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).Comment: 12 pages, no figure, to appear, Theoretical and Mathematical Physic

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.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica

High performance parallel numerical methods for Volterra equations with weakly singular kernels

Non-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations

On elliptic solutions of the cubic complex one-dimensional Ginzburg-Landau equation

The cubic complex one-dimensional Ginzburg-Landau equation is considered. Using the Hone's method, based on the use of the Laurent-series solutions and the residue theorem, we have proved that this equation has neither elliptic standing wave nor elliptic travelling wave solutions. This result amplifies the Hone's result, that this equation has no elliptic travelling wave solutions.Comment: LaTeX, 12 page

Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians

The quartic H\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\Omega_1 Q_1^2+\Omega_2 Q_2^2)/2 +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4 +(1/2)(\alpha/Q_1^2+\beta/Q_2^2) - \gamma Q_1$ passes the Painlev\'e 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 $(\alpha,\beta,\gamma)\not=(0,0,0)$. 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 Painlev\'e property. This transformation involves the stationary reduction of various partial differential equations (PDEs). The result is the same as for the three cubic H\'enon-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 Painlev\'e integrability (completeness property).Comment: 10 pages, To appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200

Ionic Liquids as Reaction Media in Catalytic Oxidations with Manganese and Iron Pyridyl Triazacyclononane Complexes

A family of bioinspired iron and manganese complexes of general formula [MII(CF3SO3)2(Me,XPyTACN)], where Me,XPyTACN = 1-[2’-(6-X-pyridyl)methyl]-4,7-dimethyl-1,4,7-triazacyclononane, and M = Fe, and Mn has been studied as efficient catalytic systems for hydrogen peroxide oxidation reactions. Previous work revealed that the manganese derivative [MnII(CF3SO3)2(Me,HPyTACN)], 1, in acetonitrile exhibits a high catalytic activity in the epoxidation of a wide range of olefins (TON: 810-4500), using acetic acid and hydrogen peroxide as primary oxidant. The analogous iron based complex [FeII(CF3SO3)2(Me,HPyTACN)], 2a and [FeII(CF3SO3)2(Me,MePyTACN)], 2b promote the high added value oxidation reaction of alkanes in mild conditions. In this work sustainability and selectivity of the oxidative system is improved with the use of the ionic liquids (ILs) as reaction medium. The possibility to recycle the catalytic phase without loss of the activity with respect to the original reaction in acetonitrile (MeCN) is reported

Multivortex Solutions of the Weierstrass Representation

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlev\'e test and analyse the possibility of deriving the B\"acklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlev\'{e} equations to construct explicit formulae in terms of the Umemura polynomials which are $\tau$-functions for rational solutions of the third Painlev\'{e} equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex Sine-Gordon equation is reduced to coupled Riccati equations.Comment: 27 pages LaTeX2e, 1 encapsulated Postscript figur

Generalized Lenard Chains, Separation of Variables and Superintegrability

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.Comment: 14 pages Revte

Correspondence Model Of Occupational Accidents

We present a new generalized model for the diagnosis and prediction of accidents among the Spanish workforce. Based on observational data of the accident rate in all Spanish companies over eleven years (7,519,732 accidents), we classified them in a new risk-injury contingency table (19x19). Through correspondence analysis, we obtained a structure composed of three axes whose combination identifies three separate risk and injury groups, which we used as a general Spanish pattern. The most likely or frequent relationships between the risk and injuries identified in the pattern facilitated the decision-making process in companies at an early stage of risk assessment. Each risk-injury group has its own characteristics, which are understandable within the phenomenological framework of the accident. The main advantages of this model are its potential application to any other country and the feasibility of contrasting different country results. One limiting factor, however, is the need to set a common classification framework for risks and injuries to enhance comparison, a framework that does not exist today. The model aims to manage work-related accidents automatically at any level