Method of Propagating Waves for a One-Dimensional Inhomogeneous Medium

The aim of this work is to develop a method of propagating waves based on the idea of a wave as a changing state of a medium. This method allows us to represent a solution of the one-dimensional wave equation in an inhomogeneous medium as the sum of two constantly deformed waves, the right wave and the left wave, transported from point to point with coefficients depending on the points and the transport time. By the propagating-wave method we obtain explicit (as far as possible) formulas for solutions of the mixed problem with homogeneous and inhomogeneous boundary conditions and solutions of the Goursat problem. The derivation of these formulas is based on special convolution formulas for the transport coefficients that are similar to the addition identities for trigonometric functions.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 3–43, 2004.     

Propagation of Reactions in Inhomogeneous Media

Consider reaction‐diffusion equation u _t =Δ u + f (x,u ) with and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0,1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the superlevel sets {u ≥ ε } and {u ≥ 1‐ε} remains uniformly bounded in time for each ε ? (0,1). Thus, u remains uniformly in time close to the characteristic function of in the sense of Hausdorff distance of superlevel sets. We also show that each {u ≥ ε} expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x ‐independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 is via construction of a counterexample for which this fails. Such results were before known for d =1 but are new for general non‐periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria of the PDE and to solutions not necessarily satisfying . © 2016 Wiley Periodicals, Inc.     

Ginzburg-Landau方程的非齐次初边值问题

研究具非线性边界条件的一类广义Ginzburg-Landau方程解的整体存在性．推导了Ginzburg-Landau方程的非齐次初边值问题光滑解的几个积分恒等式,由此得到了解的法向导数在边界上的平方模以及解的平方模和导数的平方模估计;通过逼近技巧、先验估计和取极限方法证明了Ginzburg-Landau方程的非齐次初边值问题整体弱解的存在性．     

Nonstationary Electromagnetic Gaussian Beams in Inhomogeneous Anisotropic Media

Nonstationary Gaussian beams of quasiphoton type for the Maxwell equation with an arbitrary anisotropy are constructed. The solutions of the Maxwell equations can be described as ray-type solutions with complex phases and amplitudes. Owing to a large parameter p, they are concentrated in small neighborhoods of space-time rays corresponding to different types of electromagnetic waves in an anisotropic medium. Bibliography: 6 titles.     

Inhomogeneous Plane Waves in Monoclinic Crystals subject to a Bias

Here we investigate the conditions of inhomogeneous plane waves propagation in monoclinic crystals subject to initial electromechanical fields. We obtain the components of the electroacoustic tensor for the class 2, resp. m, of the monoclinic system. For particular isotropic directional bivectors we derive the decomposition of the propagation condition, and we show that the specific coefficients are similar to the case of guided wave propagation in monoclinic crystals subject to a bias. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

NUMERICAL METHODS FOR MAXWELL＇S EQUATIONS IN INHOMOGENEOUS MEDIA WITH MATERIAL INTERFACES

In this paper, we will present some recent results on developing numerical methods for solving Maxwell‘s equations in inhomogeneous media with material interfaces. First,we will present a second order upwinding embedded boundary method - a Cartesian grid based finite difference method with special upwinding treatment near the material interfaces. Second, we will present a high order discontinuous spectral element with Dubinar orthogonal polynomials on triangles. Numerical results on electromagnetic scattering and photonic waveguide will be included.     

A Fast, Bandlimited Solver for Scattering Problems in Inhomogeneous Media

The numerical treatment of two-dimensional scattering in inhomogeneous media is considered. A novel approach in treating convolution operators with low regularity is used to construct an iterative solver for the Lippmann-Schwinger integral equation. In this way, accurate approximations within a choice of bandwidth can be obtained in a rapid manner. The performance of the method is tested on a discontinuous scattering object for which the exact solution is known.     

Waves in an Inhomogeneous Tube with a Flowing Two-Phase Fluid

Results of theoretical and mathematical justification of the problem on a pulsating flow of a two-phase barotropic bubbly fluid enclosed in an elastic semi-infinite cylindrical tube inhomogeneous along its length are presented. Linear one-dimensional equations are used. It is assumed that the tube is rigidly attached to the surrounding medium and therefore its displacement in the axial direction is absent. At infinity, the tube material is assumed to be homogeneous. To describe the pressure, flow rate, and displacement of the fluid, a pulsating pressure is given at the tube end. The problem stated is reduced to a singular Sturm-Liouville boundary-value problem, which in turn is reduced to a Volterra-type integral equation. This equation is solved by the method of successive approximations. By assuming that the corresponding potential is integrable, it is proved that these approximations converge to the exact solution of the problem. It is shown that this assumption also covers the very important practical case of piecewise inhomogeneity. For numerical realization, we consider a homogeneous tube with flowing water containing a small amount of bubbles. The effect of the volume content of bubbles on wave characteristics is revealed. In particular, it is stated that, for the oscillation regime selected, an increased bubble volume content decreases the wave velocity and considerably increases the flow speed (rate).     

Almost Planar Waves in Anisotropic Media

We investigate corners and steps of interfaces in anisotropic systems. Starting from a stable planar front in a general reaction-diffusion-convection system, we show existence of almost planar interior and exterior corners. When the interface propagation is unstable in some directions, we show that small steps in the interface may persist. Our assumptions are based on physical properties of interfaces such as linear and nonlinear dispersion, rather than properties of the modeling equations such as variational or comparison principles. We also give geometric criteria based on the Cahn–Hoffman vector, that distinguish between the formation of interior and exterior corners.     

Ray Method Expansions for Surface and Internal Waves in Inhomogeneous Oceans of Variable Depth

A ray method formalism is developed for the analysis of surface and internal waves in an inhomogeneous ocean of variable depth. In this method, we deduce from the governing system of equations a system of first order ordinary differential equations, for the group lines (rays of the ray method) and the propagation of phase and amplitude on them. The dispersion relation for these waves arises as an eigen-condition on an eigen-value problem involving an ordinary differential equation in the depth variable. The deduced equation for amplitude propagation has the interpretation of a statement of conservation of action.     

Group Classification of Eikonal Equations for the Wave Equation in Inhomogeneous Media

We present a group classification of the family of equations of the form (∇ψ)² = 1/v ²(x, y, z), called the eikonal equations, which are equations of characteristics for equations describing acoustic and electromagnetic waves in inhomogeneous media. For some cases of equations with linear and quadratic functions v(x, y, z), having the sense of the speed of waves in a medium, we present explicit solutions (eikonals of point sources) and completely describe the geometry of rays.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.     

Orbital Stability of Solitary Waves for Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov system     

Propagation of Electromagnetic Waves in Non-Homogeneous Media

We consider electromagnetic waves propagating in a periodic medium characterized by two small scales. We perform the corresponding homogenization process, relying on the modelling by Maxwell partial differential equations.     

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e^{i ω t}ϕ_ω(x) for a nonlinear Schr?dinger equation with an inhomogeneous nonlinearity V (x)|u|^{p − 1}u, where V (x) is proportional to the electron density. Here, ω > 0 and ϕ_ω(x) is a ground state of the stationary problem. When V (x) behaves like |x|^−b at infinity, where 0 < b < 2, we show that e^{i ω t}ϕ_ω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|^−b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method. Communicated by Bernard Helffer submitted 14/07/04, accepted 28/02/05     

Generalized Transition Waves and Their Properties

In this paper, we generalize the usual notions of waves, fronts, and propagation speeds in a very general setting. These new notions, which cover all usual situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces that are parametrized by time. We prove the existence of new such waves for some time‐dependent reaction‐diffusion equations, as well as general intrinsic properties, some monotonicity properties, and some uniqueness results for almost‐planar fronts. The classification results, which are obtained under some appropriate assumptions, show the robustness of our general definitions. © 2012 Wiley Periodicals, Inc.     

Simulation of Seismic Waves in Anisotropic Media

Doklady Mathematics - The problem of seismic wave propagation in a heterogeneous geological medium is considered. The dynamic behavior of the medium is described by the linear elastic system of...     

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

Boundary integral equations provide a powerful tool for the solution of scattering problems. However, often a singular kernel arises, in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision, thus special treatment is needed to handle the singular behavior. Especially, for inhomogeneous media, it is difficult if not impossible to find out an analytical expression for Green’s function. In this paper, an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media. This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient (FFT-PCG) solver. A remarkable point of this method is that there is no need to know analytical expressions for Green’s function. Numerical experiments are provided to demonstrate the advantage of the current approach, including its simplicity in implementation, its high accuracy and efficiency.