On Evolution Galerkin Methods for the Maxwell and the Linearized Euler Equations

The subject of the paper is the derivation and analysis of evolution Galerkin schemes for the two dimensional Maxwell and linearized Euler equations. The aim is to construct a method which takes into account better the infinitely many directions of propagation of waves. To do this the initial function is evolved using the characteristic cone and then projected onto a finite element space. We derive the divergence-free property and estimate the dispersion relation as well. We present some numerical experiments for both the Maxwell and the linearized Euler equations.     

Integrability and soliton-like solutions for the coupled higher-order nonlinear Schrödinger equations with variable coefficients in inhomogeneous optical fibers

Under investigation in this paper are the coupled higher-order nonlinear Schrödinger equations with variable coefficients, which represent the propagation of femtosecond soliton pulses comprising of two fields with the left and right polarization in the inhomogeneous optical fiber media. Infinitely-many conservation laws are obtained based on the Lax pair. Via the Hirota method and symbolic computation, bilinear forms, bilinear Bäcklund transformations, one- and two-soliton-like solutions are also derived. With different coefficients, bell-shaped, periodic-changing, quadratic-varying, exponential-decreasing and exponential-increasing soliton-like profiles are seen, to describe the propagation and interactions of the femtosecond soliton pulses. Head-on and overtaking elastic interactions are shown, which are decided by the directions of the velocities. We also get the bound states with periodic attraction and repulsion between two solitons.     

The plane dynamic problem of coupled thermoelasticity

A method of solving two-dimensional inner and outer boundary-value problems of coupled thermoelasticity, taking into account the finite propagation velocity of heat pulses, is proposed, based on constructed fundamental solutions of the corresponding equations. An estimate is given of the coupling of thermomechanical fields in these problems, and the hyperbolic and parabolic models of thermal conductivity are compared. It is shown that the effect of the finite propagation velocity of heat is unimportant even for very short periods of the duration of the processes (comparable with the relaxation time of the heat flux).     

Strong-coupling theory for multiband superconductors

We consider a microscopic theory of the strong coupling in multiband superconductors with an arbitrary electron-boson interaction. Based on the method of the equations of motion for two-time Green’s functions, we derive the Dyson equation with the self-energy operator in the form of the multiparticle Green’s function taking the interaction of electrons with phonons and spin fluctuations into account. We obtain a self-consistent system of equations for the normal and anomalous components of the Green’s function and the self-energy operator calculated in the approximation of noncrossing diagrams. We discuss the approximate solution of the system of equations taking only components of the self-energy operator that are diagonal with respect to the band index into account for studying superconductivity in iron-based compounds.     

On soliton and periodic solutions of Maxwell–Bloch system for two-level medium with degeneracy

Maxwell–Bloch (MB) system describing the resonant propagation of electromagnetic pulses in either two-level media with degeneracy in angle moment projection or a three-level media with equal oscillator forces is considered. An inhomogeneous broadening of energy levels and a polarization of the wave are accounted. The equations are integrated by the binary Darboux transformations technique. Pulses corresponding to a transition between levels with the largest population difference are shown to be stable. The solution describing the propagation of pulses in the medium excited by a periodic wave is obtained. The hierarchy of infinitesimal symmetries is obtained by means of Darboux transformation.     

Quantum equations of motion for multimode laser generation with a spatial dependence of the atom interaction with the field taken into account

We derive equations of motion for the electromagnetic field operators a_q′ ⁺ a_q″ for a three-level multimode laser with a spatial dependence of the interaction of atoms with the field of a standing wave in a cavity taken into account. We calculate and analyze the dynamics of means of photon numbers in the field modes and of the correlation function of field modes. We explore the effect of intermode correlations on the dynamics of establishing stationary laser generation. We find that taking the spatial dependence of the interaction of atoms with the field and the intermode correlation into account in investigating the means of photon numbers leads to revealing new properties of laser generation, such as saturation of the laser radiation intensity in a single-mode regime and generation of short light pulses of side below-threshold modes with the amplitudes depending on the initial state of the field in a cavity.     

On the approximate solution of some two‐dimensional singular integral equations

An approximation method for a wide class of two‐dimensional integral equations is proposed. The method is based on using a special function system. Orthonormality and good interaction with fundamental integral operators arising in partial differential equations are remarkable properties of this system. In addition, all the basis elements can easily be calculated by recurrence relations. Taking into account these properties we construct a numerical algorithm which does not require additional effort (such as quadrature) to compute the values of the fundamental operators on the basis elements. Copyright © 2001 John Wiley & Sons, Ltd.     

On a representation of fully nonlinear elliptic operators in terms of pure second order derivatives and its applications

We give a method of representing fully nonlinear elliptic operators given by boundedly inhomogeneous functions in terms of operators acting only on pure second order derivatives. We also discuss possible applications of such representations to using .finite di.erence approximations for solving the corresponding equations.     

Decay estimates for evolution equations with classical and fractional time-derivatives

Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving both standard and Caputo time-derivative, complex valued magnetic operators, fractional porous media equations and nonlocal Kirchhoff operators.Both local and fractional space diffusion are taken into account, possibly in a nonlinear setting. The different quantitative behaviours, which distinguish polynomial decays from exponential ones, depend heavily on the structure of the time-derivative involved in the equation.     

双正侧函数的非线性带位移边值问题

Ｃｌｉｆｆoｒｄ分析中的双正则函数是一类广义正则函数,它的研究是近年来函数论领域内的一个热门分支,本文研究双正则函数的非线性带位移的边值问题．设计积分算子,将边值问题转化成积分方程问题,借助于积分方程理论和Ｓｃｈａｕｄｅｒ不动点理论证明了边值问题解的存在性并给出了解的积分表达式．     

Fourth-order schemes for the wave equation,Maxwell equations,and linearized elastodynamic equations

With simple finite-difference operators we construct fourth-order schemes in space and in time for the wave equation, Maxwell equations, and linearized elastodynamic equations using the modified equation approach. The schemes remain stable for arbitrary heterogeneous mediums because the relevant difference operators are always positive definite. We also present some dispersion curves to show the accuracy of the schemes. © 1994 John Wiley & Sons, Inc.     

Simultaneous observability and stabilization of some uncoupled wave equations

We consider uncoupled wave equations with different speed of propagation in a bounded domain. Using a combination of the Bardos–Lebeau–Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations, we prove an observability estimate for that system. Applying Lions? Hilbert uniqueness method (HUM), one may derive simultaneous exact controllability results for the uncoupled system; the controls being locally distributed, with their supports satisfying the geometric control condition of Bardos, Lebeau and Rauch. Afterwards, we discuss the related simultaneous stabilization problem; this latter problem is solved by a combination of the new observability inequality, and a result of Haraux establishing an equivalence between observability and stabilization for second order evolution equations with bounded damping operators. Our observability and stabilization results generalize to higher space dimensions some earlier results of Haraux established in the one-dimensional setting.     

Mechanoelectromagnetic interaction in isotropic dielectrics with regard for the local displacement of mass

On the basis of the earlier proposed model of electromagnetothermomechanics of polarizable bodies, which takes into account the process of local displacement of mass, key equations for the corresponding scalar and vector potentials are written. The generalization of the Lorentz gauge, at which the equations for calculating the scalar and vector potentials of an electromagnetic field become uncoupled, is proposed. We write a resolving system of equations for potentials in the dimensionless form and obtain a parameter of interrelation of the processes of local displacement of mass and deformation. With the use of this system, the propagation of a plane harmonic wave in an infinite isotropic medium is investigated. It is shown that the model describes the dispersion of a modified elastic wave in the high-frequency region. The obtained results agree with known data presented in the literature and obtained from relations of the gradient theory of piezoelectrics.     

Time-harmonic acoustic propagation in the presence of a shear flow

This work deals with the numerical simulation, by means of a finite element method, of the time-harmonic propagation of acoustic waves in a moving fluid, using the Galbrun equation instead of the classical linearized Euler equations. This work extends a previous study in the case of a uniform flow to the case of a shear flow. The additional difficulty comes from the interaction between the propagation of acoustic waves and the convection of vortices by the fluid. We have developed a numerical method based on the regularization of the equation which takes these two phenomena into account. Since it leads to a partially full matrix, we use an iterative algorithm to solve the linear system.     

Differentialoperatoren bei partiellen Differentialgleichungen

Summary In the present paper those formally hyperbolic differential equations are characterized for which solutions can be represented by means of differential operators acting on holomorphic functions. This is done by a necessary and sufficient condition on the coefficients of the differential equation. These operators are determined simultaneously. By it a general procedure is presented to construct differential equations and corresponding differential operators which map holomorphic functions onto solutions of the differential equations. We also discuss the question under which circumstances all the solutions of a differential equation can be represented by differential operators. For the equations characterized previously we determine the Riemann function. Some special classes of differential equations are investigated in detail. Furthermore the possibility of a representation of pseudoanalytic functions and the corresponding Vekua resolvents by differential operators is discussed.

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Herrn Prof. Dr. K. W. Bauer zum 60. Geburtstag gewidmet     

A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras

The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and momenta are undetermined, we arrive at the equations that contains basis vectors and their time derivatives only. According to this approach, quantization of a classical theory, formulated in phase space, is replacement of the phase space variables with the corresponding basis vectors (operators). The basis vectors, transformed into the Witt basis, satisfy the bosonic or fermionic (anti)commutation relations, and can create spinor states of all minimal left ideals of the corresponding Clifford algebra. We consider some specific actions for point particles and fields, formulated in terms of commuting and/or anticommuting phase space variables, together with the corresponding symplectic or orthogonal basis vectors. Finally we discuss why such approach could be useful for grand unification and quantum gravity.     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.     

Wave Front Set at Infinity and Hyperbolic Linear Operators with Multiple Characteristics

We discuss a notion of wave front set which allows us tocontrol the behaviour `at infinity' of temperate distributions. Weobtain the microlocality and microellipticity properties with respect toa class of global pseudodifferential operators and a propagation theoremfor the corresponding class of Fourier Integral Operators. Through theseresults, we prove an adapted global version of the classical theoremconcerning the singularities of solutions of hyperbolic Cauchy problemsfor linear operators with multiple characteristics of constantmultiplicities. Finally, we make a comparison with the scattering wavefront set introduced by Melrose.     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.     

Evolution Galerkin methods for hyperbolic systems in two space dimensions

The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.