Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 042, 32 pages      nlin.SI/0610073      https://doi.org/10.3842/SIGMA.2007.042
Contribution to the Vadim Kuznetsov Memorial Issue

Hamiltonian Structure of PI Hierarchy

Kanehisa Takasaki
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida, Sakyo, Kyoto 606-8501, Japan

Received November 01, 2006, in final form February 13, 2007; Published online March 09, 2007

Abstract
The string equation of type (2,2g+1) may be thought of as a higher order analogue of the first Painlevé equation that corresponds to the case of g = 1. For g > 1, this equation is accompanied with a finite set of commuting isomonodromic deformations, and they altogether form a hierarchy called the PI hierarchy. This hierarchy gives an isomonodromic analogue of the well known Mumford system. The Hamiltonian structure of the Lax equations can be formulated by the same Poisson structure as the Mumford system. A set of Darboux coordinates, which have been used for the Mumford system, can be introduced in this hierarchy as well. The equations of motion in these Darboux coordinates turn out to take a Hamiltonian form, but the Hamiltonians are different from the Hamiltonians of the Lax equations (except for the lowest one that corresponds to the string equation itself).

Key words: Painlevé equations; KdV hierarchy; isomonodromic deformations; Hamiltonian structure; Darboux coordinates.

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References

  1. Brezin E., Kazakov V.A., Exactly solvable field theories of closed strings, Phys. Lett. B 236 (1990), 144-150.
  2. Douglas M.R., Shenker S.H., Strings in less than one-dimension, Nucl. Phys. B 335 (1990), 635-654.
  3. Gross D.J., Migdal A.A., Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990), 127-130.
  4. Douglas M., Strings in less than one-dimension and the generalized K-dV hierarchies, Phys. Lett. B 238 (1990), 176-180.
  5. Moore G., Geometry of the string equations, Comm. Math. Phys. 133 (1990), 261-304.
    Moore G., Matrix models of 2D gravity and isomonodromic deformation, Progr. Theoret. Phys. Suppl. 102 (1990), 255-285.
  6. Fukuma M., Kawai H., Nakayama R., Infinite dimensional Grassmannian structure of two dimensional string theory, Comm. Math. Phys. 143 (1991), 371-403.
  7. Kac V., Schwarz A., Geometric interpretation of partition function of 2D gravity, Phys. Lett. B 257 (1991), 329-334.
  8. Schwarz A., On some mathematical problems of 2D-gravity and Wh-gravity, Modern Phys. Lett. A 6 (1991), 611-616.
    Schwarz A., On solutions to the string equations, Modern Phys. Lett. A 6 (1991), 2713-2726, hep-th/9109015.
  9. Adler M., van Moerbeke P., A matrix integral solution to two-dimensional Wp-gravity, Comm. Math. Phys. 147 (1992), 25-56.
  10. Burchnall J.L., Chaundy T.W., Commutative ordinary differential operators, Proc. London Math. Soc. 21 (1992), 420-440, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 118 (1928), 557-583, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 134 (1931), 471-485.
  11. Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear equations of Korteweg-de Vries type, finite-zone linear operators and Abelian varieties, Russian Math. Surveys 31 (1976), no. 1, 59-146.
  12. Krichever I.M., Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys 32:6 (1977), 185-214.
  13. Sato M., Sato Y., Soliton equations as dynamical systems on an infinite dimensional Grassmannian manifold, in Nonlinear Partial Differential Equations in Applied Science (1982, Tokyo), Editors H. Fujita, P.D. Lax and G. Strang, North-Holland Math. Stud. 81 (1983), 259-271.
  14. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations, in Nonlinear Integrable Systems - Classical Theory and Quantum Theory, Editors M. Jimbo and T. Miwa, Singapore, World Scientific, 1983, 39-119.
  15. Segala G.B., Wilson G., Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Études Sci. 61 (1985), 5-65.
  16. Orlov A.Yu., Schulman E.I., Additional symmetries for integrable equations and conformal algebra representation, Lett. Math. Phys. 12 (1986), 171-179.
  17. Mumford D., Tata lectures on theta II, Birkhäuser, Boston, 1984.
  18. Beauville A., Jacobiennes des courbes spectrale et systémes hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235.
  19. Nakayashiki A., Smirnov F.A., Cohomologies of affine Jacobi varieties and integrable systems, Comm. Math. Phys. 217 2001, 623-652, math-ph/0001017.
  20. Eilbeck J.C., Enol'skii V.Z., Kuznetsov V.B., Tsiganov A.V., Linear r-matrix algebra for classical separable systems, J. Phys. A: Math. Gen. 27 1994, 567-578, hep-th/9306155.
  21. Magnano G, Magri F., Poisson-Nijenhuis structures and Sato hierarchy, Rev. Math. Phys. 3 (1991), 403-466.
  22. Falqui G., Magri F., Pedroni M., Zubelli J.P., A bi-Hamiltonian theory of stationary KdV flows and their separability, Regul. Chaotic Dyn. 5 (2000), 33-51, nlin.SI/0003020.
  23. Jimbo M., Miwa T., Môri Y., Sato M., Density matrix of an impenetrable Bose gas and the fifth Painlevé equations, Phys. D 1 (1980), 80-158.
  24. Flaschka H., McLaughlin D.W., Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys. 55 (1976), 438-456.
  25. Novikov S.P., Veselov A.P., Poisson brackets and complex tori, Tr. Mat. Inst. Steklova 165 (1984), 49-61.
  26. Adams M.R., Harnad J., Hurtubise J., Darboux coordinates and Liouville-Arnold integration in loop algebras, Comm. Math. Phys. 155 (1993), 385-413, hep-th/9210089.
  27. Sklyanin E.K., Separation of variables. New trends, Progr. Theoret. Phys. Suppl. 118 (1995), 35-60, solv-int/9504001.
  28. Harnad J., Dual isomonodromic deformations and moment maps to loop algebras, Comm. Math. Phys. 166 (1994), 337-365, hep-th/9301076.
  29. Harnad J., Wisse M.A., Loop algebra moment maps and Hamiltonian models for the Painlevé transcendents, Fields Inst. Commun. 7 (1996), 155-169, hep-th/9305027.
  30. Takasaki K., Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains, Comm. Math. Phys. 241 (2003), 111-142, nlin.SI/0206049.
  31. Dubrovin B., Mazzocco M., Canonical structure and symmetries of the Schlesinger equations, Comm. Math. Phys., to appear, math.DG/0311261.
  32. Okamoto K., Isomonodromic deformations and Painlevé equations, and the Garnier system, J. Math. Sci. Univ. Tokyo 33 (1986), 575-618.
  33. Kimura H., The degeneration of the two dimensional Garnier system and the polynomial Hamiltonian structure, Ann. Mat. Pura Appl. 155 1989, 25-74.
  34. Shimomura S., Painlevé property of a degenerate Garnier system of (9/2)-type and of a certain fourth order non-linear ordinary differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 1-17.
  35. Gelfand I.M., Dikii L.A., Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Russian Math. Surveys 33 (1975), no. 5, 77-113.
    Gelfand I.M., Dikii L.A., Fractional powers of operators and Hamiltonian systems, Funct. Anal. Appl. 10 (1976), 259-273.
    Gelfand I.M., Dikii L.A., The resolvent and Hamiltonian systems, Funct. Anal. Appl. 11 (1977), 93-105.
  36. Manin Yu.I., Algebraic aspects of non-linear differential equations, J. Soviet Math. 11 (1979), 1-122.
  37. Adler M., On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries equations, Invent. Math. 50 (1979), 219-248.
  38. Wilson G., Commuting flows and conservation laws, Math. Proc. Cambridge Philos. Soc. 86 (1979), 131-143.
  39. Ablowitz M.J., Ramani A., Segur H., A connection between nonlinear evolution equations adn ordinary differential equations of of P-type, J. Math. Phys. 21 (1980), 1006-1015.
  40. Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations I, Comm. Math. Phys. 76 (1980), 65-116.
  41. Mazzocco M., Mo M.-Y., The Hamiltonian structure of the second Painlevé hierarchy, nlin.SI/0610066.


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