arXiv subjects

Robert Conte

Publications and source records attributed to Robert Conte.

63 records · Page 4Linked to original sources

arXiv2001

A truncation for obtaining all the first degree birational transformations of the Painlevé transcendents

A birational transformation is one which leaves invariant an ordinary differential equation, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation of Okamoto for the master Painlevé equation P6. Then we improve it by adding a preliminary step, which is to find all the Riccati subequations of the considered Pn before performing the truncation. We discuss in some detail the main novelties of our method, taking as an example the simplest Painlevé equation for that purpose, P2. Finally, we apply the method to P5 and obtain its two inequivalent first degree birational transformations.

nlin.SI↗

arXiv2001

First degree birational transformations of the Painlevé equations and their contiguity relations

We present a consistent truncation, allowing us to obtain the first degree birational transformation found by Okamoto for the sixth Painlevé equation. The discrete equation arising from its contiguity relation is then just the sum of six simple poles. An algebraic solution is presented, which is equivalent to but simpler than the Umemura solution. Finally, the well known confluence provides a unified picture of all first degree birational transformations for the lower Painlevé equations, ranging them in two distinct sequences.

nlin.SI↗

arXiv2001

On the Schlesinger transformations of the sixth Painlevé equation

Following the recent discovery of two new Schlesinger transformations (ST) for the sixth Painlevé equation, we give the interrelations between all the known STs. We thus isolate the unique one which at the same time conserves the independent variable and is not a product of other STs.

math.CA↗

arXiv2000

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.

nlin.SI↗

arXiv2000

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.

nlin.SI↗

arXiv1996

Discrete version of the Chazy class III equation

We study the discretisation of the Chazy class III equation by two means: a discrete Painlevé test, and the preservation of a two-parameter solution to the continuous equation. We get that way a best discretisation scheme.

solv-int↗

arXiv1996

A geometrical method towards first integrals for dynamical systems

We develop a method, based on Darboux' and Liouville's works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements' form. We apply it to three dynamical systems: Lotka--Volterra, Lorenz and Rikitake.

solv-int↗

arXiv1995

Non-Fuchsian extension to the Painlevé test

We consider meromorphic particular solutions of nonlinear ordinary differential equations and perform a perturbation {\it à la} Poincaré making their linearized equation non-Fuchsian at the movable pole and Fuchsian at infinity. When the nonlinear equation possesses movable logarithms, they are detected sooner than with the perturbative (Fuchsian) Painlevé test.

solv-int↗

arXiv1993

Exact Solutions of the One-Dimensional Quintic Complex Ginzburg-Landau Equation

Exact solitary wave solutions of the one-dimensional quintic complex Ginzburg-Landau equation are obtained using a method derived from the Painlevé 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 Ansätze.

patt-sol↗