Lotka-Volterra-type equations and their explicit integration

In this Letter, we give an explicit integration of some two-dimensionalised Lotka-Volterra-type equations associated with simple Lie algebras possessing a representation without branching.     

The critical temperature and gap solution in the Bardeen-Cooper-Schrieffer theory of superconductivity

The Letter studies the problem of numerical approximations of the critical transition temperature and the energy gap function in the Bardeen-Cooper-Schrieffer equation arising in superconductivity theory. The positive kernel function leads to a phonon-dominant state at zero temperature. Much attention is paid to the equation defined on a bounded region. Two discretized versions of the equation are introduced. The first version approximates the desired solution from below, while the second, from above. Numerical examples are presented to illustrate the efficiency of the method. Besides, the approximations of a full space solution and the associated critical temperature by solution sequences constructed on bounded domains are also investigated.Part of this work was done while this author was visiting the Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA, 15213, USA     

Duality for Jacobi Group Orbit Spaces and Elliptic Solutions of the WDVV Equations

From any given Frobenius manifold one may construct a so-called ’dual’ structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the Witten– Dijkgraaf–Verlinde–Verlinde equations of associativity. Jacobi group orbit spaces naturally carry the structure of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.     

Orbital stability of peakons with nonvanishing boundary for CH and CH-γ equations

In this Letter, we consider the general expressions of peaked traveling wave solutions for CH and CH-γ equations. The orbital stability of these peakons are directly proved in the H¹ norm. Some previous results are extended.     

Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations

In this Letter, homotopy perturbation method (HPM) is directly applied to modified Camassa-Holm and Degasperis-Procesi equations to obtain their solitary wave solutions. The results show the applicability, accuracy and efficiency of HPM in solving nonlinear differential equations with fully nonlinear dispersion term. It is predicted that HPM can be widely applied in engineering problems.     

Integral invariants of the Buttin bracket

We study the densities (most general objects which may be integrated over supersurfaces in superspace), invariant with respect to supercanonical transformations which do not change the Buttin bracket. The only such nontrivial object is, in a definite sense, the odd semidensity explicitly constructed here.     

Difference schrödinger operators with linear and exponential discrete spectra

Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia.     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Syzygies among elementary string interactions in 2+1 dimensions

The elementary string interactions are the Reidemeister moves, birth/death, and fusion/fission. Such interactions have as their trace generically mapped surfaces, and these combine to form knotted surfaces in 4-space. The syzygies among these interactions are moves to such surfaces analogous to the Reidemeister moves for knots. Movie parametrizations of these syzygies are given and interpreted in dimension 2+2. A Morse theoretic argument shows there are 15 such movie moves.These moves, with appropriate choices of crossing information, are sufficient to construct any isotopy of an embedded surface on which a height function has been specified. The first seven of the movie moves are parametrized versions of those given by Roseman. The remaining eight are moves of -type.Amplitudes assigned to these interactions must satisfy relations that correspond to the movie moves. One such relation is a Zamolodchikov tetrahedral equation. We present some puzzles about the matrix formulations of these amplitudes.Supported by a grant from the University of South Alabama Research Committee.     

Laser scattering by metallic particles in two-phase nozzle flow into a vacuum

For several years the Max-Planck-Institute for Extraterrestrial Physics has performed experiments with artificial barium clouds in space by means of research rockets and satellites in order to investigate electric fields and currents in the ionosphere and in the magnetosphere. The processes which lead to the formation of these artificial clouds were simulated in laboratory in order to optimize the barium vapor yield and to obtain a better insight into the initial phase of the clouds. In these experiments the combustion of excess barium with copper oxide, the two-phase flow of the combustion products through a nozzle and their expansion into a 20 m³ vacuum chamber were investigated in detail. Purpose of this work is to measure through detection techniques of scattered laser light and thermal radiation: 1) the density of small metall particles on the particle jet axis, 2) the particle velocity, 3) the thermal radiation of larger particles, and 4) the position of the Mach disk which limits the free nozzle jet downstream. This work was carried out at the Max-Planck-Institut für extraterrestrische Physik, Garching, Germany.     

Gromov–Witten invariants of Fano hypersurfaces, revisited

The goal of this paper is to give an efficient computation of the genus zero three-point Gromov–Witten invariants of Fano hypersurfaces, starting from the Picard–Fuchs equation. This simplifies and to some extent explains the original computations of Jinzenji. The method involves solving a gauge-theoretic differential equation, and our main result is that this equation has a unique solution.     

On Generalized Non-holonomic Systems

Generalized non-holonomic mechanical systems are analyzed from a geometric point of view. The existence and uniqueness of solutions, D’Alembert principle, Gauss principle of minimal constraint, the non-holonomic momentum and Gibbs–Appell equations for such systems are studied in an invariant Lagrangian framework.      

Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity

In the present study, we have analyzed the effects of variable viscosity, variable thermocapillarity on the flow and heat transfer in a thin film on a horizontal porous shrinking sheet through a porous medium. The unsteady boundary layer equations for momentum and thermal energy are simplified by using similarity transformations. The resulted, coupled nonlinear differential equations are solved by homotopy analysis method. The results are presented graphically to interpret various physical parameters appearing in the problem.     

Suppression of Chaos and Phase Locking in Two Coupled Nonidentical Neurons under Periodic Input

Dynamical behaviour of two coupled neurons with at least one of them being chaotic is presented. Bifurcation diagrams and Lyapunov exponents are calculated to diagnose the dynamical behaviour of the coupled neurons with the increasing coupling strength. It is found that when the coupling strength increases, a chaotic neuron can be controlled by the coupling between neurons. At the same time, phase locking is studied by the maxima of the differences of instantaneous phases and average frequencies between two coupled neurons, and the inherent connection of phase locking and the suppression of chaos is formulated. It is observed that the onset of phase locking is closely related to the suppression of chaos. Finally, a way for suppression of chaos in two coupled nonidentical neurons under periodic input is suggested.     

Symmetry of Lagrangians of nonholonomic systems

In this Letter, a new symmetry of nonholonomic systems is put forward. Firstly, the definition and the criterion of the symmetry are given. Secondly, the condition under which the symmetry can lead to a conserved quantity is presented. Finally, two examples are shown to illustrate the application of the results.