We present a review of results of the so-called Painlevé singularity approach to the
investigation of the integrability of dynamical systems with finite and infinite number of …

The Landau–Lifshitz–Gilbert (LLG) equation is a fascinating nonlinear evolution equation
both from mathematical and physical points of view. It is related to the dynamics of several …

This book is designed to provide a path for the reader into an amalgamation oftwo venerable
areas ofmathematics, Dynamical Systems and Number Theory. Many of the motivating …

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 …

We present an integrability criterion for discrete-time systems that is the equivalent of the
Painlevé property for systems of a continuous variable. It is based on the observation that for …

The interplay of nonlinearity with lattice discreteness leads to phenomena and propagation
properties quite distinct from those appearing in continuous nonlinear systems. For a large …

Some simple solutions of discrete soliton equations are shown to satisfy 2D map**s. We
show that these belong to a recently introduced 18-parameter family of integrable reversible …

We present discrete forms of the Painlevé transcendental equations P III, P IV, and PV that
complement the list of the already known PI and P II. These, most likely integrable …

Abstract CONTENTS Introduction Chapter I. Integrable Lagrange systems with discrete time
§ 1. Hamiltonian theory of Lagrange systems with discrete time § 2. Discrete analogues of …

Following the work of Yorke and Li in 1975, the theory of discrete dynamical systems and
difference equations developed rapidly. The applications of difference equations also grew …