Do All Integrable Equations Satisfy Integrability Criteria?
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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 1
Laboratoire Imagerie et Mod´elisation en Neurobiologie et Canc´erologie (IMNC), CNRS UMR 8165, Universit´e Paris VII-Paris XI, Bˆatiment 104, 91406 Orsay, France 2 ´ Centre de Physique Th´eorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, France 3 Department of Mathematics, Pondicherry University, Kalapet, Puducherry 605014, India 4 Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate Studies, Puducherry 605008, India 5 Department of Theoretical Physics, Institute of Physics and Nuclear Engineering, 407 Atomistilor, Magurele, 077125 Bucharest, Romania Correspondence should be addressed to K. M. Tamizhmani, [email protected] Received 17 December 2007; Accepted 16 May 2008 Recommended by Roderick Melnik 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. Copyright q 2008 B. Grammaticos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction The study of integrable systems is intimately related to integrability criteria 1. This is due to the fact that integrable systems are very rare but also very interesting which explains the intense activity in this domain. The usefulness of integrability criteria is that they allow the proposal of conjectures which serve for integrability prediction. The integrability detectors thus elaborated to find their full usefulness in the nonconstructive approach to integrability. The opposite, constructive, approach consists in deriving integrable systems starting from the solution, or what is admittedly more customary, from an overdetermined linear system the Lax pair, the nonlinear integrable equation resulting from the compatibility of the linear one. The nonconstructive approach starts from a nonlinear system resulting from some, often
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Advances in Difference Equations
physical, usually realistic, model. It is then of the utmost importance to know whether this system is integrable since integrability conditions the long-term behaviour of its solutions. A reliable integrability detector is a most valuable tool. In the domain of continuous systems, the use of complex analysis has made possible the development of specific and efficient tools for integrability prediction, and actual integration of systems expressed as ordinary or partial differential equations. According to Poincar´e 2, to integrate a differential equation is to find for the general solution an expression, possibly multiv
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