arXiv subjects

K. Kajiwara

Publications and source records attributed to K. Kajiwara.

7 recordsLinked to original sources

arXiv2004

Point configurations, Cremona transformations and the elliptic difference Painlevé equation

A theoretical foundation for a generalization of the elliptic difference Painlevé equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl group is proposed as a group of Cremona transformations containing elliptic functions in the coefficients. For this elliptic Cremona system, a theory of $τ$-functions is developed to translate it into a system of bilinear equations of Hirota-Miwa type for the $τ$-functions on the lattice.

nlin.SI↗

arXiv1999

On the Umemura Polynomials for the Painlevé III equation

A determinant expression for the rational solutions of the Painlevé III (P$_{\rm III}$) equation whose entries are the Laguerre polynomials is given. Degeneration of this determinant expression to that for the rational solutions of P$_{\rm II}$ is discussed by applying the coalescence procedure.

solv-int↗

arXiv1999

A generalization of determinant formulas for the solutions of Painlevé II and XXXIV equations

A generalization of determinant formulas for the classical solutions of Painlevé XXXIV and Painlevé II equations are constructed using the technique of Darboux transformation and Hirota's bilinear formalism. It is shown that the solutions admit determinant formulas even for the transcendental case.

solv-int↗

arXiv1997

Two-dimensional soliton cellular automaton of deautonomized Toda-type

A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed.

solv-int↗

arXiv1996

Bilinearization of Discrete Soliton Equations and Singularity Confinement

Bilinear forms for some nonlinear partial difference equations(discrete soliton equations) are derived based on the results of singularity confinement. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.

solv-int↗

arXiv1994

Bilinear Discrete Painleve-II and its Particular Solutions

By analogy to the continuous Painlevé II equation, we present particular solutions of the discrete Painlevé II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the $τ$ function for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained and we show that they can be related by a simple gauge transformation.

solv-int↗

arXiv1994

Bilinear Structure and Exact Solutions of the Discrete Painlevé I Equation

Bilinear structure for the discrete Painlevé I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painlevé I equation is naturally extended to a discrete coupled system. Corresponding matrix model is also mentioned.

solv-int↗