Advances in Difference Equations 
Volume 2008 (2008), Article ID 317520, 16 pages
doi:10.1155/2008/317520
Review Article

Do All Integrable Equations Satisfy Integrability Criteria?

B. Grammaticos,1 A. Ramani,2 K. M. Tamizhmani,3 T. Tamizhmani,4 and A. S. Carstea5

 1Laboratoire Imagerie et Modélisation en Neurobiologie et Cancérologie (IMNC), CNRS UMR 8165, Université Paris VII-Paris XI, Bâtiment 104, 91406 Orsay, France
2Centre de Physique Théorique, École Polytechnique, CNRS, 91128 Palaiseau, France
3Department of Mathematics, Pondicherry University, Kalapet, Puducherry 605014, India
4Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate Studies, Puducherry 605008, India
5Department of Theoretical Physics, Institute of Physics and Nuclear Engineering, 407 Atomistilor, Magurele, 077125 Bucharest, Romania
Received 17 December 2007; Accepted 16 May 2008

Recommended by Roderick Melnik

Abstract

At the price of sacrificing all suspense, we can already announce that the answer to the question of the title is “no.” It is indeed our belief that one may find counterexamples to all integrability conjectures, unless one constrains the definition of integrability to the point that the integrability criterion becomes tautological. This review is devoted to a critical analysis of the situation.