Discrete versions of the Painlevé equations

Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.

Existence of zero-order meromorphic solutions of certain q-difference equations

In this paper, we consider the q-difference equation ( f ( q z ) + f ( z ) ) ( f ( z ) + f ( z / q ) ) = R ( z , f ) , where R ( z , f ) is rational in f and meromorphic in z. It shows that if the above equation assumes an admissible zero-order meromorphic solution f ( z ) , then either f ( z ) is a solution of a q-difference Riccati equation or the coefficients satisfy some conditions.

Elementary exact calculations of degree growth and entropy for discrete equations

Second-order discrete equations are studied over the field of rational functions C(z), where z is a variable not appearing in the equation. The exact degree of each iterate as a function of z can be calculated easily using the standard calculations that arise in singularity confinement analysis, even when the singularities are not confined. This produces elementary yet rigorous entropy calculations.

Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions

In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for q -P II here, is also applicable to other discrete Painlevé equations.

Using the finite domain remnant of the continuous spectrum to examine integrability: effect of boundary conditions

The aim of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE), by examining it in a finite but large domain. For monoparametric families of PDE/DDE's, that are known to possess isolated integrable points, we find that very special features occur in the finite domain remnant of the continuous ("phonon") spectrum at these "singular" points. We identify these features in the case example of a PDE and a DDE (that sustain front and pulselike solutions, respectively) for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular, integrable points and hence we propose to explore a given PDE/DDE in a finite but large domain for such traits, as a means of assessing its potential integrability.

Arithmetical method to detect integrability in maps

We develop a method to detect the presence of integrals of the motion in symplectic rational maps, by representing these maps over finite fields and examining their orbit structure. We find markedly different orbit statistics depending upon whether the map is integrable or not.