Advertisement

Theoretical and Mathematical Physics

, Volume 201, Issue 1, pp 1442–1456 | Cite as

Some Exact Solutions of the Volterra Lattice

  • V. E. AdlerEmail author
  • A. B. Shabat
Article
  • 2 Downloads

Abstract

We study solutions of the Volterra lattice satisfying the stationary equation for its nonautonomous symmetry. We show that the dynamics in t and n are governed by the respective continuous and discrete Painlevé equations and describe the class of initial data leading to regular solutions. For the lattice on the half-axis, we express these solutions in terms of the confluent hypergeometric function. We compute the Hankel transform of the coefficients of the corresponding Taylor series based on the Wronskian representation of the solution.

Keywords

Volterra lattice symmetry Painlevé equation confluent hypergeometric function Hankel transformation Catalan number 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Conflicts of interest. The authors declare no conflicts of interest.

References

  1. 1.
    A. R. Its, A. V. Kitaev, and A. S. Fokas, “The isomonodromy approach in the theory of two-dimensional quantum gravitation,” Russian Math. Surveys, 45, 155–157 (1990).ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    A. S. Fokas, A. R. Its, and A. V. Kitaev, “Discrete Painlevé equations and their appearance in quantum gravity,” Commun. Math. Phys., 142, 313–344 (1991).ADSCrossRefGoogle Scholar
  3. 3.
    V. E. Adler and A. B. Shabat, “Volterra chain and Catalan numbers,” JETP Lett., 108, 825–828 (2018).ADSCrossRefGoogle Scholar
  4. 4.
    A. N. Leznov, “On the complete integrability of a nonlinear system of partial differential equations in twodimensional space,” Theor. Math. Phys., 42, 225–229 (1980).CrossRefGoogle Scholar
  5. 5.
    M. Aigner, “Catalan-like numbers and determinants,” J. Combin. Theory Ser. A, 87, 33–51 (1999).MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. P. Stanley, Enumerative Combinatorics (Cambridge Stud. Adv. Math., vol. 62), vol. 2, Cambridge Univ. Press, Cambridge (1999).CrossRefGoogle Scholar
  7. 7.
    J. W. Layman, “The Hankel transform and some of its properties,” J. Integer Sequences, 4, Article 01.1.5 (2001).Google Scholar
  8. 8.
    I. Yu. Cherdantsev and R. I. Yamilov, “Master symmetries for differential-difference equations of the Volterra type,” Phys. D, 87, 140–144 (1995).MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. E. Adler, A. B. Shabat, and R. I. Yamilov, “Symmetry approach to the integrability problem,” Theor. Math. Phys., 125, 1603–1661 (2000).MathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Grammaticos and A. Ramani, “From continuous Painlevé IV to the asymmetric discrete Painlevé I,” J. Phys. A, 31, 5787–5798 (1998).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Grammaticos and A. Ramani, “Discrete Painlevé equations: An integrability paradigm,” Phys. Scr., 89, 038002 (2014).ADSCrossRefGoogle Scholar
  12. 12.
    M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Appl. Math. Ser., vol. 55, Dover, New York (1972).zbMATHGoogle Scholar
  13. 13.
    R. Ehrenborg, “The Hankel determinant of exponential polynomials,” Amer. Math. Monthly, 107, 557–560 (2000).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovka, Moscow OblastRussia

Personalised recommendations