The Linear Stability of Shock Waves for the Nonlinear Schrödinger–Inviscid Burgers System

We investigate the coupling between the nonlinear Schrödinger equation and the inviscid Burgers equation, a system which models interactions between short and long waves, for instance in fluids. Well-posedness for the associated Cauchy problem remains a difficult open problem, and we tackle it here via a linearization technique. Namely, we establish a linearized stability theorem for the Schrödinger–Burgers system, when the reference solution is an entropy-satisfying shock wave to Burgers equation. Our proof is based on suitable energy estimates and on properties of hyperbolic equations with discontinuous coefficients. Numerical experiments support and expand our theoretical results.     

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

Wave Dynamics of Saturated Porous Media and Evolutionary Equations

Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Biot approach. Mathematical models under research are the Biot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Biot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg–de Vries–Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Biot model and its generalization are shown.     

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.     

Wavelet method applied to specific adhesion of elastic solids via molecular bonds

We propose a wavelet method to analyze the stochastic-elastic problem of specific adhesion between two elastic solids via ligand-receptor bond clusters, which is governed by a nonlinear integro-differential equation with a singular Cauchy kernel to describe the mean-field coupling between deformation of elastic materials and stochastic behavior of the molecular bonds. To solve this problem, Galerkin method based on a wavelet approximation scheme is adopted, and special treatment which transforms the singular Cauchy kernel into a smooth one has been proposed to avoid the cumbersome calculation of singular integrals. Numerical results demonstrate that the method is fully capable of solving the specific adhesion problems with complex nonlinear and singular equations. Based on the proposed method, investigations are performed to reveal the relation between steady-state pulling force and mean surface separation under different stress concentration indexes, which is crucial for assembling the overall constitutive relations for multicellular tumor spheroids and polymer-matrix microcomposites.     

波动方程数值模拟的一种显式方法——-界面节点公式的构建

针对成层介质中标量波动的数值模拟, 基于波速有限原理和波动方程柯西问题的解, 导出了界面点在一个短时间窗内的精确解, 由此给出了具有高阶精度的界面节点显式递推公式的一种构建方法, 并以构造弹性杆界面节点的递推公式为例说明其要点. 给出了与已有内节点递推公式的精度阶匹配的二阶和四阶界面节点递推公式. 由此构成的计算格式具有``异质格式'编程简便的特性, 更合理地考虑了界面影响. 最后, 通过数值试验检验这一匹配方案的精度和稳定性.      

On some diffusive waves in non-linear heat conduction

We address the non-linear heat conduction in the presence of absorption for the case of spherical symmetry geometry. The non-linear model is based on both a temperature-dependent thermal conductivity and a non-linear generalization of the Fourier law. The governing equation belongs to a class of degenerate parabolic equations. We obtain similarity solutions in closed form for the Cauchy problem corresponding to an instantaneous point source problem. We investigate the non-linear effects on the propagation of the temperature distrubances. We find that in certain cases the temperature distribution displays travelling wave characteristics. The solution for the Cauchy problem is recovered by considering a suitable first boundary value problem.     

Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn ≪ 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M _a = v/a → 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 76–87.Original Russian Text Copyright © 2005 by Stolyarov.     

有限散射信号下二维缺陷形状识别的罚函数方法

研究在有限照射角度和频带宽度下二维缺陷的形状识别问题。首先,通过引进介质参数扰动函数,建立介质参数扰动函数和弹性波散射场之间的非线性关系,并将所关心的缺陷的形状识别问题转化为关于扰动函数的反演;然后,利用变分技术和优化方法求解,为了弥补散射数据的不足,在总的目标函数中,采用附加度量函数作为罚函数;最后,对后场散射远场测量时有限照射角度和频带宽度下几种典型缺陷进行了模拟识别,表明了;表明了罚函数法的有效性。     

The structure and asymptotic behaviour of compression waves

In the study of weak solutions to nonlinear hyperbolic partial differential equations both rarefaction waves and compression waves arise. Although the behavior of rarefaction waves is known for all time, the characteristics that determine a compression wave intersect and hence the development of the wave is not easily determined. The purpose of this paper is to study compression waves. As a first step we consider the Cauchy problem for the nonlinear wave equation. We show that if the data outside some finite interval consist of constant states, then after finite time the solution involves the same states as does the solution to the Riemann problem determined by these constant states. This result is then applied to compression waves to obtain information on the shock that arises and on the steady-state solution. The region of interaction is also described. This information is obtained via a constructive procedure.     

Fast Fourier homogenization for elastic wave propagation in complex media

In the context of acoustic or elastic wave propagation, the non-periodic asymptotic homogenization method allows one to determine a smooth effective medium and equations associated with the wave propagation in a given complex elastic or acoustic medium down to a given minimum wavelength. By smoothing all discontinuities and fine scales of the original medium, the homogenization technique considerably reduces meshing difficulties as well as the numerical cost associated with the wave equation solver, while producing the same waveform as for the original medium (up to the desired accuracy). Nevertheless, finding the effective medium requires one to solve the so-called “cell problem”, which corresponds to an elasto-static equation with a finite set of distinct loadings. For general elastic or acoustic media, the cell problem is a large problem that has to be solved on the whole domain and its resolution implies the use of a finite element solver and a mesh of the fine scale medium. Even if solving the cell problem is simpler than solving the wave equation in the original medium (because it is time and source independent, based on simple tetrahedral meshes and embarrassingly parallel) it is still a challenge. In this work, we present an alternative method to the finite element approach for solving the cell problem. It is based on a well-known method designed by H. Moulinec and P. Suquet in 1998 in structural mechanics. This iterative technique relies on Green functions of a simple reference medium and extensively uses Fast Fourier Transforms. It is easy to implement, very efficient and relies on a simple regular gridding of the medium. Through examples we show that the method gives excellent results, even, under some conditions, for discontinuous media.     

Interaction of cracks with bonds between their faces in an isotropic medium reinforced by a regular system of stringers

A problem of an elastic isotropic medium with a system of foreign (transverse with respect to crack alignment) rectilinear inclusions is considered. The medium is assumed to be attenuated by a periodic system of rectilinear cracks with zones where the crack faces interact with each other. These zones are assumed to be adjacent to the crack tips, and their sizes can be commensurable with the crack size. Interaction between the crack faces in the tip zone is modeled by introducing bonds (adhesion forces) between the cracks with a specified strain diagram. The boundary-value problem of the equilibrium of a periodic system of cracks with bonds between their faces under the action of external tensile loads and forces in the bonds is reduced to a nonlinear singular integrodifferential equation with a kernel of the Cauchy kernel type. The condition of critical equilibrium of the cracks with the tip zones is formulated with allowance for the criterion of critical tension of the bonds. A case of a stress state of the medium containing zones where the crack faces interact with each other is considered.     

GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

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Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches

This Note deals with the identification of internal planar cracks inside a three-dimensional elastic body via two approaches relying on domain decomposition using elastostatic measurements. These approaches consist in recasting the problem in terms of primal or dual Steklov–Poincaré equations. The primal approach is a straightforward continuation to the elastic Cauchy problem of the work presented in J. Ben Abdallah (2007) [1] which is devoted to the Cauchy problem for the scalar Laplace equation. The numerical performances of these formulations are compared.     

A 2.5-D dynamic model for a saturated porous medium. Part II: Boundary element method

The 3-D boundary integral equation is derived in terms of the reciprocal work theorem and used along with the 2.5-D Green’s function developed in Part I [Lu, J.F., Jeng, D.S., Williams, S., submitted for publication. A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function. Int. J. Solids Struct.] to develop the 2.5-D boundary integral equation for a saturated porous medium. The 2.5-D boundary integral equations for the wave scattering problem and the moving load problem are established. The Cauchy type singularity of the 2.5-D boundary integral equation is eliminated through introduction of an auxiliary problem and the treatment of the weakly singular kernel is also addressed. Discretisation of the 2.5-D boundary integral equation is achieved using boundary iso-parametric elements. The discrete wavenumber domain solution is obtained via the 2.5-D boundary element method, and the space domain solution is recovered using the inverse Fourier transform. To validate the new methodology, numerical results of this paper are compared with those obtained using an analytical approach; also, some numerical results and corresponding analysis are presented.     

On the hyperbolicity of 3D plasticity equations in isostatic coordinates

We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.     

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.