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

✍ Scribed by R. Sahadevan

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 521 KB
• Volume: 42
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(01)00182-1

✦ Synopsis

Abstraet--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 Lie's theory of differential equations can be extended to differentiM-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 flmctions, one of the dependent variable and of the independent variables. This is illustrated by linearising several firstand 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. (~) 2001 Elsevier Science Ltd. All rights reserved.