Geometric aspects of Painlevé equations

Kajiwara, Kenji; Noumi, Masatoshi; Yamada, Yasuhiko

doi:10.1088/1751-8121/50/7/073001

Cited by 124 publications

(266 citation statements)

References 136 publications

(256 reference statements)

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“…This relativization can be understood as passing to a Lie group from its Lie algebra, and it can be more generally formulated in terms of cluster integrable systems [21,28]. On the side of isomonodromic deformations (at least in the first nontrivial example actually used throughout this paper) one has more difference Painlevé equations than differential ones, see classification in [39,51], and similar phenomenon takes place at the level of supersymmetric gauge theories.…”

Section: Jhep02(2018)077mentioning

confidence: 99%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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We discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.We also define quantum q-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using q-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

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Section: Jhep02(2018)077mentioning

confidence: 99%

Cluster integrable systems, q-Painlevé equations and their quantization

Bershtein

Gavrylenko

Marshakov

2018

J. High Energ. Phys.

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“…where g(τ ) is an arbitrary function to be determined from consistency with the initial condition. Substituting (21) into (14), we find that g(τ ) satisfies…”

Section: An Integrable Model For Soil Water Infiltrationmentioning

confidence: 99%

“…Nishinari-Takahashi considered the Burgers equation as a traffic model and constructed discrete and ultradiscrete integrable models, through which they gave a unified view to various continuous, discrete and cellular automaton traffic models [19]. For further recent developments in discrete integrable systems, see, for example, [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

et al. 2018

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We propose an integrable discrete model of one‐dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time‐dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self‐adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.

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“…The evolution equations (3.8) are equivalent to the q-Painlevé equation of type E (1) 6 given in [4,6,8,13,19]. The 8 singular points in coordinates ( f, g) are on the two lines f = ∞ and g = ∞ and one curve f g = 1 as follows:…”

Section: (B) Contiguity Relationsmentioning

confidence: 99%

The Padé interpolation method applied to q-Painlevé equations II (differential grid version)

Nagao

2016

Lett Math Phys

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Abstract. Recently we studied Padé interpolation problems of q-grid, related to q-Painlevé equations of type E (1) 7 , E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) . By solving those problems, we could derive evolution equations, scalar Lax pairs and determinant formulae of special solutions for the corresponding q-Painlevé equations. It is natural that the q-Painlevé equations were derived by the interpolation method of q-grid, but it may be interesting in terms of differential grid that the Padé interpolation method of differential grid (i.e. Padé approximation method) has been applied to the q-Painlevé equation of type D (1) 5 by Y. Ikawa. In this paper we continue the above study and apply the Padé approximation method to the q-Painlevé equations of type E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) . Moreover determinant formulae of the special solutions for q-Painlevé equation of type E (1) 6 are given in terms of the terminating q-Appell Lauricella function. IntroductionIn this paper we continue [13] and apply the Padé approximation method to the q-Painlevé equations of type E (1) 6 , D (1) 5 , A (1) 4 and (A 2 + A 1 ) (1) .1.1. The background of discrete Painlevé equations.In Sakai's theory [21] the discrete Painlevé equations were classified on the basis of rational surfaces connected to extended affine Weyl groups. There exist three types of discrete Painlevé equations in the classification: elliptic difference (e-), multiplicative difference (q-) and additive difference (d-). The discrete Painlevé equations of q-difference type are classified as follows:Here A → B means that B is obtained from A by degeneration. method to continuous Painlevé equations of type P VI , P V , P IV and Garnier system by using differential grid (i.e. Padé approximation).The Padé method for discrete Painlevé equations has been applied to the following types:( Remark 1. On the key points of the Padé methodThere are two key points to apply the Padé approximation/interpolation method [4,13,17,33]. The first key point is the appropriate choice of approximated/interpolated functions (see Table (2.2) and Remark 4). The second key point is to consider two linear q-difference three term relations (2.6) satisfied by the error terms of the Padé approximation/interpolation problems. Then the error terms can be expressed in terms of special solutions of q-Painlevé equations. Therefore the q-difference relations are the main subject in our study, and they naturally give the evolution equations, the Lax pairs and the special solutions for the corresponding q-Painlevé equations. Remark 2. On a connection between the Padé method and the theory of semiclassical orthogonal polynomialsThe connection between semiclassical orthogonal polynomials (classical orthogonal polynomials related to a suitable weight function) and Painlevé/Garnier systems has been demonstrated in [11]. It has been shown that coefficients of three term recurrence relations, satisfied by several semiclassical orthogonal polynomials, can be expressed in terms of sol...

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Geometric aspects of Painlevé equations

Cited by 124 publications

References 136 publications

Cluster integrable systems, q-Painlevé equations and their quantization

Cluster integrable systems, q-Painlevé equations and their quantization

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

The Padé interpolation method applied to q-Painlevé equations II (differential grid version)

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