Communications in Mathematical Physics

, Volume 241, Issue 1, pp 111–142 | Cite as

Spectral Curve, Darboux Coordinates and Hamiltonian Structure of Periodic Dressing Chains

  • Kanehisa TakasakiEmail author


A chain of one-dimensional Schrödinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlevé equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.


Control Parameter Periodic Lattice Hamiltonian System Transition Matrix Poisson Structure 
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  1. 1.
    Adams, M.R., Harnad, J., Hurtubise, J.: Darboux coordinates and Liouville-Arnold integration in loop algebras. Commun. Math. Phys. 155, 385–413 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Adler, V.E.: Cutting of polygons. Funct. Anal. Appl. 27, 141–143Google Scholar
  3. 3.
    Adler, V.E.: Nonlinear chains and Painlevé equations. Phys. D73, 335–351 (1994)Google Scholar
  4. 4.
    Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. London Math. Soc. Ser. 2(21), 420–440 (1922)zbMATHGoogle Scholar
  5. 5.
    Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type. J. Soviet Math. 30, 1975–2036 (1985)zbMATHGoogle Scholar
  6. 6.
    Dubrovin, B.A.: Theta functions and non-linear equations. Russ. Math. Surv. 36(2), 11–92 (1981)zbMATHGoogle Scholar
  7. 7.
    Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Non-linear equations of Korteweg-de Vries type, finite-zone linear operators and Abelian varieties. Russ. Math. Surv. 31(1), 59–146 (1976)zbMATHGoogle Scholar
  8. 8.
    Flaschka, H., McLaughlin, D.W.: Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions. Prog. Theor. Phys. 55, 438–456 (1976)zbMATHGoogle Scholar
  9. 9.
    Garnier, R.: Sur des équations différentielles du troisième ordre dont l'intégrale est uniform et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale à ses point critiques fixés. Ann. Sci. École Norm. Sup. 29(3), 1–126 (1912)zbMATHGoogle Scholar
  10. 10.
    Harnad, J.: Dual isomonodromic deformations and moment maps to loop algebras. Commun. Math. Phys. 166, 337–365 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Harnad, J., Wisse, M.A.: Loop algebra moment maps and Hamiltonian models for the Painlevé transcendents. AMS-Fields Inst. Commun. 7, 155–169 (1996)zbMATHGoogle Scholar
  12. 12.
    Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Russ. Math. Surv. 32(6), 185–214 (1977)zbMATHGoogle Scholar
  13. 13.
    Lax, P.D.: Periodic solutions of Korteweg-de Vries equation. Comm. Pure. Appl. Math. 28, 141–188 (1975)zbMATHGoogle Scholar
  14. 14.
    McKean, M.P., van Moerbeke, P.: The spectrum of Hill's equation. Invent. Math. 30, 217–274 (1975)zbMATHGoogle Scholar
  15. 15.
    Moore, G.: Geometry of the string equations. Commun. Math. Phys. 133, 261–304 (1990); Matrix models of 2D gravity and isomonodromic deformations. Prog. Theor. Phys. Suppl. 102, 255–285 (1990)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Moser, J.: Geometry of quadrics and spectral theory. In: W.-Y. Hsiang et al. (eds.), The Chern Symposium 1979, Berlin-Heidelberg-New York: Springer-Verlag, 1980, pp. 147–188Google Scholar
  17. 17.
    Moser, J.: Finitely many mass points on the line under the influence of an exponential potential – an integrable system. In: J. Moser (ed.), Dynamical Systems, Theory and Applications, Lecture Notes in Physics, Vol. 38, Berlin-Heidelberg-New York: Springer-Verlag, 1975, pp. 467–497Google Scholar
  18. 18.
    Noumi, M., Yamada, Y.: Higher order Painlevé equations of type A (1) . Funkcial. Ekvac. 41, 483–503 (1998)Google Scholar
  19. 19.
    Noumi, M., Yamada, Y.: Affine Weyl group symmetries in Painlevé type equations. In: C.J. Howls, T. Kawai, Y. Takei (eds.), Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear, Kyoto: Kyoto University Press, 2000, pp. 245–259Google Scholar
  20. 20.
    Noumi, M., Yamada, Y.: A new Lax pair for the sixth Painlevé equation associated with \(\widehat{so(8)}\). arXiv e-print nlin.SI/0203029Google Scholar
  21. 21.
    Novikov, S.P.: Periodic problem for the Korteweg-de Vries equation I. Funct. Anal. Appl. 8, 236–246 (1974)zbMATHGoogle Scholar
  22. 22.
    Okamoto, K.: Isomonodromic deformation and Painlevé equation, and the Garnier system. J. Fac. Sci. Univ. Tokyo, Sec. IA, 33, 575–618 (1986)Google Scholar
  23. 23.
    Shabat, A.B.: The infinite-dimensional dressing dynamical system. Inverse Problems 6, 303–308 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Shabat, A.B., Yamilov, R.I.: Symmetries of nonlinear chains. Leningrad Math. J. 2, 377–400 (1991)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sklyanin, E.K.: Separation of variables – new trends. Prog. Theor. Phys. Suppl. 118, 35–60 (1995)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Veselov, A.P., Shabat, A.B.: Dressing chains and the spectral theory of the Schrödinger operator. Funct. Anal. Appl. 27, 81–96 (1993)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Willox, R., Hietarinta, J.: Painlevé equations from Darboux chains. arXiv preprint nlin.SI/0302012Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Fundamental Sciences, Faculty of Integrated Human StudiesKyoto UniversitySakyo-kuJapan

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