Integrable systems without the Painlev\'e property

We examine whether the Painlev\'e property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlev\'e continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages

B\"acklund transformations for the second Painlev\'e hierarchy: a modified truncation approach

The second Painlev\'e hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlev\'e equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev\'e analysis to ordinary differential equations. We extend these techniques in order to derive auto-B\"acklund transformations for the second Painlev\'e hierarchy. We also derive a number of other B\"acklund transformations, including a B\"acklund transformation onto a hierarchy of P34 equations, and a little known B\"acklund transformation for P2 itself. We then use our results on B\"acklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.Comment: 12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse Problem

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Rational Solutions of the Painleve' VI Equation

In this paper, we classify all values of the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the Painlev\'e VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe

Online security assessment with load and renewable generation uncertainty: The iTesla project approach

The secure integration of renewable generation into modern power systems requires an appropriate assessment of the security of the system in real-time. The uncertainty associated with renewable power makes it impossible to tackle this problem via a brute-force approach, i.e. it is not possible to run detailed online static or dynamic simulations for all possible security problems and realizations of load and renewable power. Intelligent approaches for online security assessment with forecast uncertainty modeling are being sought to better handle contingency events. This paper reports the platform developed within the iTesla project for online static and dynamic security assessment. This innovative and open-source computational platform is composed of several modules such as detailed static and dynamic simulation, machine learning, forecast uncertainty representation and optimization tools to not only filter contingencies but also to provide the best control actions to avoid possible unsecure situations. Based on High Performance Computing (HPC), the iTesla platform was tested in the French network for a specific security problem: overload of transmission circuits. The results obtained show that forecast uncertainty representation is of the utmost importance, since from apparently secure forecast network states, it is possible to obtain unsecure situations that need to be tackled in advance by the system operator

Virtual histological assessment of the prenatal life history and age at death of the Upper Paleolithic fetus from Ostuni (Italy)

The fetal remains from the Ostuni 1 burial (Italy, ca 27 ka) represent a unique opportunity to explore the prenatal biological parameters, and to reconstruct the possible patho-biography, of a fetus (and its mother) in an Upper Paleolithic context. Phase-contrast synchrotron X-ray microtomography imaging of two deciduous tooth crowns and microfocus CT measurements of the right hemimandible of the Ostuni 1b fetus were performed at the SYRMEP beamline and at the TomoLab station of the Elettra - Sincrotrone laboratory (Trieste, Italy) in order to refne age at death and to report the enamel developmental history and dental tissue volumes for this fetal individual. The virtual histology allowed to estimate the age at death of the fetus at 31–33 gestational weeks. Three severe physiological stress episodes were also identifed in the prenatal enamel. These stress episodes occurred during the last two months and half of pregnancy and may relate to the death of both individuals. Compared with modern prenatal standards, Os1b’s skeletal development was advanced. This cautions against the use of modern skeletal and dental references for archaeological fnds and emphasizes the need for more studies on prenatal archaeological skeletal samples

Higher Order Quantum Superintegrability: a new "Painlev\'e conjecture"

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N= 3$, 4 and 5 these equations have the Painlev\'e property. We conjecture that this is true for all $N\geq3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume "Integrability, Supersymmetry and Coherent States", a volume in honour of Professor V\'eronique Hussin. arXiv admin note: text overlap with arXiv:1703.0975