Theoretical and Mathematical Physics

, Volume 179, Issue 3, pp 637–648 | Cite as

Rational interpolation and solitons

  • A. B. ShabatEmail author


We consider the general construction scheme for second-order spectral problems, for which the semiclassical approximation is exact. We show that the inverse spectral problem in this case reduces to the classical interpolation problem for meromorphic functions.


inverse spectral problem semiclassical approximation interpolation of meromorphic functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York (1976).zbMATHGoogle Scholar
  2. 2.
    A. B. Shabat, “On one interpolation problem,” Paper presented at the X International Scientific Conference “Order Analyis and Related Questions of Mathematical Modeling,” Vladivostok-Nov. Tsei, Russia, 14–20 July 2013.Google Scholar
  3. 3.
    A. B. Shabat, “Symmetries of Spectral Problems,” in: Integrability (Lect. Notes. Phys., Vol. 767, A. V. Mikhailov, ed.), Springer, Berlin (2009), pp. 139–173.CrossRefGoogle Scholar
  4. 4.
    V. É. Adler and A. B. Shabat, Theor. Math. Phys., 153, 1373–1387 (2007).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    L. Martínez Alonso and A. B. Shabat, Theor. Math. Phys., 140, 1073–1085 (2004).CrossRefzbMATHGoogle Scholar
  6. 6.
    B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Russ. Math. Surveys, 31, 59–146 (1976).ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    G. A. Baker Jr. and P. Graves-Morris, “Extensions of Padé approximants,” in: Padé Approximants (Encycl. Math. and Its Appl., Vol. 59), Cambridge Univ. Press, Cambridge (1996), pp. 335–414.CrossRefGoogle Scholar
  8. 8.
    J. Drach, Compt. Rend. Acad. Sci., 168, 337–340 (1919).zbMATHGoogle Scholar
  9. 9.
    A. B. Shabat, “On potentials with a zero reflection coefficient,” in: Dynamics of Condensed Matter, Vol. 5, Inst. of Hydrodynamics, Sib. Br., USSR Acad. Sci., Novosibirsk (1970), pp. 130–145.Google Scholar
  10. 10.
    V. É. Adler, “N-solition solution of the Korteweg-de Vries equation,” in: Asymptotic Methods of Mathematical Physics, BNTs, Urals Br., USSR Acad. Sci., Ufa (1989), pp. 3–8.Google Scholar
  11. 11.
    D. H. Peregrine, J. Austral. Math. Soc. Ser. B, 25, 16–43 (1983).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    V. E. Zakharov and A. A. Gelash, Phys. Rev. Lett., 111, 054101 (2013).ADSCrossRefGoogle Scholar
  13. 13.
    V. E. Zakharov and A. B. Shabat, Soviet Phys. JETP, 34, 62–69 (1972).ADSMathSciNetGoogle Scholar
  14. 14.
    V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP, 37, 823–828 (1973).ADSGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Aliev Karachaev-Cherkess State UniversityKarachaevskRussia
  2. 2.Landau Institute for Theoretical PhysicsRAS, ChernogolovkaMoscow OblastRussia

Personalised recommendations