-Interscience, New York (1971).zbMATHGoogle Scholar
  23. 23.
    L. D. Faddeev, “Quantum inverse scattering problem. II,” in: VINITI, Series in Modern Problems of Mathematics [in Russian], 3, VINITI, Moscow (1974), pp. 93–180.Google Scholar
  24. 24.
    L. D. Faddeev and L. A. Takhtadjyan, Hamiltonian Approach to the Soliton Theory, Nauka, Moscow (1986).Google Scholar
  25. 25.
    R. Gilbert and Y. Begehr, Transformations, Transmutations and Kernel Functions, Vols. 1, 2, Pitman, Longman (1992).Google Scholar
  26. 26.
    C. Godbillon, Geometrie Differentielle et Mechanique Analytique, Hermann, Paris (1969).zbMATHGoogle Scholar
  27. 27.
    I. C. Gokhberg and M. G. Krein, Theory of Volterra Operators [in Russian], Nauka, Moscow (1984).Google Scholar
  28. 28.
    J. Golenia, Y. A. Prykarpatsky, A. M. Samoilenko, and A. K. Prykarpatsky, “The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Pt 2,” Opuscula Math., No. 24 (2004).Google Scholar
  29. 29.
    P. I. Holod and A. U. Klimyk, Theory of Symmetry [in Russian], Factorial, Moscow (2002).Google Scholar
  30. 30.
    E. Ya. Hruslov, “Asymptotics of the solution of the Cauchy problem for the KdV equation with step-like initial data,” Math. USSRSb., 28, 229–248 (1976).Google Scholar
  31. 31.
    V. V. Katrakhov and S. M. Skripnik, “Method of transformation operators and boundary-value problems for singular elliptic equations,” Sovr. Mat. Fundament. Napr., 64, No. 2, 211–428 (2018).Google Scholar
  32. 32.
    B. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectral Transform Method, World Scientific, Singapore (1993).CrossRefzbMATHGoogle Scholar
  33. 33.
    I. M. Krichever, “Algebro-geometric methods in theory of nonlinear equations,” Russ. Math. Surveys, 32, No. 6, 183–208 (1977).CrossRefzbMATHGoogle Scholar
  34. 34.
    P. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math., 21, No. 2, 467–490 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    B. M. Levitan, Theory of Operators of Generalized Translation [in Russian], Nauka, Moscow (1973).Google Scholar
  36. 36.
    B. M. Levitan and I. S. Sargsian, Sturm–Liouville and Dirac Operators [in Russian], Nauka, Moscow (1988).Google Scholar
  37. 37.
    V. B. Matveev and M. I. Salle, Darboux–Backlund Transformations and Applications, Springer, New York (1993).Google Scholar
  38. 38.
    B. M. Levitan, Sturm–Liouville Inverse Problems [in Russian], Nauka, Moscow (1984).Google Scholar
  39. 39.
    J. L. Lions, “Opérateurs de Delsarte en probl`emes mixtes,” Bull. Soc. Math. France, No. 84, 9–95 (1956).Google Scholar
  40. 40.
    J. L. Lions, “Quelques applications d’opérateurs de transmutation,” Colloq. Int. Nancy, 125–142 (1956).Google Scholar
  41. 41.
    Y. B. Lopatynski, “On harmonic fields on Riemannian manifolds,” Ukr. Math. Zh., 2, No. 1, 56–60 (1950).MathSciNetGoogle Scholar
  42. 42.
    Ya. B. Lopatinskii, Introduction to the Contemporary Theory of Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1980).Google Scholar
  43. 43.
    S. V. Manakov, “Method of inverse scattering problem and two-dimensional evolutionary equations,” Usp. Mat. Nauk, 31, No. 5, 245–246 (1976).Google Scholar
  44. 44.
    V. A. Marchenko, Spectral Theory of Sturm–Liouville Operators, Naukova Dumka, Kiev (1972).zbMATHGoogle Scholar
  45. 45.
    Ya. A. Mitropol’skii, N. N. Bogolyubov Jr., A. K. Prykarpatsky, and V. G. Samoilenko, Integrable Dynamical Systems: Differential-Geometric and Spectral Aspects [in Russian], Naukova Dumka, Kiev (1987).Google Scholar
  46. 46.
    Ya. V. Mykytiuk, “Factorization of Fredholmian operators,” Math. Stud. Proc. Lviv Math. Soc., 20, No. 2, 185–199 (2003).Google Scholar
  47. 47.
    Ya. V. Mykytiuk, “Factorization of Fredholm operators in operator algebras,” Math. Stud. Proc. Lviv Math. Soc., 21, No. 1, 87–97 (2004).MathSciNetGoogle Scholar
  48. 48.
    A. C. Newell, Solitons in Mathematics and Physics, Society for Industrial and Applied Mathematics, Philadelphia (1985).CrossRefzbMATHGoogle Scholar
  49. 49.
    R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Springer, Berlin (1989).CrossRefzbMATHGoogle Scholar
  50. 50.
    J. C. C. Nimmo, Darboux Transformations from Reductions of the KP-Hierarchy, Preprint, University Glasgow (2002).Google Scholar
  51. 51.
    L. P. Nizhnik, “Integration of nonlinear multidimensional equations by the method of inverse scattering,” Dokl. Akad. Nauk SSSR, 254, No. 2, 332–335 (1980).MathSciNetGoogle Scholar
  52. 52.
    L. P. Nizhnik, Inverse Scattering Problems for Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1991).zbMATHGoogle Scholar
  53. 53.
    L. P. Nizhnik, “The inverse scattering problems for the hyperbolic equations and their applications to nonlinear integrable equations,” Rep. Math. Phys., 26, No. 2, 261–283 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    L. P. Nizhnik, “Inverse scattering problem for the wave equation and its application,” in: J. Gottlieb and P. Duchateau (editors), Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology, Kluwer AP, Dordrecht (1996), pp. 233–238.CrossRefGoogle Scholar
  55. 55.
    L. P. Nizhnik and M. D. Pochynaiko, “The integration of a spatially two-dimensional Schrödinger equation by the inverse problem method,” Func. Anal. Appl., 16, No. 1, 80–82 (1982).CrossRefGoogle Scholar
  56. 56.
    J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton (1955).zbMATHGoogle Scholar
  57. 57.
    S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Springer (1984).Google Scholar
  58. 58.
    V. Ovsienko, Bi-Hamilton Nature of the Equation u tx = u xy u y − u yy u x , Preprint arXiv:0802.1818v1 [math-ph] (2008).Google Scholar
  59. 59.
    V. Ovsienko and C. Roger, “Looped cotangent Virasoro algebra and nonlinear integrable systems in dimension 2+1,Comm. Math. Phys., 273, 357–378 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    A. K. Prykarpatsky and D. Blackmore, “Versal deformations of a Dirac-type differential operator,” J. Nonlin. Math. Phys., 6, No. 3, 246–254 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    A. K. Prykarpatsky and I. V. Mykytiuk, Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects, Kluwer AP, Dordrecht (1998).CrossRefzbMATHGoogle Scholar
  62. 62.
    A. K. Prykarpatsky, A. M. Samoilenko, and Y. A. Prykarpatsky, “The multi-dimensional Delsarte transmutation operators, their differential-geometric structure, and applications. Pt. 1,” Opuscula Math., 23, 71–80 (2003).zbMATHGoogle Scholar
  63. 63.
    A. G. Reyman and M. A. Semenov-Tian-Shansky, Integrable Systems. Group-Theoretic Approach [in Russian], Institute of Computer Investigations, Izhevsk (2003).Google Scholar
  64. 64.
    A. M. Samoilenko and Ya. A. Prykarpats’kyi, Algebraic-Analytic Aspects of the Theory of Completely Integrable Dynamical Systems and Their Perturbations [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2002).Google Scholar
  65. 65.
    A. M. Samoilenko, Y. A. Prykarpatsky, and A. K. Prykarpatsky, “Generalized de Rham–Hodge–Skrypnik theory: differentialgeometric and spectral aspects with applications,” Ukr. Math. Bull., 2, No. 4, 550–582 (2005).MathSciNetzbMATHGoogle Scholar
  66. 66.
    A. M. Samoilenko, Y. A. Prykarpatsky, and A. K. Prykarpatsky, “The de Rham–Hodge–Skrypnik theory of Delsarte–Lions transmutations in multidimension and its applications,” Rep. Math. Phys., 55, No. 3, 351–370 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Ya. A. Prykarpats’kyi, A. M. Samoilenko, and V. H. Samoilenko, “Structure of binary transformations of Darboux type and their application to soliton theory,” Ukr. Mat. Zh., 55, No. 12, 1704–1719 (2003); English translation: Ukr. Math. J., 55, No. 12, 2041–2059 (2003).Google Scholar
  68. 68.
    A. M. Samoilenko, Y. A. Prykarpatsky, and A. K. Prykarpatsky, “The spectral and differential geometric aspects of the generalized De Rham–Hodge theory related with Delsarte transmutation operators in multidimension and its applications to spectral and soliton problems,” Nonlin. Anal., 65, 395–432 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    S. M. Sitnik, Buschman–Erdelyi Transmutations, Classification, and Applications, Preprint, arXiv:1304.2114v1 [math.CA] (2013).Google Scholar
  70. 70.
    S. M. Sitnik, “Transmutations and applications. A survey,” in: Yu. F. Korobeinik and A. G. Kusraev (editors), Advances in Modern Analysis and Mathematical Modeling, Vladikavkaz Scientific Center, Russian Academy of Sciences, Republic of North Ossetia–Alania, Vladikavkaz (2008), pp. 226–293.Google Scholar
  71. 71.
    I. V. Skrypnik, “Periods of A-closed forms,” Dokl. Akad. Nauk SSSR, 160, No. 4, 772–773 (1965).MathSciNetGoogle Scholar
  72. 72.
    I. V. Skrypnik, “Harmonic fields with singularities,” Ukr. Math. Zh., 17, No. 4, 130–133 (1965).MathSciNetCrossRefGoogle Scholar
  73. 73.
    I. V. Skrypnik, “Generalized de Rham theorem,” Dop. Akad. Nauk Ukr. SSR, 1, 18–19 (1965).MathSciNetGoogle Scholar
  74. 74.
    I. V. Skrypnik, “Harmonic forms on a compact Riemannian space,” Dop. Akad. Nauk Ukr. SSR, 2, 174–175 (1965).MathSciNetGoogle Scholar
  75. 75.
    M. Spivak, Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus, Addison-Wesley Publ. Co., Massachusetts (1965).zbMATHGoogle Scholar
  76. 76.
    M. A. Shubin, Pseudo-Differential Operators and Spectral Theory, Nauka, Moscow (1978).Google Scholar
  77. 77.
    R. Teleman, Elemente de Topologie si Varietati Diferentiabile, Bucuresti Publ. (1964).Google Scholar
  78. 78.
    Kh. Trimeche, Transmutation Operators and Mean-Periodic Functions Associated with Differential Operators, Harwood AP, London (1988).zbMATHGoogle Scholar
  79. 79.
    A. P. Veselov and S. P. Novikov, “Finite-gap two-dimensional Schrödinger operators. Potential operators,” Dokl. Akad. Nauk SSSR, 279, No. 4, 20–24 (1984).MathSciNetGoogle Scholar
  80. 80.
    F. Warner, Foundations of Differential Manifolds and Lie Groups, Academic Press, New York (1971).zbMATHGoogle Scholar
  81. 81.
    V. E. Zakharov and A. B. Shabat, “An exact theory of two-dimensional self-focusing and one-dimensional automodulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz., 61, No. 1, 118–134 (1971).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • A. M. Samoilenko
    • 1
  • Ya. A. Prykarpatsky
    • 1
    • 2
  • D. Blackmore
    • 3
  • A. K. Prykarpatsky
    • 4
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.University of Agriculture in KrakówKrakówPoland
  3. 3.New Jersey Institute of TechnologyNewarkUSA
  4. 4.Tadeusz Ko´sciuszko University of TechnologyKrakówPoland

Personalised recommendations