Gravitational law and spinning electron equation in a De Sitter symmetric space

A gravitational law is proposed on a De Sitter covariant space. Dirac's De Sitter covariant spinning electron equation is generalized to the presence of gravitational fields. The resulting equation differs from the generally covariant Dirac equation by a mass renormalization. The last result is a generalization of that of Gürsey and Lee [2] in case of the homogenous De Sitter metric, and it gives a wider outlook on the significance of this result from the point of view of gauge theory.Dedicated to Achille Papapetrou on the occasion of his retirement.     

The generalized dynamical equation and the vacuum polarization effect in hydrogenlike atoms

Based on the generalized dynamical equation, vacuum polarization effects are studied within the scope of the bound state theory in quantum electrodynamics. We find a vacuum-polarization correction to the Lamb shift for the 1S state of the hydrogen atom on the order of (α/π)²(Zα)⁴ that is not considered in the standard theory of bound states in quantum electrodynamics.     

Internal structure of a classical spinning electron

A classical model of the spinning electron in general relativity consisting of a rotating charge distribution with Poincaré stresses is set up. It is made out of a continuous superposition of thin charged shells with differential rotation. Each elementary shell is maintained in stationary equilibrium in the gravitational field created by the others. A class of interior solutions of the Kerr-Newman field is thus obtained. The corresponding stress-energy tensor naturally splits into the sum of two terms. The first one is the Maxwell tensor associated to a rotating charge distribution, and the second one corresponds to a material source having zero energy density everywhere, no radial pressure, and an isotropic transverse stress. These negative pressures or tensions are identified with the cohesive forces introduced by Poincaré to stabilize the Lorentz electron model. They are shown to be the source of a negative gravitational mass density and thereby of the violation of the energy conditions inside the electron.     

Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation

The nonlinear Boltzmann equation for a rarefied gas is investigated in the fluid dynamical limit to the level of compressible Euler equation locally in time, as the mean free path tends to zero. The nonlinear hyperbolic conservation laws obtained as the limit are also the first approximation of the Chapman-Enskog expansion.     

Small-scale properties of the KPZ equation and dynamical symmetry breaking

A functional integral technique is used to study the ultraviolet or short distance properties of the Kardar–Parisi–Zhang (KPZ) equation with white Gaussian noise. We apply this technique to calculate the one-loop effective potential for the KPZ equation. The effective potential is (at least) one-loop ultraviolet renormalizable in 1, 2, and 3 space dimensions, but non-renormalizable in 4 or higher space dimensions. This potential is intimately related to the probability distribution function (PDF) for the spacetime averaged field. For the restricted class of field configurations considered here, the KPZ equation exhibits dynamical symmetry breaking (DSB) via an analog of the Coleman–Weinberg mechanism in 1 and 2 space dimensions, but not in 3 space dimensions.     

Quasi-periodic solutions of the integrable dynamical systems related to Hill's equation

We present finite-gap solutions to the Garnier system and to the g-dimensional anisotropic harmonic oscillator in a radial quartic potential. The relationship between these solutions and solutions of Neumann-type dynamical systems is discussed.     

An accurate dynamical electron diffraction algorithm for reflection high-energy electron diffraction

The conventional multislice method (CMS) method, one of the most popular dynamical electron diffraction calculation procedures in transmission electron microscopy, was introduced to calculate reflection high-energy electron diffraction (RHEED) as it is well adapted to deal with the deviations from the periodicity in the direction parallel to the surface. However, in the present work, we show that the CMS method is no longer sufficiently accurate for simulating RHEED with the accelerating voltage 3–100 kV because of the high-energy approximation. An accurate multislice (AMS) method can be an alternative for more accurate RHEED calculations with reasonable computing time. A detailed comparison of the numerical calculation of the AMS method and the CMS method is carried out with respect to different accelerating voltages, surface structure models, Debye–Waller factors and glancing angles.     

Functional equation for dynamical zeta functions of Milnor-Thurston type

A Milnor-Thurston type dynamical zeta function _L (Z) is associated with a family of maps of the interval (–1, 1). Changing the direction of time produces a new zeta function _L (Z). These zeta functions satisfy a functional equation _L (Z) _L (Z)= ₀(Z) (where amounts to sign changes and, generically,₀1). The functional equation has non-trivial implications for the analytic properties of _L (Z).     

A non-linear dynamical response of an electron gas

The quadratic response of an electron gas to an external excitation is given in the self-consistent field framework.     

Bifurcation methods of dynamical systems for generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

By applying the bifurcation theory of dynamical system to the generalized KP-BBM equation, the phase portraits of the travelling wave system are obtained. It can be shown that singular straight line in the travelling wave system is the reason why smooth periodic waves converge to periodic cusp waves. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.      

Yangians, quantum groups and solutions of the quantum dynamical Yang-Baxter equation

We construct Drinfel’d twists that define deformed Hopf structures. In particular, we obtain deformed double Yangians and dynamical double Yangians. Presented by D. Arnaudon at the 10th Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June, 2001.     

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.     

Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation

In this paper, we give a detailed discussion about the dynamical behaviors of compact solitary waves subjected to the periodic perturbation. By using the phase portrait theory, we find one of the nonsmooth solitary waves of the mKdV equation, namely, a compact solitary wave, to be a weak solution, which can be proved. It is shown that the compact solitary wave easily turns chaotic from the Melnikov theory. We focus on the sufficient conditions by keeping the system stable through selecting a suitable controller. Furthermore, we discuss the chaotic threshold for a perturbed system. Numerical simulations including chaotic thresholds, bifurcation diagrams, the maximum Lyapunov exponents, and phase portraits demonstrate that there exists a special frequency which has a great influence on our system; with the increase of the controller strength, chaos disappears in the perturbed system. But if the controller strength is sufficiently large, the solitary wave vibrates violently.     

Nonlinear dynamical wave structures of Zoomeron equation for population models

The nonlinear dynamical exact wave solutions to the non-fractional order and the time-fractional order of the biological population models are achieved for the first time in the framwork of the Paul-Painlevé approach method (PPAM). When the variables appearing in the exact solutions take specific values, the solitary wave solutions will be easily obtained. The realized results prove the efficiency of this technique.     

A rational interpretation of the dirac equation for the electron

Rationalization of the interpretation of the Dirac equation for the electron lies beyond the conventional scope of quantum mechanics. This difficulty motivates a revision of the system of quantum mechanics through which the indeterministic trait is eliminated from the system.