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,

General solution for Hamiltonians with extended cubic and quartic potentials

We integrate with hyperelliptic functions a two-particle Hamiltonian with quartic potential and additionnal linear and nonpolynomial terms in the Liouville integrable cases 1:6:1 and 1:6:8.Comment: LaTex 2e. To appear, Theoretical and Mathematical Physics 200

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

On the exact solutions of the Bianchi IX cosmological model in the proper time

It has recently been argued that there might exist a four-parameter analytic solution to the Bianchi IX cosmological model, which would extend the three-parameter solution of Belinskii et al. to one more arbitrary constant. We perform the perturbative Painlev\'e test in the proper time variable, and confirm the possible existence of such an extension.Comment: 8 pages, no figure, standard Latex, to appear in Regular and chaotic dynamics (1998

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

Solitons from a direct point of view: padeons

AbstractA systematic approach to soliton interaction is presented in terms of a particular class of solitary waves (padeons) which are linear fractions with respect to the nonlinearity parameter ϵ. A straightforward generalization of the padeon to higher order rational fractions (multipadeon) yields a natural ansatz for N-soliton solutions. This ansatz produces multisoliton formulas in terms of an ‘interaction matrix’ A. The structure of the matrix gives some insight into the hidden IST-properties of a familiar set of ‘integrable’ equations (KdV, Boussinesq, MKdV, sine-Gordon, nonlinear Schrödinger). The analysis suggests a ‘padeon’ working definition of the soliton, leading to an explicit set of necessary conditions on the padeon equation

Integration of a generalized H\'enon-Heiles Hamiltonian

The generalized H\'enon-Heiles Hamiltonian $H=1/2(P_X^2+P_Y^2+c_1X^2+c_2Y^2)+aXY^2-bX^3/3$ with an additional nonpolynomial term $\mu Y^{-2}$ is known to be Liouville integrable for three sets of values of $(b/a,c_1,c_2)$. It has been previously integrated by genus two theta functions only in one of these cases. Defining the separating variables of the Hamilton-Jacobi equations, we succeed here, in the two other cases, to integrate the equations of motion with hyperelliptic functions.Comment: LaTex 2e. To appear, Journal of Mathematical Physic