Vladislav V. Kravchenko
We show that the Maxwell equations for arbitrary inhomogeneous media are equivalent to a single quaternionic equation which can be considered as a generalization of the Vekua equation for generalized analytic functions.
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We propose a simple quaternionic reformulation of Maxwell's equations for inhomogeneous media and use it in order to obtain new solutions in a static case.
We obtain a canonical form of a quadratic Hamiltonian for linear waves in a weakly inhomogeneous medium. This is achieved by using the WKB representation of wave packets. The canonical form of the Hamiltonian is obtained via the series of canonical Bogolyubov-type and near-identical transformations. Various examples of the application illustrating the main features of our approach are presented. The knowledge of the Hamiltonian structure for linear wave systems provides a basis for developing a theory of weakly nonlinear random waves in inhomogeneous media generalizing the theory of homogeneou…
We consider the static Maxwell system with an axially symmetric dielectric permittivity and construct complete systems of its solutions which can be used for analytic and numerical solution of corresponding boundary value problems.
General solutions to the equations of acoustics in inhomogeneous media and gas dynamics
该研究利用 Riemann 方法将常熵气体动力学方程化为变系数二阶线性双曲型方程，并借助 Euler 变换构造其解，从而得到依赖两个任意函数的声学与气体动力学新精确解。
We consider a coupled system describing the interaction between acoustic and elastic regions, where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes. Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions. The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions.
It is shown (1) that Lanczos's quaternionic formulation of Dirac's equation does not lead to a solution of the problems that plague the standard spin 3/2 theory based on the Rarita-Schwinger equation, but (2) that the four-component solutions to the quaternionic generalization of Dirac's equation proposed by Lanczos in 1929 may provide a consistent theory for spin 3/2 particles, although at the cost of giving up the postulate that there should be a one-to-one correspondence between arbitrary-high-spin unitary representations of the inhomogeneous Lorentz group and elementary particles.
We generalize Weyl's law to inhomogeneous bodies in $d$ dimensions. Using a perturbation scheme recently obtained by us in Ref. \cite{Amore09}, we have derived an explicit formula, which describes the asymptotic behavior of the eigenvalues of the negative laplacian on a closed $d$-dimensional cubic domain, either with Dirichlet or Neumann boundary conditions. For homogeneous bodies, the leading term in our formula reduces to the standard expression for Weyl's law. We have also used Weyl's conjecture to obtain a non-perturbative extension of our formula and we have compared our analytical resul…
We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with bi…
We obtain the hydrodynamic limit of a simple exclusion process in an inhomogeneous environment of divergence form. Our main assumption is a suitable version of Gamma-convergence for the environment. In this way we obtain an unified approach to recent works on the field.
We investigate the long-time evolution of branching diffusion processes (starting with a finite number of particles) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the super-critical regime, we describe the asymptotics of the number of particles in a given domain. In the sub-critical and critical regimes, we show that the limiting number of particles is finite and describe its distribution.
The problem of internal gravity waves generation by a point source moving in a stratified medium varying with respect to all spatial variables and time is considered. The fact that the characteristic horizontal scales of density variation may be large as compared with lengths of internal gravity waves makes it possible to solve this problem by means of an approximate method analogous to the WKB method. Therefore, in investigating the problem of internal gravity waves generation by a point source moving in a stratified horizontally inhomogeneous unsteady medium solution for individual mode must…
Representation of Linear Waves in Inhomogeneous Media
本文探讨了非均匀介质中的声学方程和线性化浅水方程，提出了基于Laplace级联法及降维波动方程的两种积分方法，并获得了平面波及可简化为常系数方程的解。
Electromagnetic Casimir stresses are of relevance to many technologies based on mesoscopic devices such as MEMS embedded in dielectric media, Casimir induced friction in nano-machinery, micro-fluidics and molecular electronics. Computation of such stresses based on cavity QED generally require numerical analysis based on a regularization process. A new scheme is described that has the potential for wide applicability to systems involving realistic inhomogeneous media. From a knowledge of the spectrum of the stationary modes of the electromagnetic field the scheme is illustrated by estimating n…
Diffusion in inhomogeneous materials can be described by both the Fick and Fokker--Planck diffusion equations. Here, we study a mixed Fick and Fokker-Planck diffusion problem with coefficients rapidly oscillating both in space and time. We obtain macroscopic models performing the homogenization limit by means of the unfolding technique.
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.
The accuracy and stability of the least squares finite element method (LSFEM) and the Galerkin finite element method (GFEM) for solving radiative transfer in homogeneous and inhomogeneous media are studied theoretically via a frequency domain technique. The theoretical result confirms the traditional understanding of the superior stability of the LSFEM as compared to the GFEM. However, it is demonstrated numerically and proved theoretically that the LSFEM will suffer a deficiency problem for solving radiative transfer in media with strong inhomogeneity. This deficiency problem of the LSFEM wil…
We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background $(\mathbb{R}^{3+1}, g)$ with a \textbf{time dependent metric $g$} coinciding with Minkowski metric outside the cylinder $\{(t, x)| |x|\leq R\}$. We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate these estimates in the particular case when $g$ is merely $C^1$ close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do…
A mathematical description of the spin Hall effect of light in inhomogeneous media
研究Maxwell方程在各向同性非均匀介质中的高斯波包解，推导描述几何运动修正的常微分方程组，证明光在非均匀介质中的自旋霍尔效应。
Recent progress in experimental techniques has made the quantum regime in plasmonics accessible. Since plasmons correspond to collective electron excitations, the electron-electron interaction plays an essential role in their theoretical description. Within the Random Phase Approximation, this interaction is incorporated through a system of equations of motion, which has to be solved self-consistently. For homogeneous media, an analytical solution can be found using the Fourier transform, giving rise to Lindhard theory. When the medium is spatially inhomogeneous, this is no longer possible and…
In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by the presence of zero-modes that may appear in the presence of non-trivial topology of the manifold or the boundary. An important special case is $\mathbb{R}^3$ with obstacles. In this case the zero modes have a direct interpretation in terms of the topology of the obstacle. I give a formula for the renormalised stress energy tensor in terms of an integral kernel of an operator defined by spectr…
Spectral problems in inhomogeneous media, spectral parameter power series and transmutation operators
本文综述了Sturm-Liouville理论中关于变换算子和谱参数幂级数的最新进展，并提出了一种数值求解相应谱问题的新方法。
This paper is concerned with the invisibility cloaking in electromagnetic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. Our study is based on an interior transmission eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish t…
The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see arXiv:1001.0552. In arXiv:1410.4873 using this reduction a representation of a general solution of the system was obtained in terms of a couple of Darboux-associated transmutation operators arXiv:1111.4449. In arXiv:1508.02738 a Fourier-Legendre expansion of transmutation integral kernels was obtained. This expansion is used in the present work for obtaining an exa…
In \cite{J} M. Jara has presented a method, reducing the proof of the hydrodynamic limit of symmetric exclusion processes to an homogenization problem, as unified approach to recent works on the field as \cite{N}, \cite{F1}, \cite{F2} and \cite{FJL}. Although not stated in \cite{J}, the reduction of the hydrodynamic limit to an homogenization problem was already obtained (in a different way) in \cite{N}, \cite{F1}. This alternative and very simple relation between the two problems goes back to an idea of K.\ Nagy \cite{N}, is stated in \cite{F1}[Section B] for exclusion processes on $\bbZ^d$ a…
We consider the ground state energy of the electromagnetic field in a piston geometry. In the idealised case, where the piston and the walls of the chamber are taken as perfect mirrors, the Casimir pressure on the piston is finite and independent of the small scale physics of the media that compose the mirrors; the Casimir-energy of the system can be regularised and is cutoff-independent. Yet we find that, when the body of the piston is filled with an inhomogeneous dielectric medium, the Casimir energy is cutoff-dependent, and the value of the pressure is thus inextricably dependent on the det…
The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given $α\in \R$ a medium that generates a resonance on the line $α+ ı\R$ with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints $0\leq b_1 \leq B (x) \leq b_2$ with certain constants $b_{1,2}$. The paper gives an accurate definit…
Modulated electromagnetic fields in inhomogeneous media, hyperbolic pseudoanalytic functions and transmutations
本文研究了非均匀介质中电磁波传播问题，利用双复值函数和双曲伪解析函数，通过双曲变量建立Vekua型方程，并发展了数值方法以高效计算传输调制信号。
In this work we calculate the closed time path (CTP) generating functional for the electromagnetic (EM) field interacting with inhomogeneous anisotropic matter. For this purpose, we first find a general expression for the electromagnetic field's influence action from the interaction of the field with a composite environment consisting in the quantum polarization degrees of freedom in each point of space, at arbitrary temperatures, connected to thermal baths. Then, we evaluate the generating functional for the gauge field, in the temporal gauge, by implementing the Faddeev-Popov procedure. Fina…
Upscaling the Navier-Stokes-Cahn-Hilliard model for incompressible multiphase flow in inhomogeneous porous media
本文提出一种宏观数学模型，用于描述不均匀多孔介质中不可压缩双相流的宏观行为，通过引入润湿行为并结合体积平均方法，实现了对两相流动的精确模拟。
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