The Painlevé property is closely connected to differential equations that are integrable via
related iso-monodromy problems. Many apparently integrable discrete analogues of the …

This first introductory text to discrete integrable systems introduces key notions of
integrability from the vantage point of discrete systems, also making connections with the …

This invaluable book examines qualitative and quantitative methods for nonlinear differential
equations, as well as integrability and nonintegrability theory. Starting from the idea of a …

We present a review of what is current knowledge about discrete Painlevé equations. We
start with a historical introduction which explains how the discrete Painlevé equations made …

Nonlinear differential or difference equations are encountered not only in mathematics but
also in many areas of physics (evolution equations and propagation of a signal in an optical …

For any discrete time dynamical system with a rational evolution, we define an entropy,
which is a global index of complexity for the evolution map. We analyze its basic properties …

We present a general scheme to derive higher-order members of the Painlevé VI (PVI)
hierarchy of ODE's as well as their difference analogues. The derivation is based on a …

Let w (z) be an admissible finite‐order meromorphic solution of the second‐order difference
equation w (z+ 1)+ w (z− 1)= R (z, w (z)) where R (z, w (z)) is rational in w (z) with coefficients …

An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved. Using
this result as a starting-point, a variational complex for difference equations is constructed …

We present a geometric description of the QRT map (which is an integrable map**
introduced by Quispel, Roberts and Thompson) in terms of the addition formula of a rational …