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Robert Conte

Publications and source records attributed to Robert Conte.

At least 37 records · Page 2Linked to original sources
arXiv2017

Solutions of the buoyancy-drag equation with a time-dependent acceleration

We perform the analytic study of the the buoyancy-drag equation with a time-dependent acceleration $γ(t)$ by two methods. We first determine its equivalence class under the point transformations of Roger Liouville, and thus for some values of $γ(t)$ define a time-dependent Hamiltonian from which the buoyancy-drag equation can be derived. We then determine the Lie point symmetries of the buoyancy-drag equation, which only exist for values of $γ(t)$ including the previous ones, plus additional classes of accelerations for which the equation is reducible to an Abel equation. This allows us to exhibit two régimes for the asymptotic (large time $t$) solution of the buoyancy-drag equation. %\textbf{ It is shown that they describe a mixing zone driven by the Rayleigh--Taylor instability and the Richtmyer--Meshkov instability, respectively.

nlin.SI↗
arXiv2016

Surfaces de Bonnet et équations de Painlevé

Nous montrons que les équations du repère mobile des surfaces de Bonnet conduisent à une paire de Lax matricielle isomonodromique d'ordre deux pour la sixième équation de Painlevé. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painlevé equation.

math-ph↗
arXiv2016

Analytic study of a coupled Kerr-SBS system

In order to describe the coupling between the Kerr nonlinearity and the stimulated Brillouin scattering, Mauger et alii recently proposed a system of partial differential equations in three complex amplitudes. We perform here its analytic study by two methods. The first method is to investigate the structure of singularities, in order to possibly find closed form singlevalued solutions obeying this structure. The second method is to look at the infinitesimal symmetries of the system in order to build reductions to a lesser number of independent variables. Our overall conclusion is that the structure of singularities is too intricate to obtain closed form solutions by the usual methods. One of our results is the proof of the nonexistence of traveling waves.

physics.optics↗
arXiv2016

Reductions of Gauss-Codazzi equations

We prove that conformally parametrized surfaces in Euclidean space $\Rcubec$ of curvature $c$ admit a symmetry reduction of their Gauss-Codazzi equations whose general solution is expressed with the sixth Painlevé function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko, Eitner and Kitaev 1997) can be recovered by such a reduction.

math.DG↗
arXiv2015

Exact solution of the planar motion of three arbitrary point vortices

We give an exact quantitative solution for the motion of three vortices of any strength, which Poincaré showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is periodic. There are two kinds of relative equilibrium configuration: two equilateral triangles and one or three colinear configurations, their stability conditions split the strengths space into three domains in which the sets of trajectories are topologically distinct. According to the values of the strengths and the initial positions, all possible %RC motions are classified. Two sets of strengths lead to generic motions other than biperiodic. First, when the angular momentum vanishes, besides the biperiodic regime there exists an expansion spiral motion and even a triple collision in a finite time, but the latter motion is nongeneric. Second, when two strengths are opposite, the system also exhibits the elastic diffusion of a vortex doublet by the third vortex. For given values of the invariants, the volume of the phase space of this Hamiltonian system is proportional to the period of the reduced motion, a well known result of the theory of adiabatic invariants. We then formally examine the behaviour of the quantities that Onsager defined only for a large number of interacting vortices.

nlin.SI↗
arXiv2015

Hayman's classical conjecture on some nonlinear second order algebraic ODEs

In this paper, we study the growth, in terms of the Nevanlinna characteristic function, of meromorphic solutions of three types of second order nonlinear algebraic ordinary differential equations. We give all their meromorphic solutions explicitly, and hence show that all of these ODEs satisfy the {\it classical conjecture} proposed by Hayman in 1996.

math.CV↗
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↗
arXiv2012

Nonlinear amplification of coherent waves in media with soliton-type refractive index pattern

We derive the complex Ginzburg-Landau equation for the dynamical self-diffraction of optical waves in a nonlinear cavity. The case of the reflection geometry of wave interaction as well as a medium that possesses the cubic nonlinearity (including a local and a nonlocal nonlinear responses) and the relaxation is considered. A stable localized spatial structure in the form of a "dark" dissipative soliton is formed in the cavity in the steady state. The envelope of the intensity pattern, as well as of the dynamical grating amplitude, takes the shape of a $\tanh$ function. The obtained complex Ginzburg-Landau equation describes the dynamics of this envelope, at the same time the evolution of this spatial structure changes the parameters of the output waves. New effects are predicted in this system due to the transformation of the dissipative soliton which takes place during the interaction of a pulse with a continuous wave, such as: retention of the pulse shape during the transmission of impulses in a long nonlinear cavity; giant amplification of a seed pulse, which takes energy due to redistribution of the pump continuous energy into the signal.

nlin.PS↗
arXiv2012

Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie's lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.

math.CA↗
arXiv2012

Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation

In evolution equations for a complex amplitude, the phase obeys a much more intricate equation than the amplitude. Nevertheless, general methods should be applicable to both variables. On the example of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to overcome the difficulties arising in two such methods: (i) the criterium that the sum of residues of an elliptic solution should be zero, (ii) the construction of a first order differential equation admitting the given equation as a differential consequence (subequation method).

nlin.PS↗
arXiv2010

A connection between HH3 and KdV with one source

In the system made of Korteweg-de Vries with one source, we first show by applying the Painleve' test that the two components of the source must have the same potential. We then explain the natural introduction of an additional term in the potential of the source equations while preserving the existence of a Lax pair. This allows us to prove the identity between the travelling wave reduction and one of the three integrable cases of the cubic He'non-Heiles Hamiltonian system.

nlin.SI↗
arXiv2010

Exact internal waves of a Boussinesq system

We consider a Boussinesq system describing one-dimensional internal waves which develop at the boundary between two immiscible fluids, and we restrict to its traveling waves. The method which yields explicitly all the elliptic or degenerate elliptic solutions of a given nonlinear, any order algebraic ordinary differential equation is briefly recalled. We then apply it to the fluid system and, restricting in this preliminary report to the generic situation, we obtain all the solutions in that class, including several new solutions.

nlin.PS↗
arXiv2009

Ginzburg-Landau equation for dynamical four-wave mixing in gain nonlinear media with relaxation

We consider the dynamical degenerate four-wave mixing (FWM) model in a cubic nonlinear medium including both the time relaxation of the induced nonlinearity and the nonlocal coupling. The initial ten-dimensional FWM system can be rewritten as a three-variable intrinsic system (namely the intensity pattern, the amplitude of the nonlinearity and the total net gain) which is very close to the pumped Maxwell-Bloch system. In the case of a purely nonlocal response the initial system reduces to a real damped sine-Gordon (SG) equation. We obtain a new solution of this equation in the form of a sech function with a time-dependent coefficient. By applying the reductive perturbation method to this damped SG equation, we obtain exactly the cubic complex Ginzburg Landau equation (CGL3), but with a time dependence in the loss/gain coefficient. The CGL3 describes the properties of the spatially localized interference pattern formed by the FWM.

nlin.PS↗
arXiv2009

Explicit solutions of the four-wave mixing model

The dynamical degenerate four-wave mixing is studied analytically in detail. By removing the unessential freedom, we first characterize this system by a lower-dimensional closed subsystem of a deformed Maxwell-Bloch type, involving only three physical variables: the intensity pattern, the dynamical grating amplitude, the relative net gain. We then classify by the Painleve' test all the cases when singlevalued solutions may exist, according to the two essential parameters of the system: the real relaxation time tau, the complex response constant gamma. In addition to the stationary case, the only two integrable cases occur for a purely nonlocal response (Real(gamma)=0), these are the complex unpumped Maxwell-Bloch system and another one, which is explicitly integrated with elliptic functions. For a generic response (Re(gamma) not=0), we display strong similarities with the cubic complex Ginzburg-Landau equation.

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↗
arXiv2008

Partial integrability of the anharmonic oscillator

We consider the anharmonic oscillator with an arbitrary-degree anharmonicity, a damping term and a forcing term, all coefficients being time-dependent: u" + g_1(x) u' + g_2(x) u + g_3(x) u^n + g_4(x) = 0, n real. Its physical applications range from the atomic Thomas-Fermi model to Emden gas dynamics equilibria, the Duffing oscillator and numerous dynamical systems. The present work is an overview which includes and generalizes all previously known results of partial integrability of this oscillator. We give the most general two conditions on the coefficients under which a first integral of a particular type exists. A natural interpretation is given for the two conditions. We compare these two conditions with those provided by the Painleve' analysis.

nlin.SI↗