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.     

Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front. In the case of stationary ergodic reactions, the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.     

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.     

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.     

On the Existence of Longitudinal Plane Waves in General Elastic Anisotropic Media

We give a new proof of Kolodner's result that longitudinal waves can propagate in at least three directions in a hyperelastic anisotropic medium. We give examples of an orthotropic hyperelastic tensor with exactly three such directions, of a monoclinic elastic (but not hyperelastic) tensor with only one, and of a monoclinic elastic (elliptic, but not uniformly elliptic) tensor with no direction for longitudinal waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

Simulation of Traveling Solitary Plane Waves

The possibility of propagation of solitary plane waves with the Whittaker profile in materials with a microstructure (composites) is discussed. Solitary waves are defined as aperiodic smooth waves with an initial profile that is equal to zero everywhere except for some finite interval. Functions with indices 0.0, 0.1, –1/4, and 1/4 are chosen for computer simulation. It is observed that with some restrictions on the time or distance of propagation in the material, two modes of the traveling wave with the Whittaker profile and different phase-dependent phase velocities propagate simultaneously. The discussion section focuses attention on the conditions of blanking of the second mode for small values of the phase     

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     

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.     

Circumferential Traveling Waves in Filled Cylindrical Shells

Consideration is given to deformation in the form of traveling circumferential waves in circular cylindrical shells with a liquid. The effect of the liquid on the phase velocity in the carrier shell is studied depending on the waveformation parameters and initial conditions     

A General Approach to the Asymptotic Behavior of Traveling Waves in a Class of Three-Component Lattice Dynamical Systems

In this paper, we develop a general approach to deal with the asymptotic behavior of traveling wave solutions in a class of three-component lattice dynamical systems. Then we demonstrate an application of these results to construct entire solutions which behave as two traveling wave fronts moving towards each other from both sides of x-axis for a three-species competition system with Lotka–Volterra type nonlinearity in a lattice.     

Periodic Traveling Gravity Water Waves with Discontinuous Vorticity

We use elliptic theory to prove the existence of steady two-dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity. This is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context.     

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.     

Traveling Waves in Porous Media Combustion: Uniqueness of Waves for Small Thermal Diffusivity

We study traveling wave solutions arising in Sivashinsky’s model of subsonic detonation which describes combustion processes in inert porous media. Subsonic (shockless) detonation waves tend to assume the form of a reaction front propagating with a well defined speed. It is known that traveling waves exist for any value of thermal diffusivity [5]. Moreover, it has been shown that, when the thermal diffusivity is neglected, the traveling wave is unique. The question of whether the wave is unique in the presence of thermal diffusivity has remained open. For the subsonic regime, the underlying physics might suggest that the effect of small thermal diffusivity is insignificant. We analytically prove the uniqueness of the wave in the presence of non-zero diffusivity through applying geometric singular perturbation theory. Dedicated to Mr. Brunovsky in honor of his 70th birthday.     

Traveling Waves in Discrete Periodic Media for Bistable Dynamics

This paper is concerned with the existence, uniqueness, and global stability of traveling waves in discrete periodic media for a system of ordinary differential equations exhibiting bistable dynamics. The main tools used to prove the uniqueness and asymptotic stability of traveling waves are the comparison principle, spectrum analysis, and constructions of super/subsolutions. To prove the existence of traveling waves, the system is converted to an integral equation which is common in the study of monostable dynamics but quite rare in the study of bistable dynamics. The main purpose of this paper is to introduce a general framework for the study of traveling waves in discrete periodic media.     

Traveling Waves in a Convolution Model for Phase Transitions

The existence, uniqueness, stability and regularity properties of traveling-wave solutions of a bistable nonlinear integrodifferential equation are established, as well as their global asymptotic stability in the case of zero-velocity continuous waves. This equation is a direct analog of the more familiar bistable nonlinear diffusion equation, and shares many of its properties. It governs gradient flows for free-energy functionals with general nonlocal interaction integrals penalizing spatial nonuniformity. (Accepted October 16, 1995)     

旋转薄圆盘的行波动力学特性分析

旋转圆盘是广泛应用于旋转机械装置中的基本结构元件,圆盘在高速旋转状态下会表现出与低速或非旋转状态下迥异的力学性能.本文对高速旋转薄圆盘横向振动的行波动力学特性进行了分析,建立了考虑离心力引起的薄膜内力影响下的动力学控制方程以及相应的边界条件.采用伽辽金法数值模拟了旋转圆盘前、后行波振动频率和动力屈曲失稳临界转速随着圆盘几何参数如半径比、厚度的变化规律,以及材料参数对于振动频率和临界转速的影响等.本文的数值计算可以同时给出圆盘旋转的前、后行波频率,并且结果与实验结果吻合良好.     

Global Existence for Elastic Waves with Memory

We treat the Cauchy problem for nonlinear systems of viscoelasticity with a memory term. We study the existence and the time decay of the solution to this nonlinear problem. The kernel of the memory term includes integrable singularity at zero and polynomial decay at infinity. We prove the existence of a global solution for space dimensions n3 and arbitrary quadratic nonlinearities.     

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.     

Maintenance of Oscillations in Three-Dimensional Traveling Waves in Plane Poiseuille Flows

Two solutions of three-dimensional Navier–Stokes equations are studied numerically. These solutions describe the fluid motion in a plane channel, are of the traveling-wave form, and are periodic in the streamwise and spanwise directions. It is shown that, in each solution, the oscillations arise as a result of linear instability in the streamwise averaged velocity field. This instability is due to the existence of streamwise streaks known as the regions where the velocity is higher or lower than the mean velocity. A mechanism for the maintenance of streamwise vortices causing the formation of streaks is revealed. The obtained results confirm and extend the existing knowledge about the mechanism for the formation of near-wall turbulent structures.     

The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c0. Convergence results for solutions are obtained at the singular perturbation limit c 0.