Rational solutions for the discrete Painlevé II equation

Kajiwara, Kenji; Yamamoto, Kazushi; Ohta, Yuichi

doi:10.1016/s0375-9601(97)00397-6

Cited by 23 publications

(29 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Typical examples of this class are obtained by applying Bäcklund transformations to the simple solutions characterized by the invariance with respect to the Dynkin diagram automorphisms. Many of such solutions are interpreted as simple specialization of the Schur functions or the universal characters [47,59,60,63,67,72,88,85,124].…”

Section: (1) Applying Bäcklund Transformations (Birational Transformamentioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

Self Cite

123

250

View full text Add to dashboard Cite

In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

show abstract

Section: (1) Applying Bäcklund Transformations (Birational Transformamentioning

confidence: 99%

Geometric aspects of Painlevé equations

Kajiwara¹,

Noumi²,

Yamada³

2017

J. Phys. A: Math. Theor.

Self Cite

123

250

View full text Add to dashboard Cite

show abstract

“…Note here that these phenomena cannot be seen for the algebraic (or rational) solutions. For example, it is known that substituting d = 0 into the determinant expression of the rational solutions to (1.4) yields those to (1.1); see [20,23,24]. The τ function is one of the most important objects in the theory of integrable systems and is regarded as carrying the underlying fundamental mathematical structures.…”

Section: Introductionmentioning

confidence: 99%

Projective Reduction of the Discrete Painleve System of Type (A2 + A1)(1)

Kajiwara

Nakazono

Tsuda

2010

International Mathematics Research Notices

View full text Add to dashboard Cite

We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type (A 2 + A 1 ) (1) . We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions.

show abstract

“…This multi-linear form (often referred to as a "bilinear" form) is particularly effective in the derivation of multisoliton solutions for integrable partial differential and difference equations. In the case of ordinary difference equations, the Hirota form has been used to obtain explicit solutions (see for example [12,13,18]) and to provide a natural framework for the derivation of auto-Bäcklund transformations [15].…”

Section: Introductionmentioning

confidence: 99%

Hirota bilinear formalism and ultra-discrete singularity analysis

2009

View full text Add to dashboard Cite

Ultra-discrete equations are equations in which both the dependent and independent variables can be restricted to take only integer values. In this paper, we show that it is possible to use singularity analysis to obtain the Hirota bilinear form of ultradiscrete versions of integrable equations. This method is applied to ultra-discrete Painlevé equations and to integrable ultra-discrete equations in 1 + 1 dimensions.

show abstract

Rational solutions for the discrete Painlevé II equation

Cited by 23 publications

References 18 publications

Geometric aspects of Painlevé equations

Geometric aspects of Painlevé equations

Projective Reduction of the Discrete Painleve System of Type (A2 + A1)(1)

Hirota bilinear formalism and ultra-discrete singularity analysis

Contact Info

Product

Resources

About