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3 - Definitions and Predictions of Integrability for Difference Equations

Published online by Cambridge University Press:  05 July 2011

By
Jarmo Hietarinta
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Jarmo Hietarinta
Affiliation:
University of Turku
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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Summary

Abstract

In these lectures we take a look at various meanings of integrability for difference equations, and the possibility of algorithmic methods to identify (partial) integrability. Analogies with continuum equations are used when possible, but the world of discrete equations is richer and many new things enter.

Preliminaries

Points of view on integrability

One can approach the integrability of dynamical equations from two opposite directions:

Top down: In this approach one first chooses some high level mathematical structure and then derives its consequences/manifestations for dynamical equations. The underlying mathematical structure is there from the start and therefore as a result one gets a method for generating equations with good properties.

Bottom up: In this case the equation is given, for example: “In my application I found this equation, what can you say about it?” In order to answer questions of this type we need a toolbox of (algorithmic) methods that can be applied. With them we can, hopefully, identify the equation or at least say whether it is integrable or partially integrable or chaotic. If the equation is at least partially integrable we may be able to say something about its structure, construct some solutions, conserved quantities etc. And although complete integrability itself is structurally unstable, many properties persist in nearby non-integrable systems, and one can try a perturbative expansion around the integrable one.

Type
Chapter
Information
Symmetries and Integrability of Difference Equations , pp. 83 - 114
Publisher: Cambridge University Press
Print publication year: 2011

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References

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