Robert Conte

Link between solitary waves and projective Riccati equations

Many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective (hence linearizable) Riccati system. From that property, the authors deduce a method for building these solutions by determining only a finite number of coefficients. This method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation. They handle several examples. For the Henon-Heiles Hamiltonian system, they obtain several exact solutions; one of them defines a new solitary wave solution for a coupled system of Boussinesq and nonlinear Schrodinger equations. For a third order dispersive equation with two monomial nonlinearities, they isolate all cases where the general solution is single valued.

Invariant painlevé analysis of partial differential equations

Abstract The whole Painleve analysis of PDEs is shown to be invariant under an arbitrary homographic transformation of the function ϕ defining the singularity manifold. The best expansion function is ϰ = ( ϕ x ϕ − ϕ xx 2ϕ x ) -1 . This solves the ques tion of invariance under the Mobius group in Painleve analysis and explains naturally Backlund transformation between solutions.

A perturbative Painleve´ approach to nonlinear differential equations

Abstract We further improve the Painleve test so that negative indices (“resonances”) can be treated: we demand single valuedness not only for any pole-like expansion as in the usual Painleve test, but also for every solution close to it, represented as a perturbation series in a small parameter e. Order zero is the usual test. Order one, already treated in a preliminary paper, reduces to a (linear) Fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients. Higher orders lead to the analysis of a linear, Fuchsian type inhomogeneous equation. We obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points, arising at every integer index, whether positive or negative, and at every order; those arising at negative indices, including -1, are new, while some conditions may not arise before some high perturbation order. We present several illustrative examples. We discuss the understanding of negative indices, and conclude that they are indistinguishable from positive indices, just as in the Fuchs theory. In particular, negative indices give rise to doubly infinite Laurent series.

Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation

Abstract Exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equations are obtained using a method derived from the Painleve test for integrability. These solutions are expressed in terms of hyperbolic functions, and include the pulses and fronts found by van Saarloos and Hohenberg. We also find previously unknown sources and sinks. The emphasis is put on the systematic character of the method which breaks away from approaches involving somewhat ad hoc Ansatze.

Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation

Abstract Systems of two linear partial differential equations (PDEs) with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear PDEs. For the one-dimensional complex Ginzburg- Landau equation, the usual representations of the complex field (Re A , or ( A , Ā ) are multivalued, which makes difficult the search for exact solutions in this manner. Fortunately, the Painleve analysis naturally introduces an elementary single valued-like representation of A by two fields ( Z ,grad Θ), respectively complex abd real, uniquely defined by an explicit expression for Θ and A = Ze iΘ . Another important feature is the invariance by parity on A , which increases the class of expected solutions. This allows to retrieve quite easily the four famous solutions of Nozaki and Bekki, represented by the constant coefficients of two linear partial differential equations and a finite set of constants.

The Painlevé Approach to Nonlinear Ordinary Differential Equations

The “Painleve analysis” is quite often perceived as a collection of tricks reserved to experts. The aim of this chapter is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject that is, in fact, the theory of the (explicit) integration of nonlinear differential equations.

The Bianchi IX (mixmaster) cosmological model is not integrable

Abstract The perturbation of an exact solution exhibits a movable transcendental essential singularity, thus proving the nonintegrability. Then, all possible exact particular solutions which may be written in closed form are isolated with the perturbative PainlevtE test; this proves the inexistence of any vacuum solution other than the three known ones.

Analytic solitary waves of nonintegrable equations

Abstract A major drawback of most methods to find analytic expressions for solitary waves is the a priori restriction to a given class of expressions. To overcome this difficulty, we present a new method, applicable to a wide class of autonomous equations, which builds as an intermediate information the first order autonomous ordinary differential equation (ODE) satisfied by the solitary wave. We discuss its application to the cubic complex one-dimensional Ginzburg–Landau equation, and conclude to the elliptic nature of the yet unknown most general solitary wave.

Universal invariance properties of Painlevé analysis and Bäcklund transformation in nonlinear partial differential equations

Abstract We define two transformations, independent of the polynomial PDE, making all the compatibility conditions at resonances and all the equations defining the Backlund transformation invariant by homographic transformation of the expansion function. The resulting dramatic shortening of invariant equations makes their solution feasible by hand, leading to new analytic solutions.

Exact solutions of nonlinear partial differential equations by singularity analysis

Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool is the appropriate use of the singularities of the solutions, and this can be done without knowing these solutions in advance. Since the elaboration of the \textit{singular manifold method} by Weiss et al., many improvements have been made. After some basic recalls, we give an interpretation of the method allowing us to understand why and how it works. Next, we present the state of the art of this powerful technique, trying as much as possible to make it a (computerizable) algorithm. Finally, we apply it to various PDEs in 1+1 dimensions, mostly taken from physics, some of them chaotic: sine-Gordon, Boussinesq, Sawada-Kotera, Kaup-Kupershmidt, complex Ginzburg-Landau, Kuramoto-Sivashinsky, etc.