arXiv subjects

Nobutaka Nakazono

Publications and source records attributed to Nobutaka Nakazono.

27 records · Page 2Linked to original sources

arXiv2016

Reflection groups and discrete integrable systems

We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the $n$-dimensional real Euclidean space $\mathbb{R}^n$. The "multi-dimensional consistency" property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics of the system is described by the affine translations of the polytopes on the weight lattices of the Weyl groups. The connections between some well-known discrete systems such as the multi-dimensional consistent systems of quad-equations \cite{abs:03} and discrete Painlevé equations \cite{sak:01} are obtained via the geometric constraints that relate the polytope of one symmetry group to that of another symmetry group, a procedure which we call geometric reduction.

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arXiv2016

Geometry of an elliptic-difference equation related to Q4

In this paper, we investigate a nonlinear non-autonomous elliptic difference equation, which was constructed by Ramani, Carstea and Grammaticos by integrable deautonomization of a periodic reduction of the discrete Krichever-Novikov equation, or Q4. We show how to construct it as a birational mapping on a rational surface blown up at eight points in $\mathbb P^1\times \mathbb P^1$, and find its affine Weyl symmetry, placing it in the geometric framework of the Painlevé equations. The initial value space is ell-$A_0^{(1)}$ and its symmetry group is $W(F_4^{(1)})$. We show that the deautonomization is consistent with the lattice-geometry of Q4 by giving an alternative construction, which is a reduction from Q4 in the usual sense. A more symmetric reduction of the same kind provides another example of a second-order integrable elliptic difference equation.

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arXiv2015

Lattice equations arising from discrete Painlevé systems (I): $(A_2+A_1)^{(1)}$ and $(A_1+A_1')^{(1)}$ cases

We introduce the concept of $ω$-lattice, constructed from $τ$ functions of Painlevé systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painlevé systems corresponding affine Weyl group symmetries are of $(A_2+A_1)^{(1)}$- and $(A_1+A_1)^{(1)}$-types, respectively.

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arXiv2015

A systematic approach to reductions of type-Q ABS equations

We present a class of reductions of Möbius type for the lattice equations known as Q1, Q2, and Q3 from the ABS list. The deautonomised form of one particular reduction of Q3 is shown to exist on the $A_1^{(1)}$ surface which belongs to the multiplicative type of rational surfaces in Sakai's classification of Painlevé systems. Using the growth of degrees of iterates, all other mappings that result from the class of reductions considered here are shown to be linearisable. Any possible linearisations are calculated explicitly by constructing a birational transformation defined by invariant curves in the blown up space of initial values for each reduction.

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arXiv2014

Geometric Reductions of ABS equations on an $n$-cube to discrete Painlevé systems

In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlevé equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlevé equation, which is a dynamical system on a rational surface of type $A_5^{(1)}$ with the extended affine Weyl group $\widetilde{\mathcal W}\bigl((A_2+A_1)^{(1)}\bigr)$. We provide general theorems to show that this reduction also extends to other discrete Painlevé equations at least of type A.

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arXiv2014

Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlevé IV Equation

We consider a $q$-Painlevé IV equation which is the $A_4^{(1)}$-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ${}_2φ_1$ basic hypergeometric series and the other is given by ${}_2ψ_2$ bilateral basic hypergeometric series.

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arXiv2013

Hypergeometric solutions to the symmetric q-Painlevé equations

We consider the symmetric q-Painlevé equations derived from the birational representation of affine Weyl groups by applying the projective reduction and construct the hypergeometric solutions. Moreover, we discuss continuous limits of the symmetric q-Painlevé equations to Painlevé equations together with their hypergeometric solutions.

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arXiv2010

Hypergeometric $τ$ Function of the $q$-Painlevé Systems of Type $(A_2+A_1)^{(1)}$

We consider a $q$-Painlevé III equation and a $q$-Painlevé II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $τ$ functions.

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arXiv2010

Projective reduction of the discrete Painlevé system of type $(A_2+A_1)^{(1)}$

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.

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