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

A brief review on the recent results of nonlinear differential-difference and difference equations toward its complete integrability and exact solvability is presented. In particular, we show how Lieʹs theory of differential equations can be extended to differential-difference and pure difference equations and illustrate its applicability through the discrete Korteweg-deVries equation as an example. Also, we report that an autonomous nonlinear difference equation of an arbitrary order with one or more independent variables 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, one of the dependent variable and of the independent variables. This is illustrated by linearising several first- and second-order ordinary nonlinear difference equations. A possible connection between the Lie symmetry analysis and the onset of chaos with reference to first-order mappings is explored.