Searcharxiv⌕ Search

arXiv subjects

Nobutaka Nakazono

Publications and source records attributed to Nobutaka Nakazono.

At least 19 recordsLinked to original sources
arXiv2026

Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables

In this paper, we study special solutions of five autonomous integrable partial difference equations (P$Δ$Es). More precisely, we show that these P$Δ$Es admit special solutions that are described by non-autonomous ordinary difference equations arising from Bäcklund transformations of the third and sixth Painlevé equations and the Garnier system in two variables. This result provides a new perspective on the relationship between autonomous integrable P$Δ$Es and Painlevé-type dynamics.

nlin.SI↗
arXiv2026

Solutions to an autonomous discrete KdV equation via Painlevé-type ordinary difference equations

Hirota's discrete KdV equation is a well-known integrable two-dimensional partial difference equation regarded as a discrete analogue of the KdV equation. In this paper, we show that a variation of Hirota's discrete KdV equation with an additional parameter admits two types of exact solutions: discrete Painlevé transcendent solutions and periodic solutions described by Painlevé-type ordinary difference equations.

nlin.SI↗
arXiv2024

A higher-order generalization of an $A_4^{(1)}$-surface type $q$-Painlevé equation with $\widetilde{W}\left((A_{2N}\rtimes A_1)^{(1)}\times A_1^{(1)}\right)$ symmetry

Recently, a birational representation of an extended affine Weyl group of $(A_{2N}\rtimes A_1)^{(1)}$-type, which gives a higher-order generalization of an $A_4^{(1)}$-surface type $q$-Painlevé equation, was obtained. In this paper, we extend it to a birational representation of an extended affine Weyl group of $(A_{2N}\rtimes A_1)^{(1)}\times A_1^{(1)}$-type. Moreover, we provide conjectures on periodic reductions from systems of P$Δ$Es with the CAC property to Painlevé type $q$-O$Δ$Es.

nlin.SI↗
arXiv2022

Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property

The lattice sine-Gordon equation is an integrable partial difference equation on ${\mathbb Z}^2$, which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property.

nlin.SI↗
arXiv2021

On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries Equation

Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.

nlin.SI↗
arXiv2020

Classification of quad-equations on a cuboctahedron

In this paper, we consider polynomials associated with faces and internal quadrilaterals of a cuboctahedron and classify them under the requirement that they are consistent. These polynomials give rise to a system of partial difference equations on a face-centred cubic lattice. Our results were motivated by $τ$-functions related to discrete Painlevé equations.

nlin.SI↗
arXiv2020

Reduction of quad-equations consistent around a cuboctahedron I: additive case

In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)\ast}$-type discrete Painlevé equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.

nlin.SI↗
arXiv2019

A review of elliptic difference Painlevé equations

Discrete Painlevé equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of $n$. In this paper, we focus on the elliptic type and give a review of the construction of such equations on the $E_8$ lattice. The first such construction was given by Sakai \cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painlevé equations.

nlin.SI↗
arXiv2017

Geometric description of discrete power function associated with the sixth Painlevé equation

In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with $\widetilde{W}(3A_1^{(1)})$ symmetry. By constructing the action of $\widetilde{W}(3A_1^{(1)})$ as a subgroup of $\widetilde{W}(D_4^{(1)})$, i.e., the symmetry group of P$_{\rm VI}$, we show how to relate $\widetilde{W}(D_4^{(1)})$ to the symmetry group of the lattice. Moreover, by using translations in $\widetilde{W}(3A_1^{(1)})$, we explain the odd-even structure appearing in previously known explicit formulas in terms of the $τ$ function.

math-ph↗
arXiv2017

Elliptic Painlevé equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice

The well known elliptic discrete Painlevé equation of Sakai is constructed by a standard translation on the $E_8^{(1)}$ lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlevé equation obtained by translations along next-nearest-neighbor vectors. This equation is a generic (8-parameter) version of a 2-parameter elliptic difference equation found by reduction from Adler's partial difference equation, the so-called Q4 equation. We also provide a projective reduction of the well known equation of Sakai.

math-ph↗
arXiv2016

Lax pairs of discrete Painlevé equations: $(A_2+A_1)^{(1)}$ case

In this paper, we provide a comprehensive method for constructing Lax pairs of discrete Painlevé equations by using a reduced hypercube structure. In particular, we consider the $A_5^{(1)}$-surface $q$-Painlevé system which has the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$. Two new Lax pairs are found.

math-ph↗