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Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property

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Book cover Integrability of Nonlinear Systems

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

Abstract

The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable equations have the Painlevé property, that is, all solutions are single-valued around all movable singularities. In this expository article, we review methods for analysing such singularity structure. In particular, we describe well known techniques of nonlinear regular-singular-type analysis, i.e., the Painlevé tests for ordinary and partial differential equations. Then we discuss methods of obtaining sufficiency conditions for the Painlevé property. Recently, extensions of irregular singularity analysis to nonlinear equations have been achieved. Also, new asymptotic limits of differential equations preserving the Painlevé property have been found. We discuss these also.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    M. D. Kruskal

  2. Department of Pure Mathematics, University of Adelaide, 5005, Adelaide, SA, Australia

    N. Joshi

  3. Department of Applied Mathematics, University of Colorado, 80309-529, Boulder, CO, USA

    R. Halburd

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  1. M. D. Kruskal
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  2. N. Joshi
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  3. R. Halburd
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Y. Kosmann-Schwarzbach B. Grammaticos K. M. Tamizhmani

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© 1997 Springer-Verlag

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Kruskal, M.D., Joshi, N., Halburd, R. (1997). Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property. 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/BFb0113696

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  • DOI: https://doi.org/10.1007/BFb0113696

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