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Integrability — and how to detect it

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Integrability of Nonlinear Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 495))

Abstract

We present a physicist’s approach to integrability and its detection. Starting from specific examples we present a working definition of what is meant by “integrability”. The integrability detector on which this whole course in based is the “Painlevé method” which links the integrable character of a (differential) system to the singularity structure of its solutions. Recent results on integrable discrete systems are also discussed here. They are, for the major part, obtained through the application of the “singularity confinement” approach that is the discrete equivalent of the Painlevé method. Foremost among these results are the discrete Painlevé equations that generalize in the discrete domain the transcendental functions introduced by Painlevé and which have so many interesting applications in the domain of nonlinear physics.

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Authors and Affiliations

  1. LPN, Université Paris VII, Tour 24-14, 5ème étage, F-75251, Paris, France

    B. Grammaticos

  2. CPT, Ecole Polytechnique, CNRS, UPR 14, F-91128, Palaiseau, France

    A. Ramani

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  1. B. Grammaticos
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  2. A. Ramani
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Y. Kosmann-Schwarzbach B. Grammaticos K. M. Tamizhmani

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© 1997 Springer-Verlag

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Grammaticos, B., Ramani, A. (1997). Integrability — and how to detect it. In: Kosmann-Schwarzbach, Y., Grammaticos, B., Tamizhmani, K.M. (eds) Integrability of Nonlinear Systems. Lecture Notes in Physics, vol 495. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0113693

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  • DOI: https://doi.org/10.1007/BFb0113693

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