Group Classification of Generalized Eikonal Equations

By using a new approach to a group classification, we perform a symmetry analysis of equations of the form u_au_a = F(t, u, u_t) that generalize the well-known eikonal and Hamilton–Jacobi equations.     

Eikonal Equation for Anisotropic Media

The methods of group analysis are applied to establish a classification of eikonal equations for anisotropic stationary media, g ^ij (x)ψ _i ψ _j ?=?1. The equivalence group and the groups of symmetries are described. The classification is based on the fact that the Riemannian space (with the metric ds ²?=?g _ij (x) dx ⁱ dx ^j ) associated with the equation has a special structure, namely, that of a semi-homogeneous space: the metric form can be represented as $ds^2=g_{\hat \imath\hat \jmath}(\hat x)\,dx^{\hat \imath}\,dx^{\hat \jmath}+ G^2(\hat x) g_{\tilde \imath\tilde \jmath}(x)\,dx^{\tilde \imath}\,dx^{\tilde \jmath}$ , where the principal part $g_{\hat \imath\hat \jmath}(\hat x)\,dx^{\hat \imath}\,dx^{\hat \jmath}$ is the metric of a Riemannian space of constant curvature.     

Group Classification and Exact Solutions of Nonlinear Wave Equations

We perform complete group classification of the general class of quasi linear wave equations in two variables. This class may be seen as a broad generalization of the nonlinear d'Alembert, Liouville, sin/sinh-Gordon and Tzitzeica equations. In this way we derived a number of new genuinely nonlinear invariant models with high symmetry properties. In particular, we obtain four classes of nonlinear wave equations admitting five-dimensional invariance groups. Applying the symmetry reduction technique we construct multi-parameter families of exact solutions of these equations.     

Stability of Fronts in Inhomogeneous Wave Equations

Acta Applicandae Mathematicae - This paper presents an introduction to the existence and stability of stationary fronts in wave equations with finite length spatial inhomogeneities. The main focus...     

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.     

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.     

Regularity of Inhomogeneous Quasi-Linear Equations on the Heisenberg Group

We establish Hölder continuity of the horizontal gradient of weak solutions to quasi-linear $p$-Laplacian type non-homogeneous equations in the Heisenberg Group.     

对波动方程初边值问题边界条件齐次化函数的一个注解

基于传统的齐次化边界条件方法,采用傅里叶级数法讨论了波动方程初边值问题第一类非齐次边界条件齐次化函数问题,分析表明:对同一定解问题,在不同齐次化函数下的解在适定意义下是等价的.     

Optimal Besov Differentiability for Entropy Solutions of the Eikonal Equation

In this paper we study the eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production, and the validity of a kinetic formulation. © 2019 Wiley Periodicals, Inc.     

Decoupling of Modes for the Elastic Wave Equation in Media of Limited Smoothness

We establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twice-differentiable Lamé parameters. Precisely, we show that the P?S components of the wave propagation operator are regularizing of order one on L ² data, by establishing the diagonalization of the elastic system modulo a L ²-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.     

一类非线性波动方程的势对称分类

先给出了含有一个任意函数的线性波动方程的古典和势对称的完全分类.然后,在此基础上给出了含有两个任意函数的一类非线性波动方程的两种情形势对称分类,得到了该方程的新势对称.在处理对称群分类问题的难点-求解确定方程组时我们提出了微分形式吴方法算法,克服了以往难于处理的困难.在整个计算过程中反复使用了吴方法,吴方法起到了关键的作用.     

Generalized Eigenfunctions for Waves in Inhomogeneous Media

Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrödinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrödinger form, leading to the study of the spectral theory of its classical wave operator, a self-adjoint, partial differential operator on a Hilbert space of vector-valued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure.     

y-Regularization-Operator Splitting Method for the Numerical Solution of a Scalar Eikonal Equation

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.     

Eikonal型方程粘性解的表达式

对于圆锥型和棱锥型Hamiltonian的Eikonal型方程,本文给出了一种几何方法,得出其初值问题解的表达式并且说明由此式给出的解为原初值问题的粘性解．首先用一个凸函数序列逼近Eikonal型方程中的Hamiltonian,再由Hopf-Lax公式给出方程序列的粘性解,最后证明了该粘性解序列会收敛到Eikonal方程的粘性解．     

拟线性三阶演化方程的初步群分类

利用古典无穷小算法、等价性变换技巧和有限维抽象李代数的分类理论,给出了一般拟线性三阶演化方程在半单和一维至四维可解李代数下不变的群分类.证明了只存在3个不等价的方程在三维单李代数下不变,而且进一步证明在所有半单李代数下不变的不等价方程只有这3个.另外,还证明了存在2个、5个、29个和26个不等价的方程,分别在一维至四维可解李代数下不变.     

The Third-Order Nonlinear Evolution Equation Governing Wave Propagation in Relaxing Media

Initial value problem for the third-order nonlinear evolution equation governing wave propagation in relaxing media is considered for the case of two space dimensions and small initial data. Existence and uniqueness of the classical solution is established and the solution itself is constructed in the form of a series in the small parameter present in the initial conditions. Long time asymptotic representation is found, which shows that the nonlinearity does not contribute to its major term. The latter consists of two parts corresponding to isotropic and nonisotropic transfer of small perturbations in space.     

Stochastic Heat and Wave Equations on a Lie Group

We study nonlinear heat and wave equations on a Lie group. The noise is assumed to be a spatially homogeneous Wiener process. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.