Nonlinear differential-difference and difference equations: Integrability and exact solvability

A brief review on the recent results of nonlinear differential-difference a nd difference equations toward its complete integrability and exact solvabi lity is presented. In particular, we show how Lie's theory of differential equations can be extended to differential-difference and pure difference eq uations and illustrate its applicability through the discrete Korteweg-deVr ies equation as an example. Also, we report that an autonomous nonlinear di fference equation of an arbitrary order with one or more independent variab les can be linearised by a point transformation if and only if it admits a symmetry vector field whose coefficient is the product of two functions, on e of the dependent variable and of the independent variables. This is illus trated by linearising several first- and second-order ordinary nonlinear di fference equations. A possible connection between the Lie symmetry analysis and the onset of chaos with reference to first-order mappings is explored. (C) 2001 Elsevier Science Ltd. All rights reserved.