We provide a Lax pair for the surfaces of Voss and Guichard, and we show that such particular surfaces considered by Gambier are characterized by a third Painleve function.
Nous montrons que les equations du rep`ere mobile des surfaces de Bonnet conduisent `a une paire de Lax matricielle isomonodromique dordre deux pour la sixi`eme equation de Painleve. We show that the moving frame equations of Bonnet surfaces can be
extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painleve equation.
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 tw
o 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.
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 Painleve 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.
We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player
turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)
This short survey presents the essential features of what is called Painleve 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{Th
e Painleve handbook} or in various lecture notes posted on arXiv.
We investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order and modal languages without labels on transitions or atomic propositions on markings. We consi
der several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.
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 l
ocal 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.
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 cub
ic-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).
We show that all meromorphic solutions of the stationary reduction of the real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We then obtain them all explicitly by the subequation method, and one of them appears to be a new elliptic solution.
