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Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions

Original Paper

Abstract

We show that a system of Hirota bilinear equations introduced by Jimbo and Miwa defines tau-functions of the modified KP (MKP) hierarchy of evolution equations introduced by Dickey. Some other equivalent definitions of the MKP hierarchy are established. All polynomial tau-functions of the KP and the MKP hierarchies are found. Similar results are obtained for the reduced KP and MKP hierarchies.

Keywords and phrases

KP and MKP hierarchies tau-functions wave functions formal pseudodifferential operators Schur polynomials 

Mathematics Subject Classification (2010)

14M15 17B10 17B65 17B67 20G43 22E70 35Q53 35R03 47G30 

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References

  1. [1]
    M. Adler and P. van Moerbeke, Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials, Duke Math. J., 80 (1995), 863–911.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    M. Adler and P. van Moerbeke, Vertex operator solutions to the discrete KP-hierarchy, Comm. Math. Phys., 203 (1999), 185–210.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    L.-L. Chau, J.C. Shaw and H.C. Yen, Solving the KP hierarchy by gauge transformations, Comm. Math. Phys., 149 (1992), 263–278.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M.M. Crum, Associated Sturm–Liouville systems, Quart. J. Math. Oxford Ser. (2), 6 (1955), 121–127.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations, In: Nonlinear Integrable Systems—Classical Theory and Quantum Theory, World Sci. Publ., River Edge, NJ, 1983, pp. 39–119.Google Scholar
  6. [6]
    L.A. Dickey, Modified KP and discrete KP, Lett. Math. Phys., 48 (1999), 277–289.MathSciNetCrossRefGoogle Scholar
  7. [7]
    L.A. Dickey, Soliton Equations and Hamiltonian Systems. Second ed., Adv. Ser. Math. Phys., 26, World Sci. Publ., River Edge, NJ, 2003.CrossRefMATHGoogle Scholar
  8. [8]
    A. Givental, An-1 singularities and nKdV hierarchies, Mosc. Math. J., 3 (2003), 475–505.MathSciNetMATHGoogle Scholar
  9. [9]
    G.F. Helminck and J.W. van de Leur, Geometric Bäcklund–Darboux transformations for the KP hierarchy, Publ. Res. Inst. Math. Sci., 37 (2001), 479–519.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Jimbo and T. Miwa, Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci., 19 (1983), 943–1001.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    V.G. Kac and D.H. Peterson, Lectures on the infinite wedge-representation and the MKP hierarchy, In: Systèmes dynamiques non linéaires: intégrabilité et comportement qualitatif, Sém. Math. Sup., 102, Presses Univ. Montréal, Montreal, QC, 1986, pp. 141–184.Google Scholar
  12. [12]
    V.G. Kac and J.W. van de Leur, The geometry of spinors and the multicomponent BKP and DKP hierarchies, In: The Bispectral Problem, Montreal, PQ, 1997, CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, RI, 1998, pp. 159–202.Google Scholar
  13. [13]
    V.G. Kac and J.W. van de Leur, The n-component KP hierarchy and representation theory, J. Math. Phys., 44 (2003), Special Issue: Integrability, Topological Solitons and Beyond, 3245–3293.Google Scholar
  14. [14]
    V.B. Matveev and M.A. Salle, Darboux Transformations and Solitons, Springer Ser. Nonlinear Dynam., Springer-Verlag, 1991.CrossRefMATHGoogle Scholar
  15. [15]
    M. Sato, Soliton equations as dynamical systems on a infinite dimensional Grassmann manifold, RIMS Kôkyûroku, 439 (1981), 30–46.Google Scholar
  16. [16]
    T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math., 83 (1986), 333–382.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    V.V. Sokolov and A.B. Shabat, (L,A)-pairs and Ricatti type substitution, Funct. Anal. Appl., 14 (1980), 148–150.CrossRefMATHGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Mathematical InstituteUtrecht UniversityUtrechtThe Netherlands

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