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 reduce
s to $A_2^{(1)ast}$-type discrete Painleve equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.
The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from the discre
te symmetry of the Garnier system in two variables.
Discrete Painleve 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 th
e 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 Painleve equations.
In this paper, we consider the discrete power function associated with the sixth Painleve 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 embe
dded 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 $tau$ function.
We study the asymptotic behaviour of solutions of the fourth Pain-leve equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show that the lim
it set of each solution is compact and connected and, moreover, that any non-special solution has an infinite number of poles and infinite number of zeroes.
We construct the initial-value space of a $q$-discrete first Painleve equation explicitly and describe the behaviours of its solutions $w(n)$ in this space as $ntoinfty$, with particular attention paid to neighbourhoods of exceptional lines and irred
ucible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.
We introduce the concept of $omega$-lattice, constructed from $tau$ functions of Painleve systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painleve systems corresponding affi
ne Weyl group symmetries are of $(A_2+A_1)^{(1)}$- and $(A_1+A_1)^{(1)}$-types, respectively.
The classical Painleve equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domai
n. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamotos geometric space of initial values for the first and second Painleve equations. In this paper, we review this method and indicate how it may be extended to the discrete Painleve equations.
We study the solutions of the second Painleve equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as
the thirty-fourth Painleve equation. By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repellor for the dynamics in Okamotos space. Moreover, we show that the limit set of the solutions exists and is compact and connected.
The lattice potential Korteweg-de Vries equation (LKdV) is a partial difference equation in two independent variables, which possesses many properties that are analogous to those of the celebrated Korteweg-de Vries equation. These include discrete so
liton solutions, Backlund transformations and an associated linear problem, called a Lax pair, for which it provides the compatibility condition. In this paper, we solve the initial value problem for the LKdV equation through a discrete implementation of the inverse scattering transform method applied to the Lax pair. The initial value used for the LKdV equation is assumed to be real and decaying to zero as the absolute value of the discrete spatial variable approaches large values. An interesting feature of our approach is the solution of a discrete Gelfand-Levitan equation. Moreover, we provide a complete characterization of reflectionless potentials and show that this leads to the Cauchy matrix form of N-soliton solutions.
