On the Algebro-Geometric Integration¶of the Schlesinger Equations

Deift, Percy; Its, Alexander; Kapaev, Andrei; Zhou, Xin

doi:10.1007/s002200050037

Cited by 48 publications

(56 citation statements)

References 11 publications

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“…The assertions of Garnier in [15] on his generalisation of the Fuchs' approach might form a starting point for extending this elliptic connection to the Garnier systems, in which case we should expect to be able to find a realisation of those systems in terms of hyperelliptic integrals rather than elliptic ones. This might eventually lead to the construction of algebraic solutions of those systems, possibly in the spirit of the recent papers [27,28]. It would be of interest to further investigate the role of the discrete systems in connection with the Garnier systems: we expect them to constitute the superposition formulae for the underlying higher root systems of the corresponding affine Weyl groups.…”

Section: ð3:2þmentioning

confidence: 97%

The discrete and continuous Painlevé VI hierarchy and the Garnier systems

Nijhoff¹,

Walker²

2001

Glasgow Math. J.

219

338

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Abstract. We present a general scheme to derive higher-order members of the Painleve´VI (P VI ) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the P VI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.1991 Mathematics Subject Classification. 34E99, 33E30, 58F07, 39A10.1. Introduction. In recent years there has been a growing interest in discrete analogues of the famous Painleve´equations; i.e. nonlinear nonautonomous ordinary difference equations tending to the continuous Painleve´equations in a well-defined limit and which are integrable in their own right; cf. In a recent paper [3] we established a connection between the continuous Painleve´VI (P VI ) equation and a non-autonomous ordinary difference equation depending on four arbitrary parameters. This novel example of a discrete Painleveé quation arises on the one hand as the nonlinear addition formula for the P VI transcendents, in fact what is effectively a superposition formula for its Ba¨cklund-Schlesinger transforms, on the other hand from the similarity reduction on the lattice (cf. [4,5]), of a system of partial difference equations associated with the lattice KdV family. In subsequent papers [6,7] some more results on these systems were established, namely the existence of the Miura chain and the discovery of a novel Schwarzian PDE generating the entire (Schwarzian) KdV hierarchy of nonlinear evolution equations and whose similarity reduction is exactly the P VI equation, this being to our knowledge the first example of an integrable scalar PDE that reduces to full P VI with arbitrary parameters.In the present note we extend these results to multidimensional systems associated with higher-order generalisations of the P VI equation. Already in [3] we noted that the similarity reduction of the lattice KdV system could be generalised in a natural way to higher-order differential and difference equations without, however, clarifying in detail the nature of such equations. We shall argue here that, in fact, such equations constitute what one could call the Painleve´VI hierarchy and its discrete counterpart. Whilst the idea of constructing hierarchies of Painleve´equations

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Section: ð3:2þmentioning

confidence: 97%

The discrete and continuous Painlevé VI hierarchy and the Garnier systems

Nijhoff¹,

Walker²

2001

Glasgow Math. J.

219

338

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“…Note that in [4,13], solutions in terms of theta functions are obtained. Now we consider the case Q = 0.…”

Section: (See Eq(38)) Then a Solution To Eq(341) Is Written As λmentioning

confidence: 99%

The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Takemura

2008

Math. Z.

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Abstract. Several results including integral representation of solutions and HermiteKrichever Ansatz on Heun's equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich-Verdier potential and additional apparent singularities of exponents −1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schrödinger operator on our potential. IntroductionIt is well known that a Fuchsian differential equation with three singularities is transformed to a Gauss hypergeometric equation, and plays important roles in substantial fields in mathematics and physics. Several properties of solutions to the hypergeometric equation have been explained in various textbooks.A canonical form of a Fuchsian equation with four singularities is written aswith the conditionand is called Heun's equation. Heun's equation frequently appears in physics, i.e., general relativity [21], fluid mechanics [3] and so on. Despite that Heun's equation was resolved in the 19th century; several results of solutions have only been recently revealed. Namely, integral representations of solutions, global monodromy in terms of hyperelliptic integrals, relationships with the theory of finite-gap potential and the Hermite-Krichever Ansatz for the case γ, δ, ǫ, α − β ∈ Z + 1/2 are contemporary (see [1,6,19,23,24,25,26,28] etc.), though they are not written in a textbook on Heun's equation [17].2000 Mathematics Subject Classification. 82B23,34M55,33E10,33E15.

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“…As a final remark on the theme of special classes of solutions to the P VI system, we note that the specification of the parameters (3.19) is a particular example which permits elliptic solutions [26,5,17]. More generally the latter occur when…”

Section: Jimbo-miwa-okamoto τ -Functions and Orthogonal Polynomialsmentioning

confidence: 99%

“…In the infinite system there are some celebrated instances of such evaluations. In particular Jimbo et al [22] related the problem of evaluating ρ ∞ (x; 0) to integrable systems theory, and consequently were able to derive the formula 5) where σ V satisfies the non-linear equation…”

Section: Introductionmentioning

confidence: 99%

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

Forrester

Frankel²,

Garoni³

et al. 2003

Commun. Math. Phys.

112

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Abstract:The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

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On the Algebro-Geometric Integration¶of the Schlesinger Equations

Cited by 48 publications

References 11 publications

The discrete and continuous Painlevé VI hierarchy and the Garnier systems

The discrete and continuous Painlevé VI hierarchy and the Garnier systems

The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

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