Advertisement

Theoretical and Mathematical Physics

, Volume 122, Issue 1, pp 17–28 | Cite as

An elementary approach to the polynomial τ-functions of the KP Hierarchy

  • G. Falqui
  • F. Magri
  • M. Pedroni
  • J. P. Zubelli
Article

Abstract

We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.

Keywords

Riccati Equation Central System Pseudodifferential Operator Laurent Series Soliton Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. J. Ablowitz and J. Satsuma,J. Math. Phys.,19, 2180–2186 (1978).CrossRefADSMathSciNetGoogle Scholar
  2. 2.
    H. Airault, H. P. McKean, and J. Moser,Commun. Pure Appl. Math.,30, 95–148 (1977).ADSMathSciNetGoogle Scholar
  3. 3.
    F. Calogero,Nuovo Cimento B,43, 177–241 (1978).ADSMathSciNetGoogle Scholar
  4. 4.
    D. V. Chudnovsky and G. V. Chudnovsky,Nuovo Cimento B,40, 339–353 (1977).MathSciNetGoogle Scholar
  5. 5.
    I. M. Krichever,Funct. Anal. Appl.,12, 59–61, (1978).CrossRefGoogle Scholar
  6. 6.
    M. D. Kruskal, “The Korteweg-de Vries equation and related evolution equations,” in:Nonlinear Wave Motion (A. C. Newell, ed.) (Lect. Appl. Math., Vol. 15), Am. Math. Soc., Providence, RI (1974), pp. 61–83.Google Scholar
  7. 7.
    V. B. Matveev,Lett. Math. Phys.,3, 503–512 (1979).CrossRefMathSciNetGoogle Scholar
  8. 8.
    T. Shiota,J. Math. Phys.,35, 5844–5849 (1994).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    G. Wilson,Invent. Math.,133, 1–41 (1998).CrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold,” in:Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo) (P. Lax and H. Fujita, eds.), North-Holland, Amsterdam (1982), pp. 259–271.Google Scholar
  11. 11.
    Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro,Progr. Theor. Phys. Suppl.,94, 210–241 (1988).ADSMathSciNetGoogle Scholar
  12. 12.
    J. P. Zubelli,Lett. Math. Phys.,24, 41–48 (1992).CrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Wilson,J. Reine Angew. Math.,442, 177–204 (1993).MathSciNetGoogle Scholar
  14. 14.
    J. Harnad and A. Kasman, eds.,The Bispectral Problem (Montreal, PQ, 1997), Am. Math. Soc., Providence, RI (1998).zbMATHGoogle Scholar
  15. 15.
    G. Falqui, F. Magri, and M. Pedroni,Commun. Math. Phys.,197, 303–324 (1998).CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    K. Takasaki,Rev. Math. Phys.,1, 1–46 (1989).CrossRefMathSciNetGoogle Scholar
  17. 17.
    I. V. Cherednik,Funct. Anal. Appl.,12, 195–203 (1979).CrossRefGoogle Scholar
  18. 18.
    G. Wilson,Quart. J. Math. Oxford,32, 491–512 (1981).CrossRefGoogle Scholar
  19. 19.
    E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Transformation groups for soliton equations,” in:Proc. RIMS Symp. Nonlinear Integrable Systems: Classical Theory and Quantum Theory (M. Jimbo and T. Miwa, eds.), World Scientific, Singapore (1983), pp. 39–119.Google Scholar
  20. 20.
    L. A. Dickey,Soliton Equations and Hamiltonian Systems (Adv. Series in Math Phys, Vol. 12), World Scientific, Singapore (1991).zbMATHGoogle Scholar
  21. 21.
    P. Casati, G. Falqui, F. Magri, and M. Pedroni,J. Math. Phys.,38, 4606–4628 (1997).CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    M. F. de Groot, T. J. Hollowood, and J. L. Miramontes,Commun. Math. Phys.,145, 57–84 (1992).CrossRefADSGoogle Scholar
  23. 23.
    G. Falqui, F. Magri, and G. Tondo,Theor. Math. Phys. (forthcoming).Google Scholar
  24. 24.
    K. Takasaki, “Integrable systems as deformations ofD-modules”, in:Theta Functions—Bowdoin 1987 (L. Ehrenpreis and R. C. Gunning, eds.) (Proc. Symp. Pure Math., Vol. 49, Part 1), Am. Math. Soc., Providence, RI (1989), pp. 143–168.Google Scholar
  25. 25.
    M. Mulase, “Algebraic theory of the KP equations,” in:Perspectives in Mathematical Physics (R. Penner and S.-T. Yau, eds.), International Press, Boston (1994), pp. 151–217.Google Scholar
  26. 26.
    F. Magri, M. Pedroni, and J. P. Zubelli,Commun. Math. Phys.,188, 305–325 (1997).CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    W. T. Reid,Riccati Differential Equations, Acad. Press, New York (1972).zbMATHGoogle Scholar
  28. 28.
    V. G. Kac,Infinite Dimensional Lie Algebras (3rd ed.), Cambridge Univ. Press, Cambridge (1990).zbMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 2000

Authors and Affiliations

  • G. Falqui
    • 1
  • F. Magri
    • 2
  • M. Pedroni
    • 3
  • J. P. Zubelli
    • 4
  1. 1.SISSATriesteItaly
  2. 2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly
  3. 3.Dipartimento di MatematicaUniversità di GenovaGenovaItaly
  4. 4.IMPA-CNPqRio de JaneiroBrazil

Personalised recommendations