Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries

Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

Given an equation arising from some application or theoretical consideration one of the first questions one might ask is: What is its behavior? It is integrable? In these lectures we will introduce two different ways for establishing (and in some sense also defining) integrability for difference equations: Algebraic Entropy and Generalized Symmetries. Algebraic Entropy deals with the degrees of growth of the solution of any kind of discrete equation (ordinary, partial or even differential-difference) and usually provides a quick test to establish if an equation is or not integrable. The approach based on Generalized Symmetries also provides tools for investigating integrable equations and to find particular solutions by symmetry reductions. The main focus of the lectures will be on the computational tools that allow us to calculate Generalized Symmetries and extract the value of the Algebraic Entropy from a finite number of iterations of the map.

Notes

Acknowledgements

The author would like to thank Prof. D. Levi, Prof. R.I. Yamilov and Prof. C.-M. Viallet for the many helpful and fruitful discussions during the preparation of these notes. The author also acknowledges financial support from INFN IS-CSN4 Mathematical Methods of Nonlinear Physics.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomaItaly

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