Traveling Waves in Diffusive Random Media

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves. AMS Subject Classification: 35K55, 35K57, 35B50     

Stability of Concatenated Traveling Waves

We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is individually asymptotically stable. We show that the sequence of traveling waves is itself asymptotically stable: as \(t\rightarrow \infty \), the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Essentially the same result was previously proved by Wright (J Dyn Differ Equ 21:315–328, 2009) and Selle (Decomposition and stability of multifronts and multipulses, 2009), who regarded the concatenated wave pattern as a sum of traveling waves. In contrast to their work, we regard the pattern as a sequence of traveling waves restricted to subintervals of \(\mathbb {R}\) and separated at any finite time by small jump discontinuities. Our proof uses spatial dynamics and Laplace transform.     

Traveling Waves for a Biological Reaction-Diffusion Model

We investigate the existence of traveling wave solutions for a system of reaction–diffusion equations that has been used as a model for microbial growth in a flow reactor and for the diffusive epidemic population. The existence of traveling waves was conjectured early but only has been proved recently for sufficiently small diffusion coefficient by the singular perturbation technique. In this paper we show the existence of traveling waves for an arbitrary diffusion coefficient. Our approach is a shooting method with the aid of an appropriately constructed Liapunov function.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.Wenzhang Huang-Research was supported in part by NSF Grant DMS-0204676.     

Traveling Waves in Spatial SIRS Models

We study traveling wavefront solutions for two reaction–diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument.     

Classification of Traveling Waves for a Class of Nonlinear Wave Equations

We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.     

Propagation Regimes of Self-Supported Light-Detonation Waves

The light-detonation wave structure is investigated. It is shown that self-supported laser radiation absorption waves can propagate in the Jouguet detonation or undercompressed detonation regimes. The conditions of realization of these regimes are found numerically. It is shown that the undercompressed detonation regime is realized if the radiation flux is sufficiently powerful. In the case of a light-detonation wave this regime is theoretically detected and investigated for the first time.     

正向爆轰驱动高焓激波风洞的数值模拟

对充满氢氧可燃气体、带扩容腔的正向爆轰驱动的激波风洞进行了数值模拟。计算采用了欧拉方程,频散可控耗散差分格式（DCD）和改进的二阶段化学反应模型。在扩容腔附近采用二维轴对称计算模型,而在驱动段和被驱动段的直管道部分则采用一维计算模型。本文分析了爆轰波在管道中的传播、反射和绕射过程。计算结果表明扩容腔的尺寸对爆轰波的传播、反射、汇聚等起着决定性的作用;带扩容腔的正向爆轰驱动的激波风洞能够得到平稳的持续时间较长的气流,提高了实验的精确度和可重复性。     

超音速来流中爆轰波衍射和二次起爆过程研究

预爆管技术被广泛地应用在爆轰波发动机的起爆过程中,但是在超音速来流中基于预爆管技术起始爆轰波的研究并未被广泛地开展。基于此,本文中数值研究了横向超音速来流对半自由空间内爆轰波的衍射和自发二次起爆、及管道内的衍射和壁面反射二次起爆两种现象的影响。数值模拟的控制方程为二维欧拉方程,空间上使用五阶WENO格式进行数值离散,采用带有诱导步的两步链分支化学反应模型。所模拟的爆轰波具有规则的胞格结构,对应于用惰性气体高度稀释过的可爆混合物中形成的爆轰波。结果表明:在半自由空间内,在本文所模拟的几何尺寸下,爆轰波并未成功发生二次起爆现象,但是爆轰波的自持传播距离随着横向超音速来流强度的增强而增加。在核心的三角形流动区域外,波面诱导产生了更多的横波结构;在管道内,横向的超音速来流在逆流侧对出口气流产生了压缩作用,能有效提高波面压力,因此反射后的激波压力也比较高。在同样的几何尺寸下,爆轰波在静止和超音速(Ma=2.0)气流中分别出现了二次起爆失败和成功两种现象,这是由于在超音速来流中化学反应面的褶皱诱导产生了横波结构,横波与管壁以及其他横波之间的碰撞提高了前导激波的强度,并最终促进了爆轰波在超声速流主管道内的成功起始。     

Stability and Instability of Fourth-Order Solitary Waves

We study ground-state traveling wave solutions of a fourth-order wave equation. We find conditions on the speed of the waves which imply stability and instability of the solitary waves. The analysis depends on the variational characterization of the ground states rather than information about the linearized operator.     

Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

气相爆轰波在障碍物上Mach反射的实验验证

本文公布了气相爆轰波沿收缩管道传播时发生Mach反射的实验证据。在爆轰波通过的管道中安装不同楔角的楔块,形成管道的收缩。爆轰波在通过楔块时会发生Mach反射。利用烟熏玻璃片记录到了爆轰波Mach反射时形成的三波点迹线及其两侧胞格尺寸和密度的变化。据我们掌握的资料,这是首次用胞格结构变化的记录证实,气相爆轰波与无化学反应的空气中的冲击波一样,在一定的入射条件下会发生Mach反射。这一实验结果可使我们更深入了解爆轰波的本质,也为数值模拟气相爆轰波在障碍物上Mach反射现象提供了可对比的依据。     

Existence of Surface Waves and Band Gaps in Periodic Heterogeneous Half-spaces

We find a sufficient condition for the existence of surface (Rayleigh) waves based on the Rayleigh-Ritz variational method. When specialized to a homogeneous half-space, the sufficient condition recovers the known criterion for the existence of subsonic surface waves. A simple existence criterion in terms of material properties is obtained for periodic half-spaces of general anisotropic materials. Further, we numerically compute the dispersion relation of the surface waves for a half-space of periodic laminates of two materials and demonstrate the existence of surface wave band gaps.     

Existence,Uniqueness, and Stability of Generalized Traveling Waves in Time Dependent Monostable Equations

The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic, it is almost periodic in the sense that its wave profile and wave speed are almost periodic.     

Modeling Chemotactic Waves in Saturated Porous Media using Adaptive Mesh Refinement

Bacterial transport is heavily influenced by chemical gradients and interfaces that exist in the subsurface. The main aim of this article is to describe a method of simulating the propagation of a traveling bacterial wave in a contaminated region and the resulting degradation of the contaminant. The presence of the chemotactic term and the relatively small bacterial diffusion means that the wave contains a very sharp wavefront. We, therefore, use an upwind conservative numerical scheme to obtain accurate and numerically stable solutions. The accuracy of the method is verified by comparisons with an exact one-dimensional solution of a simplified problem to give the same wavespeed. The method is then used to simulate the propagation of a realistic chemotactic wave in one dimension. We then use adaptive mesh refinement (AMR) to compute the propagation of chemotactic waves in two dimensions using the simplified model calibrated to give the same wavespeed as the full model.     

Uniqueness of the Bistable Traveling Wave for Mutualist Species

For a system of reaction–diffusion equations that models the interaction of n mutualist species, the existence of the bistable traveling wave solution has been proved where the nonlinear reaction terms possess a certain type of monotonicity. However the problem of whether there can be two distinct traveling waves remains open. In this paper we use a homotopy approach incorporated with the Liapunov–Schmidt method to show that the bistable traveling wave solution is unique. Our method developed in this paper can also be applied to study the existence and uniqueness of traveling wave solutions for some competition models.     

Time Periodic Traveling Waves for a Periodic and Diffusive SIR Epidemic Model

In this paper, we study the time periodic traveling wave solutions for a periodic SIR epidemic model with diffusion and standard incidence. We establish the existence of periodic traveling waves by investigating the fixed points of a nonlinear operator defined on an appropriate set of periodic functions. Then we prove the nonexistence of periodic traveling via the comparison arguments combined with the properties of the spreading speed of an associated subsystem.     

Theory of the detonation of weak condensed explosives

The possibilities of detonation taking place in a material characterized by a shock adiabatic containing a sharp break (leading to a double shock-wave configuration) are examined. The range of possible velocities D of a self-sustaining detonation in the second shock wave is determined; D may be subsonic with respect to the original material. However, even for an arbitrarily low velocity of sound the range of subsonic D values above the break point on the adiabat is extremely limited: The minimum detonation velocity D_min coincides (apart from a factor of 0.5–0.8) with the velocity of a longitudinal sound wave in the original material below the break point. This limitation with regard to D is associated with the formation of a wave of rarefaction in the reaction products, For D < D_min the shock wave of rarefaction reaches the Jouguet point and breaks the steady-state complex of the detonation wave. The results obtained are valid not only for weak, but also for powerful, explosive substances, if (by virtue of any kind of losses) low-velocity forms of detonation are realized in these materials.     

New Solitary Waves in Elastic Materials: Existence and Nonlinear Interaction

Waves mentioned in the title were revealed in composite materials that are described by the microstructural theory of the second order — the theory of two-phase mixtures. For harmonic periodic waves, a mixture is always a dispersive medium. This medium admits existence of other waves — waves with profiles described by functions of mathematical physics (the Chebyshov–Hermite, Whittaker, Mathieu, and Lamé functions). If the initial profile of a plane wave is chosen in the form of the Chebyshev–Hermite or Whittaker function, then the wave may be regarded as an aperiodic solitary wave. The dispersivity of a mixture as a nonlinear frequency dependence of phase velocities transforms for nonperiodic solitary waves into a nonlinear phase-dependence of wave velocities. This and some other properties of such waves permit us to state that these waves fall into a new class of waves in materials, which is intermediate between the classical simple waves and the classical dispersion traveling waves. The existence of these new waves is proved in a computer analysis of phase-velocity-versus-phase plots. One of the main results of the interaction study is proof of the existence of this interaction itself. Some features of the wave interaction — triplets and the concept of synchronization — are commented on