有限变形弹性杆中的几何非线性波

利用有限变形理论的Lagrange描述,借助非保守系统的Hamilton型变分原理,导出了描述弹性杆中几何非线性波的波动方程．为了使非线性波动方程有稳定的行波解,计及了粘性效应引入的耗散和横向惯性效应导致的几何弥散．运用多重尺度法将非线性波动方程简化为KdV-Bergers方程,这个方程在相平面上对应着异宿鞍-焦轨道,其解为振荡孤波解．如果略去粘性效应或横向惯性,方程将分别退化为KdV方程或Bergers方程,由此得到孤波解或冲击波解,它们在相平面上对应着同宿轨道或异宿轨道．     

An Approximation for Slowly Varying Waves in Elastic Rods

A systematic method of approximation, involving series expansionsin terms of a small parameter , is used to study the propagationof slowly varying (long wave) disturbances in elastic rods.It is shown that the bar equations are the result in the limitas 0. This result is independent of the form of the cross-sectionof the rod. Further terms may be calculated, but these dependon this shape. Near the driven end this approximation breaksdown and a boundary layer is introduced. The boundary layerequations are displayed and discussed.     

有限变形弹性杆中三种非线性弥散波

在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上,分别计入有限变形和弥散效应,借助Hamilton变分原理,由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明,这些方程在相平面上存在同宿轨道或异宿轨道,分别相应于孤波解或冲击波解．根据齐次平衡原理,用Jacobi椭圆函数展开对这些演化方程进行了求解,在一定的条件下它们均可能存在孤立波解或冲击波解,这与方程的定性分析完全一致．     

Torsion Waves on the Cylindrical Surface of a Semiinfinite Elastic Medium

Torsion waves on a cylindrical cavity in a semiinfinite elastic medium are investigated. The waves are generated by steady-state torsional vibrations of a flat, circular punch coupled with the half-space. A technique of contour transformation of the integrals involved in problems of this kind is described. An algorithm and numerical results are given for calculating the modulus of the complex amplitude of the displacement vector on the cylindrical surface as a function of the vertical coordinate in both the near and far fields.     

Solitary-Wave Interactions in Elastic Rods

The propagation of longitudinal deformation waves in an elastic rod is modelled by the nonlinear partial differential equation with p = 3 or 5. This equation is first derived under a range of possible constraints. We then show that this equation and even certain generalizations do not pass the Painlevé test, and hence are probably not completely integrable. Finally, we study the head-on collision of two equal solitary waves numerically and also asymptotically for small and large amplitude.     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

Scattering of Plane Harmonic Waves in an Infinite Elastic Solid by an Arbitrary Configuration of Two-dimensional Obstacles

The problem of the multiple scattering of plane harmonic wavesin an infinite elastic solid by arbitrary configurations oftwo-dimensional obstacles is considered. Sets of integral equationsrelating the multiple scattering functions to the single scatteringfunctions are derived. The integral equations are replaced bya set of algebraic equations which are then solved by an iterativeprocedure. Some results are given for the scattering of a planeharmonic P or SV wave by an identical pair of parallel circularcylindrical cavities.     

Periodic Traveling Waves in Diatomic Granular Chains

We study bifurcations of periodic traveling waves in diatomic granular chains from the anti-continuum limit, when the mass ratio between the light and heavy beads is zero. We show that every limiting periodic wave is uniquely continued with respect to the mass ratio parameter, and the periodic waves with a wavelength larger than a certain critical value are spectrally stable. Numerical computations are developed to study how this solution family is continued to the limit of equal mass ratio between the beads, where periodic traveling waves of homogeneous granular chains exist.     

一类具有扩散项的流行性传染病模型中的行波解

讨论了一类具有扩散项的流行性传染病模型中的行波解的存在性.首先,将对该模型所对应的反应扩散系统的行波解的讨论转化为对二阶常微分系统的上下解的讨论;然后,通过上下解方法建立了这个具有扩散项的传染病模型中行波解的存在性条件,并进一步讨论了扩散因素对行波解的波速的影响,得到被感染人群的流动对病毒的传播有一定的影响.     

Regularity of Traveling Free Surface Water Waves with Vorticity

We prove real analyticity of all the streamlines, including the free surface, of a gravity- or capillary-gravity-driven steady flow of water over a flat bed, with a Hölder continuous vorticity function, provided that the propagating speed of the wave on the free surface exceeds the horizontal fluid velocity throughout the flow. Furthermore, if the vorticity possesses some Gevrey regularity of index s, then the stream function of class C ^2,μ admits the same Gevrey regularity throughout the fluid domain; in particular if the Gevrey index s equals 1, then we obtain analyticity of the stream function. The regularity results hold not only for periodic or solitary-water waves, but also for any solution to the hydrodynamic equations of class C ^2,μ .     

A Barrier Principle for Surfaces with Prescribed Mean Curvature and Arbitrary Codimension

We consider two dimensional surfaces ${X : \Omega\to\mathbb R^{n+2}, \Omega\subset \mathbb C, w=u+iv\mapsto X(w)}$ with arbitrary codimension n and prove a barrier principle for strong (possibly branched) subsolutions ${X\in C^1(\Omega, \mathbb {R}^{n+2})\cap H_{2,{\rm loc}}^2(\Omega,\mathbb R^{n+2})}$ of the integral inequality $$\int_{\Omega} \Big\lbrace \langle \nabla X, \nabla \varphi\rangle +2W \sum_{k=1}^n H_k \langle N_k,\varphi \rangle \Big\rbrace \; dudv\ge 0$$ with mean curvature functions (H _k ) _k=1,...,n which lie locally on one side of a supporting hypersurface S. We show under suitable assumption on the 2-mean curvature of the supporting surface S that X is locally contained in S. This generalizes a corresponding result for surfaces in ${\mathbb R^3}$ , cf. (Dierkes et al., Regularity of Minimal Surfaces, §4.4, 2010).     

A Constructive Approach to Traveling Waves in Chemotaxis

In this paper we study the existence of one-dimensional and multidimensional traveling wave solutions for general chemotaxis or so-called Keller-Segel models without reproduction of the chemotactic species. We present a constructive approach to give modelers a choice of chemotactic sensitivity functionals, production, and degradation terms for the chemical signal at hand. The main aim is to understand the type of functionals and the interplay between them that are needed for the traveling wave and pulse patterns to occur.     

Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.     

Some New Expressions of Compactons in Compressible Elastic Rods

In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.     

含奇异线的广义KdV方程的行波解

研究了一个广义KdV方程的行波解,在行波变换下,该方程转化成含奇异线的平面系统,通过平衡点分析定性地得到不同参数条件下系统解的特性.特别的,由于相平面上的奇异线的存在,系统具有一些特殊结构的解,例如compactons、kink-compactons、anti-kink-compactons,给出了这些解的积分表达式,并且由椭圆函数积分求出了精确解.     

Traveling Waves of a Reaction-Diffusion SIRQ Epidemic Model with Relapse

This paper is concerned with the traveling waves of a reaction-diffusion SIRQ epidemic model with relapse. We find that the existence and nonexistence of traveling waves are determined by the basic reproduction number of the system and the minimal wave speed. This threshold dynamics is proved by Schauder''s fixed-point theorem combining Lyapunov functional with the theory of asymptotic spreading. Moreover, the numerical simulations are provided to illustrate our analytical results and the effect of the relapse is also discussed.     

嵌入弹性半空间的弹性迴转轴的扭转*

本文用线载荷积分方程法(LLIEM)研究嵌在弹性半空间的弹性迴转轴的扭转问题.将“点环力偶(PRC)”和“半空间点环力偶(PRCHS)”分别分布于迴转轴内和外的轴线上,就能将本问题归结为一维的Fredholm第一种积分方程组.直接用离散法求解时,会发现有时解是不稳定的,也就是病态情形.本文采用以带小参数的Fredholm第二种积分方程代替病态的Fredholm第一种积分方程的方法可以得到稳定的解,此法比Tikhonov正规化法简单,易于在计算机上运行.文中给出圆维、圆柱、圆锥-圆柱、抛物线轴等数值例子.     

一类带有扩散与时滞的流行性传染病模型的行波解

讨论了一类带有扩散与时滞的流行性传染病模型的行波解的存在性.首先,将系统的行波解的存在性问题转化为一个二阶常微分方程组的单调解的存在性问题;应用单调方法和不动点方法,进一步地将问题转化为方程组的上下解的构造问题;应用所建立的引理与定理,通过构造适合的上下解,证明了系统单调行波解的存在性.     

一类弧形弹性杆大变形的多解性与分支

基于大变形理论建立弧形弹性杆大变形的数学模型,弹性杆的一端固定,另一端自由且在中间受一竖直向下的集中力,所建立的模型可变形为摆动方程的边值问题.利用流形法得到数学模型的分支图,进而分析弹性杆变形的多解性.