Related papers: A Lax Formalism for the Elliptic Difference Painle…
An interpolation problem related to the elliptic Painlev\'e equation is formulated and solved. A simple form of the elliptic Painlev\'e equation and the Lax pair are obtained. Explicit determinant formulae of special solutions are also…
All $q$-Painlev\'e equations which are obtained from the $q$-analog of the sixth Painlev\'e equation are expressed in a Lax formalism. They are characterized by the data of the associated linear $q$-difference equations. The degeneration…
Discrete Painlev\'e 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…
A $\tau$ function formalism for Sakai's elliptic Painlev'e equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta-Ramani-Grammaticos. We also give a simple geometric description of the…
We construct the elliptic Painlev\'e equation and its higher dimensional analogs as the action of line bundles on 1-dimensional sheaves on noncommutative surfaces.
An explicit form of the Lax pair for the q-difference Painleve equation with affine Weyl group symmetry of type E^{(1)}_8 is obtained. Its degeneration to E^{(1)}_7, E^{(1)}_6 and D^{(1)}_5 cases are also given.
An algebro-geometric setting for the study of the Painlev\'e VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two.…
The solutions of the discrete Painlev\'e equation I were constructed in terms of elliptic and hyperelliptic $\psi$ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference…
It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of…
The 8-parameter elliptic Sakai difference Painlev\'e equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schr\"odinger equation for the $BC_1$ 8-parameter…
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev\'e equations, with a particular emphasis on the discrete Painlev\'e equations. The theory is controlled by the…
A Lax pair for the additive difference Painlev\'e equation of type $E_7^{(1)}$ is explicitly obtained as certain linear difference equations of scalar form. The compatibility of the Lax pair is proved by using certain characterization of…
Considering a certain interpolation problem, we derive a series of elliptic difference isomonodromic systems together with their Lax forms. These systems give a multivariate extension of the elliptic Painlev\'e equation.
A discretization of Painlev\'e VI equation was obtained by Jimbo and Sakai in 1996. There are two ways to quantize it: 1) use the affine Weyl group symmetry (of $D_5^{(1)}$) (Hasegawa, 2011), 2) Lax formalism i.e. monodromy preserving point…
The well known elliptic discrete Painlev\'e 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 Painlev\'e equation…
Let $P$ be an arbitrary point on an elliptic curve over the complex numbers of the form $y^2=x^3+a_4\,x+a_6$ or of the form $y^2=x^3+a_2\,x^2+a_4\,x$. We provide explicit formulae to compute the points $P/2$, i.e., the points $Q$ such that…
We investigate the structure of $\tau$-functions for the elliptic difference Painlev\'e equation of type $E_8$. Introducing the notion of ORG $\tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed…
We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at $4m+12$ points for $m \geq 1$, which appear in pairs due to a symmetry condition. We parameterize…
In this paper, we construct a new relation between ABS equations and Painlev\'e equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painlev\'e equations.
In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…