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## On a Poisson structure on the space of Stokes matrices

Venue: | Internat. Math. Res. Notices 1999 |

Citations: | 19 - 0 self |

### Citations

502 |
Hamiltonian methods in the theory of solitons
- Faddeev, Takhtajan
- 1989
(Show Context)
Citation Context ...ransformation of the Hamiltonian system to the action-angle variables [ZF]. Further development of these ideas was very important for development of the Hamiltonian approach to the theory of solitons =-=[FT]-=- and for the creation of a quantum version of this theory. In the general theory of MPDE it remains essentially an open question to understand the Hamiltonian nature of the monodromy transform, i.e., ... |

377 |
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients.II,
- Jimbo, Miwa
- 1981
(Show Context)
Citation Context ...can be explicitly done in a very few cases, one can extract certain important information regarding the analytic properties of the solution; see more detailed discussions of these properties in [IN], =-=[JM]-=-, [JMU], [Si]. One can write the MPDE as a Hamiltonian system on the space of the skewsymmetric matrices V with the standard linear Poisson bracket for V = (vab) ∈ so(n): Indeed, the Lax equation (1.4... |

213 |
Geometry of 2D Topological Field Theories
- Dubrovin
(Show Context)
Citation Context ...trices. A complete and detailed description of the phenomenon can be found in [BJL1], [Si], [IN],[U]; here we will concentrate our attention on the particular operator Λ(z) = d V dz − U − z (see also =-=[D]-=-). Following [D] we define an admissible line for the system (1.1) as a line l through the origin on the z-plane such that Rez(u i − u j )|z∈l ̸= 0 ∀i ̸= j. We denote the half-lines where ψ is a fixed... |

79 | Poisson structure on moduli of flat connections on Riemann surfaces and the r-matrix
- Fock, Rosly
- 1999
(Show Context)
Citation Context ...his corresponds to taking, for every ui, the residue Ai ∈ gl(n, lC) with the natural Poisson bracket on gl(n, lC). The residues relative to different singular points commute. In other words (see [KS],=-=[FR]-=-,[A]) this corresponds to read the matrices Ai as residues of a flat connection (with values in the Lie algebra g = gl(n, lC)) on the Riemann surface with n + 1 punctures: A = n∑ Ai dλ λ − ui i=0 (in ... |

76 |
The Isomonodromic Deformation Method in the Theory of Painlevé Equations
- Novokshenov
- 1986
(Show Context)
Citation Context ...famous six Painlevé equations are known [G] to be of this type. MPDE were included in the framework of the general theory of integrable systems much later, at the end of 70s [ARS, FN1, JMU]; see also =-=[IN]-=-). Many authors were inspired by the parallelism between the technique of soliton theory based on the spectral transform and that of the MPDE theory based on the monodromy transform. Another issue of ... |

75 | Painlevé transcendents in two-dimensional topological field theory - Dubrovin |

71 |
Korteweg-de Vries equation is a completely integrable Hamiltonian system,
- Zakharov, Faddeev
- 1971
(Show Context)
Citation Context ...ecall that one of the first steps in soliton theory was understanding of the Hamiltonian nature of the spectral transform as the transformation of the Hamiltonian system to the action-angle variables =-=[ZF]-=-. Further development of these ideas was very important for development of the Hamiltonian approach to the theory of solitons [FT] and for the creation of a quantum version of this theory. In the gene... |

52 | Symplectic structure of the moduli space of flat connection on a Riemann surface - Alekseev, Malkin - 1995 |

47 | Sur quelques equations differentielles lineares du second ordre,” - Fuchs - 1905 |

45 | Nonlinear evolution equations and ordinary differential equations of Painleve type, - Ablowitz, Ramani, et al. - 1978 |

36 |
D.A.Lutz: Birkhoff Invariants and Stokes’ Multipliers for Meromorphic Linear
- Balser
- 1979
(Show Context)
Citation Context ...d this fact is known as Stokes phenomenon. The matrices connecting the solutions in different sectors are called Stokes matrices. A complete and detailed description of the phenomenon can be found in =-=[BJL1]-=-, [Si], [IN],[U]; here we will concentrate our attention on the particular operator Λ(z) = d V dz − U − z (see also [D]). Following [D] we define an admissible line for the system (1.1) as a line l th... |

33 |
D.A.Lutz: On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities
- Balser
- 1981
(Show Context)
Citation Context ...n singularities: dΦ dλ = n∑ Bi Φ, (2.1) λ − ui where and Bi = −Ei i=1 ( V + 1 2 1l ) , i = 1, . . .,n B∞ = V + 1 2 1l. Such a relation is well known in the domain of differential equations, see, e.g. =-=[BJL]-=-, [Sch]. Now we will briefly describe the monodromy data of the system (2.1). In this case uj is a Fuchsian singular points and, as in (1.2), the general solution near uj can be expressed as Φj(λ) = W... |

21 |
K.Ueno: Monodromy Preserving Deformations of Linear Ordinary Differential Equations with
- Jimbo
- 1981
(Show Context)
Citation Context ... explicitly done in a very few cases, one can extract certain important information regarding the analytic properties of the solution; see more detailed discussions of these properties in [IN], [JM], =-=[JMU]-=-, [Si]. One can write the MPDE as a Hamiltonian system on the space of the skewsymmetric matrices V with the standard linear Poisson bracket for V = (vab) ∈ so(n): Indeed, the Lax equation (1.4) can b... |

14 |
differential equations in the complex domain: problems of analytic continuation, volume 82 of Translations of Mathematical Monographs
- Linear
- 1990
(Show Context)
Citation Context ...act is known as Stokes phenomenon. The matrices connecting the solutions in different sectors are called Stokes matrices. A complete and detailed description of the phenomenon can be found in [BJL1], =-=[Si]-=-, [IN],[U]; here we will concentrate our attention on the particular operator Λ(z) = d V dz − U − z (see also [D]). Following [D] we define an admissible line for the system (1.1) as a line l through ... |

9 | Quantization of coset space σ-models coupled to two-dimensional gravity
- Korotkin, Samtleben
- 1996
(Show Context)
Citation Context ... (see formula (3.2) below). The resulting Poisson bracket does not depend on U since is involved in the Hamiltonian description of the isomonodromy deformations of the operator Λ(z). The technique of =-=[KS]-=- was important in the derivation of this main result of the present paper. We hope that this interesting new class of polynomial Poisson brackets and their quantization (cf. [R, Ha2]) deserves a furth... |

2 | A.C.Newell: Monodromy–and Spectrum–Preserving–Deformations - Flaschka - 1980 |

2 |
The inverse monodromy transform is a canonical transformation, Nonlinear problems: present and future
- Flaschka, Newell
- 1981
(Show Context)
Citation Context ...tand the Hamiltonian nature of the monodromy transform, i.e., of the map associating the monodromy data to the linear differential operator with rational coefficients. This question was formulated in =-=[FN2]-=- and solved in an example of a MPDE of a particular second order linear differential operator. However, the general algebraic properties of the arising class of Poisson brackets on the spaces of monod... |

1 |
M.Audin: Lectures on Integrable Systems and Gauge theory, In ”Gauge theory and symplectic geometry
- Dordrecht
- 1995
(Show Context)
Citation Context ...e authors of the papers [AM, FR, KS, Hi] consider the important case of MPDE of Fuchsian systems in a more general setting of symplectic structures on the moduli space of flat connections (see, e.g., =-=[A]-=-) not writing, however, the Poisson bracket on the space of monodromy data in a closed form. MPDE of non-Fuchsian operators and Poisson structure on their monodromy data were not considered in these p... |

1 | E.L.Ince: Ordinary differential equations, London–New York etc.,Longmans - Co - 1972 |

1 | N.Reshetikhin: The Knizhnik–Zamolodchikov System as a deformation of the Isomonodromy Problem - Phys - 1992 |