Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.Supported partly by the Wittgenstein 2000 Award of Peter Markowich, funded by the Austrian Science Foundation (FWF), research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).On leave Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. Supported partly by Max Planck Institute for the Mathematics in Sciences (Leipzig) and the Austrian Science Foundation (FWF) START Project (Y-137-TEC) of Norbert Mauser.     

Adiabatic Limit for Some Nonlinear Equations of Gauge Field Theory

We consider the adiabatic limit for nonlinear dynamic equations of gauge field theory. Our main example of such equations is given by the Abelian (2+1)-dimensional Higgs model. We show next that the Taubes correspondence, which assigns pseudoholomorphic curves to solutions of Seiberg--Witten equations on symplectic 4-manifolds, may be interpreted as a complex analogue of the adiabatic limit construction in the (2+1)-dimensional case.     

Nonrelativistic Limit of Scattering Theory for Nonlinear Klein-Gordon Equations

We study the nonrelativistic limit of the nonlinear Klein-Gordon equation and prove that the wave operators, the inverses of them, and the scattering operator for the naturally modulated equation converge to those for the singular limit, the nonlinear Schrödinger equation.     

Erratum to: Static Limit and Penrose Effect in Rotating Reference Frames

Theoretical and Mathematical Physics - Reference [3] on page 1125, 3. A. A. Grib and Yu. V. Pavlov, “Comparison of particle properties in Kerr metric and in rotating coordinates,” Gen....     

Finite Energy Scattering for the Lorentz–Maxwell Equation

In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz–Maxwell equation (Abraham model) for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation. Submitted: October 31, 2007. Accepted: April 21, 2008.     

Dynamical Locality of the Free Maxwell Field

The extent to which the non-interacting and source-free Maxwell field obeys the condition of dynamical locality is determined in various formulations. Starting from contractible globally hyperbolic spacetimes, we extend the classical field theory to globally hyperbolic spacetimes of arbitrary topology in two ways, obtaining a ‘universal’ theory and a ‘reduced’ theory of the classical free Maxwell field and their corresponding quantisations. We show that the classical and the quantised universal theory fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical pre-symplectic spaces and non-trivial centres in the quantised *-algebras. The classical and the quantised reduced theory are both locally covariant and dynamically local, thus closing a gap in the discussion of dynamical locality and providing new examples relevant to the question of how theories should be formulated so as to describe the same physics in all spacetimes.     

大型自行车共享系统的平均场极限理论与排队模型研究

人口的快速增长与空间的高度城市化带来了汽车尾气污染等环境污染问题,这已成为影响社会可持续发展的主要制约因素。基于此,近十年来自行车共享系统在世界多个国家的许多重要城市获得了高度重视并取得了迅速发展。然而,相比于自行车共享系统的快速发展,其相关研究却并未取得较大进展,主要原因在于它是一个大型的复杂系统,涉及密集的城市交通、异构的运营环境、多重的顾客偏好选择以及多渠道的收益管理等多种关联因素。在这种背景下,本文建立了一个通用的大型自行车共享系统,并提出了一种基于平均场极限理论与非时齐排队模型相结合的有效随机模型分析方法,包括利用平均场理论建立了非时齐排队系统、构建了经验测度过程(Empirical measure process)的非线性生灭过程、给出了分段结构下生灭过程的固定点的“几何之和”算法以及提供了问题站点稳态概率的数值计算等等。本文为研究大型自行车共享系统的随机模型提供了一个重要的发展途径,并有望能够用于分析更加一般的大型自行车共享系统。     

Rotating Loop Soliton of the Coupled Dispersionless Equations

We investigate soliton solutions of the coupled dispersionless equations that describe a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space with bilinear equations. We obtain a new type of loop soliton solutions that rotate around the Z axis. We also investigate the two-soliton interaction.     

Particles in a Singular Asymptotic Limit of a Classical Field Theory

We give a detailed discussion of the singular asymptotic limit of a classical field theory as a small parameter goes to zero. The phenomenon under investigation is the appearance of particles in the limit. The particles are rescaled kinks of a sine-Gordon field acted on by a second field. As the parameter goes to zero, these kinks shrink while their energy is invariant, leaving point-like particles accelerated by the second field. The validity of this picture of the asymptotics hinges on the dispersive nature of the radiative correction to the kink produced by the perturbation. This is analysed using a perturbation theory based on complete integrability.     

Dispersion Relations for the Forward Elastic Scattering Amplitude in Noncommutative Quantum Field Theory

We show that the forward elastic scattering amplitude of two massive spinless particles in noncommutative quantum field theory has the same analytic properties as in the commutative case if the noncommutativity condition involves only the spatial variables.     

Static Limit and Penrose Effect in Rotating Reference Frames

Theoretical and Mathematical Physics - We show that effects similar to those for a rotating black hole arise for an observer using a uniformly rotating reference frame in a flat space-time: a...     

Homogenization of the Maxwell Equations: Case I. Linear Theory

The Maxwell equations in a heterogeneous medium are studied. Nguetseng's method of two-scale convergence is applied to homogenize and prove corrector results for the Maxwell equations with inhomogeneous initial conditions. Compactness results, of two-scale type, needed for the homogenization of the Maxwell equations are proved.     

非线性耦合标量场方程新的精确行波解

对于具有丰富物理意义和众多应用价值的非线性耦合标量场方程,通过将所求方程约化为初等积分形式,再利用多项完全判别系统对被积函数中的多项式的根进行分类,得到该方程的丰富的精确解,其中包含有理函数型解,孤波解,三角函数型周期解,椭圆函数型周期解,这其中有许多是新解.这一方法是非常简洁而又有力的,对于椭圆方程,它能得到所有可能的解,对于更高阶方程,与其它方法相比,多项式完全判别系统方法是更有力的.     

Theory of the Maxwell effect for large velocity gradients

Translated from Issledovaniya po Prikladnoi Matematike, Kazan', No. 14, pp. 62–66, 1987.     

One Non-Relativistic Particle Coupled to a Photon Field

We investigate the ground state energy of an electron coupled to a photon field. First, we regard the self-energy of a “free” electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant a \alpha , the leading order term is represented by 2ap^-1(L - ln[1 + L]) 2\alpha\pi^{-1}(\Lambda - \textrm{ln}[1 + \Lambda]) .¶ Next we put the electron in the field of an arbitrary external potential V , such that the corresponding Schrödinger operator p² + V has at least one eigenvalue, and show that by coupling to the radiation field the binding energy increases, at least for small enough values of the coupling constant a \alpha . Moreover, we provide concrete numbers for a \alpha , the ultraviolet cut-off, and the radiative correction for which our procedure works.     

On the lnviscid Theory of Rotating Fluids

Conservation theorems for “energy” and mean circulation in an inviscid rotating fluid are stated in their most general forms (which do not presume the existence of inertial modes). The geostrophic motion that contains all the mean circulation of an initial velocity field is identified as the time average of the flow. A preliminary study of the in viscid motion inside a closed cone reveals many singular features and indicates that the inertial wave spectrum may be continuous.     

Field Theory Approach to the Dynamics of a Continuous Medium: A Perturbation Theory

We propose a perturbation theory that allows solving the equations of motion for the displacement vector in the body of the Earth in the framework of the linear theory of elasticity. We show that tectonic processes are primarily determined by tidal actions. We analyze the tidal effects in the Earth–Moon–Sun system.     

巴氏空间中随机元的极限理论及其应用

本文综述了巴拿赫空间中随机元的极限理论及其应用的一些结果，主要内容为：极限理论的某些新结果；等周方法的优化技巧；巴氏空间几何的概率方法；在经验过程研究中的应用。重点是近十年来有关问题研究的一些进展。