Searcharxiv⌕ Search

arXiv subjects

Micheline Musette

Publications and source records attributed to Micheline Musette.

18 recordsLinked to original sources
arXiv2025

Methods for exact solutions of nonlinear ordinary differential equations\

In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by Eremenko, which can be turned into algorithms, thus avoiding \textit{ad hoc} assumptions, able to provide \textit{all} (as opposed to some) solutions in a precise class. To illustrate these methods, we present some new such exact solutions, physically relevent.

nlin.SI↗
arXiv2023

All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations

For both cubic and quintic nonlinearities of the one-dimensional complex Ginzburg-Landau evolution equation, we prove by a theorem of Eremenko the finiteness of the number of traveling waves whose squared modulus has only poles in the complex plane, and we provide all their closed form expressions. Among these eleven solutions, five are provided by the method used. This allows us to complete the list of solutions previously obtained by other authors.

nlin.PS↗
arXiv2022

New solutions to the complex Ginzburg-Landau equations

The various régimes observed in the one-dimensional complex Ginzburg-Landau equation result from the interaction of a very small number of elementary patterns such as pulses, fronts, shocks, holes, sinks. We provide here three exact such patterns observed in numerical calculations but never found analytically. One is a quintic case localized homoclinic defect, observed by Popp et alii, the two others are bound states of two quintic dark solitons, observed by Afanasyev et alii.

nlin.PS↗
arXiv2014

Introduction to the Painlevé property, test and analysis

This short survey presents the essential features of what is called Painlevé analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found in \textit{The Painlevé handbook} or in various lecture notes posted on arXiv.

nlin.SI↗
arXiv2009

Elliptic general analytic solutions

In order to find analytically the travelling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: "projective Riccati method", "tanh-method", "exponential method", "Jacobi expansion method", "new ...", etc. The common default to all these "truncation methods" is to only provide some solutions, not all of them. By implementing three classical results of Briot, Bouquet and Poincare', we present an algorithm able to provide in closed form \textit{all} those travellingz waves which are elliptic or degenerate elliptic, i.e. rational in one exponential or rational. Our examples here include the Kuramoto-Sivashinsky equation and the cubic and quintic complex Ginzburg-Landau equations.

math.CA↗
arXiv2006

Painleve' property of the He'non-Heiles Hamiltonians

Time independent Hamiltonians of the physical type H = (P_1^2+P_2^2)/2+V(Q_1,Q_2) pass the Painleve' test for only seven potentials $V$, known as the He'non-Heiles Hamiltonians, each depending on a finite number of free constants. Proving the Painleve' property was not yet achieved for generic values of the free constants. We integrate each missing case by building a birational transformation to some fourth order first degree ordinary differential equation in the classification (Cosgrove, 2000) of such polynomial equations which possess the Painleve' property. The properties common to each Hamiltonian are: (i) the general solution is meromorphic and expressed with hyperelliptic functions of genus two, (ii) the Hamiltonian is complete (the addition of any time independent term would ruin the Painleve' property).

nlin.SI↗
arXiv2005

Hamiltonians with two degrees of freedom admitting a singlevalued general solution

Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians $H=(p_1^2+p_2^2)/2+V(q_1,q_2)$ whose some integer power $q_j^{n_j}(t)$ of the general solution is a singlevalued function of the complex time $t$. In addition to the well known rational potentials $V$ of Hénon-Heiles, this selects possible cases with a trigonometric dependence of $V$ on $q_j$. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven (three ``cubic'' plus four ``quartic'') rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.

nlin.SI↗
arXiv2005

Completeness of the cubic and quartic Hénon-Heiles Hamiltonians

The quartic Hénon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(Ω_1 Q_1^2+Ω_2 Q_2^2)/2 +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4 +(1/2)(α/Q_1^2+β/Q_2^2) - γQ_1$ passes the Painlevé test for only four sets of values of the constants. Only one of these, identical to the traveling wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the three others are not yet integrated in the generic case $(α,β,γ)\not=(0,0,0)$. We integrate them by building a birational transformation to two fourth order first degree equations in the classification (Cosgrove, 2000) of such polynomial equations which possess the Painlevé property. This transformation involves the stationary reduction of various partial differential equations (PDEs). The result is the same as for the three cubic Hénon-Heiles Hamiltonians, namely, in all four quartic cases, a general solution which is meromorphic and hyperelliptic with genus two. As a consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlevé integrability (completeness property).

nlin.SI↗
arXiv2004

Explicit integration of the Hénon-Heiles Hamiltonians

We consider the cubic and quartic He'non-Heiles Hamiltonians with additional inverse square terms, which pass the Painleve' test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painleve' sense (impossibility to add any term to the Hamiltonian without destroying the Painleve' property).

nlin.SI↗
arXiv2004

Solitary waves of nonlinear nonintegrable equations

Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first class, which includes the well known so-called truncation methods, one \textit{a priori} assumes a given class of expressions (polynomials, etc) for the unknown solution; the involved work can easily be done by hand but all solutions outside the given class are surely missed. In the second class, instead of searching an expression for the solution, one builds an intermediate, equivalent information, namely the \textit{first order} autonomous ODE satisfied by the solitary wave; in principle, no solution can be missed, but the involved work requires computer algebra. We present the application to the cubic and quintic complex one-dimensional Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.

nlin.PS↗
arXiv2002

The Painlevé methods

This short review is an introduction to a great variety of methods, the collection of which is called the Painlevé analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether ordinary, partial, or discrete.

nlin.SI↗
arXiv2001

New contiguity relation of the sixth Painlevé equation from a truncation

For the master Painlevé 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.

nlin.SI↗
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↗
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↗
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↗