The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operato

An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods Mod. Phy

The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation

The general construction of quasi-classically concentrated solutions to the Hartree-type equation, based on the complex WKB-Maslov method, is presented. The formal solutions of the Cauchy problem for this equation, asymptotic in small parameter \h (\h\to0), are constructed with a power accuracy of O(\h^{N/2}), where N is any natural number. In constructing the quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations (equations for middle or centered moments) is essentially used. The nonlinear superposition principle has been formulated for the class of quasi-classically concentrated solutions of the Hartree-type equations. The results obtained are exemplified by the one-dimensional equation Hartree-type with a Gaussian potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class: Accelerator PhysicsComment: 36 pages, LaTeX-2

The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

Symmetry operators and separation of variables in the (2+1)(2+1)-dimensional Dirac equation with external electromagnetic field

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.Comment: 24 pages, version accepted for publication in Int. J. Geom. Methods Mod. Phy

Star Formation History in the Galactic Thin Disk

The behavior of the relative magnesium abundances in the thin-disk stars versus their orbital radii suggests that the star formation rate in the thin disk decreases with increasing Galactocentric distance, and there was no star formation for some time outside the solar circle while this process was continuous within the solar circle. The decrease in the star formation rate with increasing Galactocentric distance is responsible for the existence of a negative radial metallicity gradient in the thin disk. At the same time the relative magnesium abundance exhibits no radial gradient. It is in detail considered the influence of selective effects on the form of both age - metallicity and age - relative magnesium abundance diagrams. It is shown that the first several billion years of the formation of the thin disk interstellar medium in it was on the average sufficiently rich in heavy elements ( = -0.22), badly mixed (\sigma_[Fe/H] = 0.21), and the average relative magnesium abundance was comparatively high ( = 0.10). Approximately 5 billion years ago average metallicity began to systematically increase, and its dispersion and the average relative magnesium abundance - to decrease. These properties may be explained by an increase in star formation rate with the simultaneous intensification of the processes of mixing the interstellar medium in the thin disk, provoke possible by interaction the Galaxy with the completely massive by satellite galaxy

Injuries of the floating crew of the Northern water pool in a state of alcoholic intoxication

The analysis of injuries of floating crew of the Northern water pool in a state of alcoholic intoxication havebeen based on the 180 accidents on board of ships with temporary loss of ability to work, and 1686 casehistories with alcoholic injuries, which demanded treatment in a surgical department. Among persons,the received injuries on the ship in the state of alcoholic intoxication were 8.1% of the victims: in thestrength of the transport fleet — 8.9%; fishing — 8.9% and river ones — 4.1%. Masters of fish production,skippers, rulers of the radio stations and masters of fish processing were most frequently injured afterthe consumption of the alcoholic beverages. Alcoholic injuries have been recorded at the time of walkingon the catwalk and the decks (54.2%), mooring operations (15.1%), maintenance and repairing the deckmachinery, water preparation (6.6%), as well as boat and loading-unloading works (4.3%). Falls from heightconstituted 36.6% of the injuries. Alcohol in 3.2 times increases the weight of the combined injuries. Thedeaths from the alcohol related injuries in marine conditions (43.4%) significantly exceed the indicators inthe group of non-alcoholic injuries (7.0%). Alcoholic intoxication has been noted in 35.0% of the cases ofthe floating crew injuries, hospitalised in the surgical department. Victims with alcoholic injuries receivedduring performing ship’s works were hospitalised 10 times less, than those with non-productive types ofinjuries. In the structure of non-industrial injuries, household injuries prevail (78.2%) over those receiveddue to falls on the street, in pedestrian flows (10.3%), transport and traffic accidents (6.7%), intentionalinjuries (4.1%) or those connected with sports games and competitions (0.7%). Fishermen are a professionalgroup of seamen, subject to the high social vulnerability to the alcoholic beverages consumptionand related injuries