A perturbative Painlevé approach to nonlinear differential equations

Conte, Robert; Fordy, Allan P.; Pickering, Andrew

doi:10.1016/0167-2789(93)90179-5

Cited by 156 publications

(207 citation statements)

References 45 publications

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“…It allows to extract the information contained in the negative indices [52], thus building infinitely many necessary conditions for the absence of movable critical singularities of the logarithmic type [31].…”

Section: The Fuchsian Perturbative Methodsmentioning

confidence: 99%

“…In the case of a single equation, since indices must be distinct integers, the condition Q (1) ρ = 0 at the smallest Fuchs index i = ρ is identically satisfied. Nevertheless, the frequently encountered statement "resonance −1 is always compatible" is erroneous, and numerous nonzero stability conditions Q (n) −1 = 0 can be found in the examples of [31]. Indeed, even at first perturbation order, the stability condition at index −1 may not be satisfied : just like Fuchs index i = ρ provides an identically satisfied stability condition, Painlevé "resonance" −1 has the same property if and only if ρ = −1, i.e.…”

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confidence: 99%

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The Painlevé Approach to Nonlinear Ordinary Differential Equations

Conte¹

1999

The Painlevé Property

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The "Painlevé analysis" is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory of the (explicit) integration of nonlinear differential equations.To achieve our goal, we will not start the exposition with a more or less precise "Painlevé test". On the contrary, we will finish with it, after a gradual introduction to the rich world of singularities of nonlinear differential equations, so as to remove any cooking recipe.The emphasis is put on embedding each method of the test into the well known theorem of perturbations of Poincaré. A summary can be found at the beginning of each chapter.

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Section: The Fuchsian Perturbative Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

The Painlevé Approach to Nonlinear Ordinary Differential Equations

Conte¹

1999

The Painlevé Property

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166

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“…In [7], R. Conte, A. P. Fordy and A. Picking have further improved the Painlevé test such that negative indices can be treated.…”

Section: Perturbative Painlevé Analysis For Gckdv Equationsmentioning

confidence: 99%

“…In order to solve this problem, in the next section we use the perturbative Painlevé analysis [7] to find an arbitrary function for the resonance k = −2, which extends the particular solution (4) into a general one.…”

Section: Standard and Invariant Painlevé Analysis For Gckdv Equationsmentioning

confidence: 99%

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Painlevé Properties And Exact Solutions Of The Generalized Coupled Kdv Equations

Li-Jun

Lin

2005

Zeitschrift Für Naturforschung A

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The generalized coupled Korteweg-de Vries (GCKdV) equations as one case of the four-reduction of the Kadomtsev-Petviashvili (KP) hierarchy are studied in details. The Painlevé properties of the model are proved by using the standard Weiss-Tabor-Carnevale (WTC) method, invariant, and perturbative Painlevé approaches. The meaning of the negative index k = −2 is shown, which is indistinguishable from the index k = −1. Using the standard and nonstandard Painlevé truncation methods and the Jacobi elliptic function expansion approach, some types of new exact solutions are obtained.

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Symmetry and the Chazy Equation

Clarkson

Olver

1996

Journal of Differential Equations

102

161

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There are three different actions of the unimodular Lie group SL(2, C) on a twodimensional space. In every case, we show how an ordinary differential equation admitting SL(2) as a symmetry group can be reduced in order by three, and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation. A particular example is the Chazy equation, whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation. The reduction method leads to an alternative formula in terms of solutions to the Lame equation, resulting in a surprising transformation between the Lame and hypergeometric equations. Finally, we discuss the Painleve analysis of the singularities of solutions to the Chazy equation. 1996Academic Press, Inc.It arises in the study of third order ordinary differential equations having the``Painleve property'' that the solutions have only poles for moveable singularities (see also [3,4]). The Chazy equation is important since it is the simplest example of an ordinary differential equation whose solutions

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A perturbative Painlevé approach to nonlinear differential equations

Cited by 156 publications

References 45 publications

The Painlevé Approach to Nonlinear Ordinary Differential Equations

The Painlevé Approach to Nonlinear Ordinary Differential Equations

Painlevé Properties And Exact Solutions Of The Generalized Coupled Kdv Equations

Symmetry and the Chazy Equation

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