Hypergeometric solutions to the q-Painlev\'e equation of type A4(1)A_4^{(1)}

We consider the q-Painlev\'e equation of type $A_4^{(1)}$ (a version of q-Painlev\'e V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for the solutions.Comment: 16 pages, IOP styl

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP

Smooth rationally connected threefolds contain all smooth curves

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.Comment: Version 1 was called "Any smooth toric threefold contains all curves". This version is completely rewritten and proves a much stronger result, following suggestions of Janos Kolla

The discrete potential Boussinesq equation and its multisoliton solutions

An alternate form of discrete potential Boussinesq equation is proposed and its multisoliton solutions are constructed. An ultradiscrete potential Boussinesq equation is also obtained from the discrete potential Boussinesq equation using the ultradiscretization technique. The detail of the multisoliton solutions is discussed by using the reduction technique. The lattice potential Boussinesq equation derived by Nijhoff et al. is also investigated by using the singularity confinement test. The relation between the proposed alternate discrete potential Boussinesq equation and the lattice potential Boussinesq equation by Nijhoff et al. is clarified.Comment: 17 pages,To appear in Applicable Analysis, Special Issue of Continuous and Discrete Integrable System

Bilinear Discrete Painleve-II and its Particular Solutions

By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e 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 $\tau$ 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.Comment: 9 pages in plain Te

Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ on a lattice with spacing $h$, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the Yablonskii-Vorob'ev polynomials, to which they reduce when $h=0$. They also provide rational solutions for a particular discretisation of $P_{II}$, namely the so called {\it alternate discrete} $P_{II}$, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation ($P_{III}$). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete $P_{II}$, which recovers Jimbo and Miwa's Lax pair for $P_{II}$ in the continuum limit $h\to 0$.Comment: 23 pages, IOP style. Title changed, and connection with Umemura polynomials adde

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.Comment: 11 pages, LaTeX fil

A remark on the Hankel determinant formula for solutions of the Toda equation

We consider the Hankel determinant formula of the $\tau$ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the $\tau$ functions in the framework of the KP theory. Similar phenomena that have been observed for the Painlev\'e II and IV equations are recovered. The case of finite lattice is also discussed.Comment: 14 pages, IOP styl

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.Comment: 19 page