Hamiltonian structure of isomonodromic deformation dynamics in linear systems of PDE’s. (English) Zbl 07919719
Kielanowski, Piotr (ed.) et al., Geometric methods in physics XL, workshop, Białowieża, Poland, June 20–25, 2023. Cham: Birkhäuser. Trends Math., 349-366 (2024).
Summary: The Hamiltonian approach to isomonodromic deformation systems for generic rational covariant derivative operators on the Riemann sphere having any matrix dimension \(r\) and any number of isolated singularities of arbitrary Poincaré rank is derived using the split classical rational \(R\)-matrix Poisson bracket structure on the dual space \(L^*\mathfrak{gl}(r)\) of the loop algebra \(L\mathfrak{gl}(r)\). Nonautonomous isomonodromic counterparts of isospectral systems are obtained by identifying the deformation parameters as Casimir elements on the phase space. These are shown to coincide with the higher Birkhoff invariants determining the local asymptotics near to irregular singular points, together with the pole loci. They appear as the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve, while the corresponding dual spectral invariant Hamiltonians appear as the “mirror image” positive power terms. Infinitesimal isomonodromic deformations are generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation.
For the entire collection see [Zbl 07882604].
For the entire collection see [Zbl 07882604].
MSC:
| 34M56 | Isomonodromic deformations for ordinary differential equations in the complex domain |
| 37Kxx | Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems |
| 37Jxx | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 14Dxx | Families, fibrations in algebraic geometry |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 32Sxx | Complex singularities |
References:
| [1] | Adams, M.R., Harnad, J., Hurtubise, J.: Darboux coordinates and Liouville-Arnol’d integration in loop algebras. Comm. Math. Phys. 155(2), 385-413 (1993). http://projecteuclid.org/euclid.cmp/1104253285 · Zbl 0791.58047 |
| [2] | Balser, W., Jurkat, W.B., Lutz, D.A.: Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations. J. Math. Anal. Appl. 71(1), 48-94 (1979). https://doi.org/10.1016/0022-247X(79)90217-8 · Zbl 0415.34008 |
| [3] | Bertola, M., Harnad, J., Hurtubise, J.: Hamiltonian structure of rational isomonodromic deformation systems. Journal of Mathematical Physics 64(8), 083502 (2023). https://doi.org/10.1063/5.0142532 · Zbl 1520.37042 |
| [4] | Birkhoff, G.D.: The generalized riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations. Proceedings of the American Academy of Arts and Sciences 49(9), 521-568 (1913). http://www.jstor.org/stable/20025482 |
| [5] | Boalch, P.: Symplectic manifolds and isomonodromic deformations. Adv. Math. 163(2), 137-205 (2001). https://doi.org/10.1006/aima.2001.1998 · Zbl 1001.53059 |
| [6] | Boalch, P.: Quasi-Hamiltonian geometry of meromorphic connections. Duke Math. J. 139(2), 369-405 (2007). https://doi.org/10.1215/S0012-7094-07-13924-3 · Zbl 1126.53055 |
| [7] | Flaschka, H., Newell, A.C.: Monodromy- and spectrum-preserving deformations. I. Comm. Math. Phys. 76(1), 65-116 (1980). http://projecteuclid.org/euclid.cmp/1103908189 · Zbl 0439.34005 |
| [8] | Flaschka, H., Newell, A.C.: The inverse monodromy transform is a canonical transformation. In: Nonlinear problems: present and future (Los Alamos, N.M., 1981), North-Holland Math. Stud., vol. 61, pp. 65-89. North-Holland, Amsterdam-New York (1982) · Zbl 0555.35107 |
| [9] | Fuchs, R.: Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63(3), 301-321 (1907). https://doi.org/10.1007/BF01449199 · JFM 38.0362.01 |
| [10] | Gaiur, I., Mazzocco, M., Rubtsov, V.: Isomonodromic deformations: confluence, reduction and quantisation. Comm. Math. Phys. 400(2), 1385-1461 (2023). https://doi.org/10.1007/s00220-023-04650-8 · Zbl 1517.32031 |
| [11] | Gambier, B.: Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33(1), 1-55 (1910). https://doi.org/10.1007/BF02393211 · JFM 40.0377.02 |
| [12] | Garnier, R.: Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Ann. Sci. École Norm. Sup. (3) 29, 1-126 (1912). http://www.numdam.org/item?id=ASENS_1912_3_29__1_0 · JFM 43.0382.01 |
| [13] | Harnad, J.: Dual isomonodromic deformations and moment maps to loop algebras. Comm. Math. Phys. 166(2), 337-365 (1994). http://projecteuclid.org/euclid.cmp/1104271613 · Zbl 0822.58018 |
| [14] | Harnad, J.: Hamiltonian theory of the general rational isomonodromic deformation problem. presentation at Fields Institute workshop on integrable and near-integrable Hamiltonian systems, May 17-21, 2004 (2004). http://www.fields.utoronto.ca/audio/03-04/integrable/harnad/ |
| [15] | Harnad, J., Balogh, F.: Tau functions and their applications. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2021). https://doi.org/10.1017/9781108610902 · Zbl 1482.37001 |
| [16] | Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and \(\tau \)-function. Physica D 2(2), 306-352 (1981). https://doi.org/10.1016/0167-2789(81)90013-0 · Zbl 1194.34167 |
| [17] | Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D 2(3), 407-448 (1981). https://doi.org/10.1016/0167-2789(81)90021-X · Zbl 1194.34166 |
| [18] | Malmquist, J.: Sur les équations différentielles du second ordre dont l’intégrale général a ses points critiques fixes. Ark. Mat. Astr. Fys. 17, 1-89 (1922-1923) · JFM 49.0305.02 |
| [19] | Marchal, O., Orantin, N., Alameddine, M.: Hamiltonian representation of isomonodromic deformations of general rational connections on \(\mathfrak{gl}_2(\mathbb{C}) (2023)\). [math-ph/2212.04833] |
| [20] | Marchal, O., Alameddine, M.: Isomonodromic and isospectral deformations of meromorphic connections: the \(\mathfrak{sl}_2(\mathbb{C})\) case (2023). [math-ph/2306.07378] |
| [21] | Mazzocco, M., Mo, M.Y.: The Hamiltonian structure of the second Painlevé hierarchy. Nonlinearity 20(12), 2845-2882 (2007). https://doi.org/10.1088/0951-7715/20/12/006 · Zbl 1156.34079 |
| [22] | Okamoto, K.: Polynomial Hamiltonians associated with Painlevé equations. I. Proc. Japan Acad. Ser. A Math. Sci. 56(6), 264-268 (1980). http://projecteuclid.org/euclid.pja/1195516808 · Zbl 0476.34010 |
| [23] | Painlevé, P.: Mémoire sur les équations différentielles dont l’intégrale générale est uniforme. Bull. Soc. Math. France 28, 201-261 (1900). http://www.numdam.org/item?id=BSMF_1900__28__201_0 · JFM 31.0337.03 |
| [24] | Painlevé, P.: Sur les équations différentielles du second ordre aux points critiques fixes. C. R. Acad. Sc. Paris 143, 1111-1117 (1906) · JFM 37.0341.04 |
| [25] | Picard, E.: Mémoire sur la théorie des functions algébriques de deux variable. J. Liouville pp. 135-319 (1989) · JFM 21.0775.01 |
| [26] | Schlesinger, L.: Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten. J. Reine Angew. Math. 141, 96-145 (1912). https://doi.org/10.1515/crll.1912.141.96 · JFM 43.0385.01 |
| [27] | Semenov-Tyan-Shanskii, M.A.: What is a classical r-matrix? Functional Analysis and Its Applications 17(4), 259-272 (1983). https://doi.org/10.1007/BF01076717 · Zbl 0535.58031 |
| [28] | Yamakawa, D.: Tau functions and Hamiltonians of isomonodromic deformations. Josai Math. Monog. p. 139160 (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
