Ukrainian Mathematical Journal

, Volume 71, Issue 6, pp 921–955 | Cite as

Theory of Multidimensional Delsarte–Lions Transmutation Operators. II

  • A. M. Samoilenko
  • Ya. A. Prykarpatsky
  • D. Blackmore
  • A. K. PrykarpatskyEmail author

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand–Levitan–Marchenko equations that describe these operators are studied by using suitable differential de Rham–Hodge–Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties of the Delsarte transmutation operators is established. Some applications to multidimensional differential operators are presented, including the three-dimensional Laplace operator, the two-dimensional classical Dirac operator, and its multidimensional affine extension associated with self-dual Yang–Mills equations. The soliton solutions are discussed for a certain class of dynamical systems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Abraham and J. Marsden, Foundations of Mechanics, Cummings, New York (1978).zbMATHGoogle Scholar
  2. 2.
    V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York (1978).CrossRefGoogle Scholar
  3. 3.
    Yu. Berezansky, Expansion in Eigenfunction of Self-Adjoint Operators, American Mathematical Society, Providence, RI (1968).Google Scholar
  4. 4.
    F. A. Berezin and M. A. Shubin, The Schr¨odinger Equation, Springer (1991).Google Scholar
  5. 5.
    A. L. Bukhgeim, Volterra Equations and Inverse Problems [in Russian], Nauka, Moscow (1983).Google Scholar
  6. 6.
    S. Donaldson, “An application of gauge theory to four-dimensional topology,” J. Different. Geom., 18, 279–315 (1983).MathSciNetzbMATHGoogle Scholar
  7. 7.
    D. L. Blackmore, Ya. A. Prykarpatsky, and R. V. Samulyak, “The integrability of Lie-invariant geometric objects generated by ideals in Grassmann algebras,” J. Nonlin. Math. Phys., 5, 54–67 (1998).MathSciNetzbMATHGoogle Scholar
  8. 8.
    D. Blackmore, A. K. Prykarpatsky, and V. HR. Samoylenko, Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Differential-Geometrical Integrability Analysis, World Scientific Publishing, New Jersey (2011).zbMATHGoogle Scholar
  9. 9.
    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).MathSciNetzbMATHGoogle Scholar
  10. 10.
    N. N. Bogoliubov, Jr., Ya. A. Prykarpatsky, A. M. Samoilenko, and A. K. Prykarpatsky, “A generalized de Rham–Hodge theory of multidimensional Delsarte transformations of differential operators and its applications for nonlinear dynamic systems,” Phys. Particles Nuclei, 36, No. 1, 110–121 (2005).Google Scholar
  11. 11.
    E. Cartan, Lecons sur Invariants Integraux, Hermann (1971).Google Scholar
  12. 12.
    S. S. Chern, Complex Manifolds, Chicago Univ. Publ., Chicago (1956).zbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    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).Google Scholar
  15. 15.
    J. Delsarte, “Sur certaines transformations fonctionelles relative aux equations lineaires aux derives partielles du second ordre,” C. R. Acad. Sci. Paris, 206, 178–182 (1938).Google Scholar
  16. 16.
    J. Delsarte and J. L. Lions, “Transmutations d’operateurs differentielles dans le domain complexe,” Comment. Math. Helv., 52, 113–128 (1957).zbMATHGoogle Scholar
  17. 17.
    L. D. Faddeev, “Inverse problem in quantum scattering theory,” J. Soviet Math., 5, No. 3 (1976).Google Scholar
  18. 18.
    L. Faddeev and L. Takhtadjyan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (2007).Google Scholar
  19. 19.
    I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol. 2: Spaces of Fundamental and Generalized Functions, American Mathematical Society, Providence, RI (2016).Google Scholar
  20. 20.
    C. Godbillon, Geometrie Differentielle et Mechanique Analytique, Hermann, Paris (1969).zbMATHGoogle Scholar
  21. 21.
    I. C. Gokhberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, American Mathematical Society, Providence, RI (2004).Google Scholar
  22. 22.
    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
  23. 23.
    M. Gromov, Partial Differential Relations, Springer, New York (1986).zbMATHGoogle Scholar
  24. 24.
    C. H. Gu, “Generalized self-dual Yang–Mills flows, explicit solutions, and reductions,” Acta Appl. Math., 39, 349–360 (1995).MathSciNetzbMATHGoogle Scholar
  25. 25.
    O. Ya. Hentosh, M. M. Prytula, and A. K. Prykarpatsky, Differential-Geometric Integrability Fundamentals of Nonlinear Dynamical Systems on Functional Manifolds, Lviv University, Lviv (2006).Google Scholar
  26. 26.
    E. J. Hruslov, “Asymptotics of the solution of the Cauchy problem for the KdV equation with step-like initial data,” Math. USSR-Sb., 28, 229–248 (1976).Google Scholar
  27. 27.
    S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. 1,Wiley, New York (1963); Vol. 2,Wiley, New York (1969).Google Scholar
  28. 28.
    B. G. Konopelchenko, “On the integrable equations and degenerate dispersion laws in multidimensional spaces,” J. Phys. A: Math. Gen., 16, L311–L316 (1983).MathSciNetzbMATHGoogle Scholar
  29. 29.
    D. Levi, L. Pilloni, and P. M. Santini, “Backlund transformations for nonlinear evolution equations in (2 + 1)-dimensions,” Phys. Lett. A, 81, No. 8, 419–423 (1981).MathSciNetGoogle Scholar
  30. 30.
    W. Liu, Darboux Transformations for a Lax Integrable Systems in 2n-Dimensions, Preprint arXive: solve-int/9605002 v1 (1996).Google Scholar
  31. 31.
    Y. B. Lopatynski, “On harmonic fields on Riemannian manifolds,” Ukr. Mat. Zh., 2, No. 1, 56–60 (1950).MathSciNetGoogle Scholar
  32. 32.
    Ya. B. Lopatinskii, Introduction to the Contemporary Theory of Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1980).Google Scholar
  33. 33.
    J. D. Moore, Lectures on Seiberg–Witten Invariants, Springer, Berlin (2001).zbMATHGoogle Scholar
  34. 34.
    V. B. Matveev and M. I. Salle, Darboux–Backlund Transformations and Applications, Springer, New York (1993).Google Scholar
  35. 35.
    A. S. Mishchenko and A. T. Fomenko, Introduction to Differential Geometry and Topology, Moscow University, Moscow (1983).Google Scholar
  36. 36.
    V. A. Marchenko, Sturm–Liouville Operator and Applications, Birkh¨auser, Basel (1986).Google Scholar
  37. 37.
    Ya. V. Mykytiuk, “Factorization of Fredholmian operators,” Math. Stud. Proc. Lviv Math. Soc., 20, No. 2, 185–199 (2003).Google Scholar
  38. 38.
    Ya. V. Mykytiuk, “Factorization of Fredholmian operators in operator algebras,” Math. Stud. Proc. Lviv Math. Soc., 21, No. 1, 87–97 (2004).Google Scholar
  39. 39.
    A. C. Newell, Solitons in Mathematics and Physics, Society for Industrial and Applied Mathematics, Arizona (1985).zbMATHGoogle Scholar
  40. 40.
    R. G. Newton, Inverse Schr¨odinger Scattering in Three Dimensions, Springer, Berlin (1989).Google Scholar
  41. 41.
    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
  42. 42.
    L. P. Nizhnik, Inverse Scattering Problem for Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1991).Google Scholar
  43. 43.
    L. P. Nizhnik and M. D. Pochynaiko, “Integration of the nonlinear two-dimensional spatial Schr¨odinger equation by the inverseproblem method,” Funct. Anal. Appl., 16, No. 1, 66–69 (1982).Google Scholar
  44. 44.
    L. P. Nizhnik, “The inverse scattering problems for the hyperbolic equations and their applications to nonlinear integrable equations,” Rept. Math. Phys., 26, No. 2, 261–283 (1988).zbMATHGoogle Scholar
  45. 45.
    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 Academic Publishers, Dordrecht (1996), pp. 233–238.Google Scholar
  46. 46.
    J. C. C. Nimmo, Darboux Transformations from Reductions of the KP-Hierarchy, Preprint, University of Glasgow (2002).Google Scholar
  47. 47.
    S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Springer (1984).zbMATHGoogle Scholar
  48. 48.
    M. D. Pochynaiko and Yu. M. Sydorenko, “Integrating some (2 + 1)-dimensional integrable systems by methods of inverse scattering problem and binary Darboux transformations,” Mat. Stud., No. 20, 119–132 (2003).MathSciNetGoogle Scholar
  49. 49.
    A. K. Prykarpatsky and D. Blackmore, “Versal deformations of a Dirac type differential operator,” J. Nonlinear Math. Phys., 6, No. 3, 246–254 (1999).MathSciNetzbMATHGoogle Scholar
  50. 50.
    Y. A. Prykarpatsky, A. M. Samoilenko, and A. K. Prykarpatsky, “The multi-dimensional Delsarte transmutation operators, their differential-geometric structure and applications. Pt. 1,” Opuscula Math., 23, 71–80 (2003).zbMATHGoogle Scholar
  51. 51.
    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).MathSciNetzbMATHGoogle Scholar
  52. 52.
    A. M. Samoilenko, Ya. A. Prykarpatsky, D. Blackmore, and A. K. Prykarpatsky, “Theory of multidimensional Delsarte–Lions transmutation operators. I,” Ukr. Mat. Zh., 70, No. 12, 1660–1695 (2018); English translation: Ukr. Math. J., 70, No. 12, 1913–1952 (2019).MathSciNetzbMATHGoogle Scholar
  53. 53.
    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).MathSciNetzbMATHGoogle Scholar
  54. 54.
    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
  55. 55.
    Y. A. Prykarpatsky, A. M. Samoilenko, A. K. Prykarpatsky, and V. Hr. Samoylenko, The Delsarte–Darboux Type Binary Transformations and Their Differential-Geometric and Operator Structure, Preprint arXiv: math-ph/0403055 v 1 (2004).Google Scholar
  56. 56.
    A. K. Prykarpatsky and I. V. Mykytiuk, Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects, Kluwer Acad. Publ., Dordrecht (1998).zbMATHGoogle Scholar
  57. 57.
    G. de Rham, Varietes Differentielles, Hermann, Paris (1955).zbMATHGoogle Scholar
  58. 58.
    G. de Rham, “Sur la theorie des formes differentielles harmoniques,” Ann. Univ. Grenoble, 22, 135–152 (1946).MathSciNetzbMATHGoogle Scholar
  59. 59.
    I. V. Skrypnik, “Periods of A-closed forms,” Dokl. Akad. Nauk SSSR, 160, No. 4, 772–773 (1965).MathSciNetGoogle Scholar
  60. 60.
    I. V. Skrypnik, “Harmonic fields with singularities,” Ukr. Math. Zh., 17, No. 4, 130–133 (1965).MathSciNetGoogle Scholar
  61. 61.
    I. V. Skrypnik, “Generalized de Rham theorem,” Dop. Akad. Nauk Ukr. SSR, 1, 18–19 (1965).MathSciNetGoogle Scholar
  62. 62.
    I. V. Skrypnik, “Harmonic forms on a compact Riemannian space,” Dop. Akad. Nauk Ukr. SSR, 2, 174–175 (1965).MathSciNetGoogle Scholar
  63. 63.
    S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs (1956).zbMATHGoogle Scholar
  64. 64.
    R. Teleman, Elemente de Topologie si Varietati Diferentiabile, Bucuresti Publ. (1964).Google Scholar
  65. 65.
    F. Warner, Foundations of Differential Manifolds and Lie Groups, Academic Press, New York (1971).zbMATHGoogle Scholar
  66. 66.
    V. E. Zakharov and A. B. Shabat, “A scheme of integration of nonlinear equations of mathematical physics via the inverse scattering problem. I,” Funct. Anal. Appl., 8, No. 3, 226–235 (1974); 13, No. 3, 166–174 (1979).Google Scholar
  67. 67.
    V. E. Zakharov, “Integrable systems in multidimensional spaces,” in: Lecture Notes in Physics, 153 (1982), pp. 190–216.Google Scholar
  68. 68.
    V. E. Zakharov and S. V. Manakov, “On a generalization of the inverse scattering problem,” Theor. Math. Phys., 27, No. 3, 283–287 (1976).MathSciNetGoogle Scholar

Copyright information

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

Authors and Affiliations

  • A. M. Samoilenko
    • 1
  • Ya. A. Prykarpatsky
    • 2
    • 3
  • D. Blackmore
    • 4
  • A. K. Prykarpatsky
    • 5
    Email author
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  2. 2.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine
  3. 3.Agricultural University in KrakówKrakówPoland
  4. 4.New Jersey Institute of TechnologyNewarkUSA
  5. 5.Kościuszko University of TechnologyKrakówPoland

Personalised recommendations