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Painlevé Analysis for Nonlinear Partial Differential Equations

  • Chapter
The Painlevé Property

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

Abstract

The Painlevé analysis introduced by Weiss, Tabor, and, Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDEs) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODEs) without movable critical points. In this chapter we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated Bäcklund transformations. Many remarkable properties are shared by these so-called integrable equations, but they are generically no longer valid for equations modeling physical phenomena. Belonging to this second class, some equations called “partially integrable” sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed-form analytic solutions, which necessarily agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and systems of Riccati equations that are linearizable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.

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  1. Micheline Musette
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  1. Service de Physique de l’etat Condensé, CEA Saclay, F91191, Gif-sur-Yvette Cedex, France

    Robert Conte

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Musette, M. (1999). Painlevé Analysis for Nonlinear Partial Differential Equations. In: Conte, R. (eds) The Painlevé Property. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1532-5_8

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  • DOI: https://doi.org/10.1007/978-1-4612-1532-5_8

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