The Painlev\'e methods

This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether ordinary, partial, or discrete.Comment: LaTex 2e, subject index, Nonlinear integrable systems: classical and quantum, ed. A. Kundu, Special issue, Proceedings of Indian Science Academy,

On the Links Between Some Generalized Pinney and Nonlinear Gambier Equations

AbstractTwo generalized forms of the Pinney equation, recently derived by Rogers, Schief, and Winternitz from Lie group considerations, are here connected with two linearizable second-order ODE's belonging to the Gambier classification

Analytic solitary waves of nonintegrable equations

Even if it is nonintegrable, a differential equation may nevertheless admit particular solutions which are globally analytic. On the example of the dynamical system of Kuramoto and Sivashinsky, which is generically chaotic and presents a high physical interest, we review various methods, all based on the structure of singularities, allowing us to characterize the analytic solution which depends on the largest possible number of constants of integration.Comment: LaTex 2e. To appear, Physica

New contiguity relation of the sixth Painlev\'e equation from a truncation

For the master Painlev\'e equation P6(u), we define a consistent method, adapted from the Weiss truncation for partial differential equations, which allows us to obtain the first degree birational transformation of Okamoto. Two new features are implemented to achieve this result. The first one is the homography between the derivative of the solution $u$ and a Riccati pseudopotential. The second one is an improvement of a conjecture by Fokas and Ablowitz on the structure of this birational transformation. We then build the contiguity relation of P6, which yields one new second order nonautonomous discrete equation.Comment: LaTex 2e. To appear, Physica

A new method to test discrete Painlev\'e equations

Necessary discretization rules to preserve the Painlev\'e property are stated. A new method is added to the discrete Painlev\'e test, which perturbs the continuous limit and generates infinitely many no-log conditions.Comment: 12 pages, no figure, standard Latex, to appear in Physics Letters

Analytic expressions of hydrothermal waves

When subjected to a horizontal temperature difference, a fluid layer with a free surface becomes unstable and hydrothermal waves develop in the bulk. Such a system is modelized by two coupled amplitude equations of the one-dimensional, complex, cubic Ginzburg-Landau type. By transposing the method developed for one CGL3 equation, we obtain several new exact solutions expressed by closed form, singlevalued, analytic expressions. Some of them are the analogue of the famous amplitude hole solution of Bekki and Nozaki.Comment: LaTeX, 12 pages, no figure, to appear, Reports on Math. Physic

A reduction of the resonant three-wave interaction to the generic sixth Painleve' equation

Among the reductions of the resonant three-wave interaction system to six-dimensional differential systems, one of them has been specifically mentioned as being linked to the generic sixth Painleve' equation P6. We derive this link explicitly, and we establish the connection to a three-degree of freedom Hamiltonian previously considered for P6.Comment: 13 pages, 0 figure, J. Phys. A Special issue "One hundred years of Painleve' VI

Collisions élastiques spin 0-spin 1/2 à haute énérgie. Un modèle en pôles de Regge et diffusion multiple

Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe