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Discovery Projects - Grant ID: DP130100967

Title
Discovery Projects - Grant ID: DP130100967
Funding
ARC | Discovery Projects
Contract (GA) number
DP130100967
Start Date
2013/01/01
End Date
2015/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/DP130100967

 

  • Quicksilver Solutions of a q-difference first Painlev\'e equation

    In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a $q$-difference Painlev\'e equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlev\'e equation ($q\Pon$), whose phase space (space of initial values) is a rational surface of type $A_7^{(1)}$. We describe four families of almost stationary behaviours, but focus on the most complicated case, which i...

    Geometric Reductions of ABS equations on an $n$-cube to discrete Painlev\'e systems

    Joshi, Nalini; Nakazono, Nobutaka; Shi, Yang (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100967 (DP130100967)
    In this paper, we show how to relate $n$-dimensional cubes on which ABS equations hold to the symmetry groups of discrete Painlev\'e equations. We here focus on the reduction from the 4-dimensional cube to the $q$-discrete third Painlev\'e 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 P...

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

    Joshi, Nalini; Nakazono, Nobutaka; Shi, Yang (2014)
    Projects: ARC | Discovery Projects - Grant ID: DP130100967 (DP130100967)
    We introduce the concept of $\omega$-lattice, constructed from $\tau$ functions of Painlev\'e systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painlev\'e systems corresponding affine Weyl group symmetries are of $(A_2+A_1)^{(1)}$- and $(A_1+A_1)^{(1)}$-types, respectively.

    An Overview of Geometric Asymptotic Analysis of Continuous and Discrete Painlev\'e Equations

    The classical Painlev\'e 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 domain. 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 bou...
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