Painlevé Analysis for Nonlinear Partial Differential Equations

Abstract

La propriété de Painlevé, un siècle après The Painlevé property, one century laterAbstract. The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) 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 (ODE's) without movable critical points. In these lectures 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 transformation. A lot of remarkable properties are shared by these so-called "integrable" equations but they are generically no more valid for equations modelising physical phenomema. 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 which are linearisable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.The Painlevé property, one century later, ed. R. Conte, CRM series in mathematical physics (Springer-Verlag, Berlin, 1998) 31 March 1998 solv-int/9804003

On Existence of Solitons for the Second Harmonic Equations of a Laser Beam

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New contiguity relation of the sixth Painlevé equation from a truncation

“…Each Pn equation which admits a birational transformation has one or several (four for P6) couples of families of movable simple poles with opposite residues ±u 0 , therefore both the onefamily truncation and the two-family truncation [22,14,15] are applicable. In the present paper, we consider only the one-family truncation, whose assumption is…”

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Exact solutions of nonlinear partial differential equations by singularity analysis

“…As a general bibliography on the subject of these lectures, we recommend Cargèse lecture notes [84] and a shorter subset of these with emphasis on the various so-called truncations [24].…”