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.     

Propagation Phenomena for A Reaction–Advection–Diffusion Competition Model in A Periodic Habitat

This paper is devoted to the study of propagation phenomena for a Lotka–Volterra reaction–advection–diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trivial steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction–diffusion model.     

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay

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.     

Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.     

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.     

On a Free Boundary Problem for a Two-Species Weak Competition System

We study a Lotka–Volterra type weak competition model with a free boundary in a one-dimensional habitat. The main objective is to understand the asymptotic behavior of two competing species spreading via a free boundary. We also provide some sufficient conditions for spreading success and spreading failure, respectively. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the competition model on the whole real line without a free boundary.     

The Minimal Speed of Traveling Fronts for the Lotka–Volterra Competition System

We study the minimal speed for a two species competition system with monostable nonlinearity. We are interested in the linear determinacy for the minimal speed in the sense defined by (Lewis et al. J Math Biol 45:219–233, 2002). We provide more general cases for the linear determinacy than that of (Lewis et al. J Math Biol 45:219–233, 2002). For this, we study the minimal speed for the corresponding lattice dynamical system. Our approach gives one new way to study the traveling waves of the parabolic equations through its discretization which can be applied to other similar problems.     

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.     

Axisymmetric Thermoelastic Interactions without Energy Dissipation in an Unbounded Body with Cylindrical Cavity

The linear theory of thermoelasticity without energy dissipation is employed to study thermoelastic interactions in a homogeneous and isotropic unbounded body containing a cylindrical cavity. The interactions are supposed to be due to a constant step in radial stress or temperature applied to the boundary of the acvity, which is maintained at a constant temperature or zero radial stress (as the case may be). By using the Laplace transform technique, it is found that the interactions consist of two coupled waves both of which propagate with a finite speed but with no attenuation. The discontinuities that occurs at the wavefronts are computed. Numerical results applicable to a copper-like material are presented.     

In-situ experiment tests and particulate simulations on powder paving process of additive manufacturing

In this work, in-situ experimental tests are first performed to investigate the powder spreading process of additive manufacturing, where different kinds of scrapers and spreading speeds are employed. Detailed kinetic behaviours of individual powder particles are discussed by discrete element method simulations. It is found that the decrease of inclination angle of the scraper improves the powder pressure and compaction in the spreading process, leading to a denser powder flow and thus a denser powder bed. The increase of spreading speed also improves the powder pressure and compaction in the spreading process. However, the powder flow becomes looser due to the volume dilation, and thus the quality of the paved powder bed decreases. In industrial applications, if the higher powder spreading speed is employed to improve the processing efficiency, the scraper with a smaller inclination angle can be used to ensure the powder bed quality.     

Elastic pulse generation by tractions applied to a spherical cavity

The generation of an elastic pulse in an infinite homogeneous, isotropic elastic solid by the application of axisymmetrical time-dependent loading to the surface of a spherical cavity is considered. Formal Laplace transformed general solutions are obtained, before studies are made of cases in which tractions are applied only to a cap of the cavity wall. The patterns of wavefronts which develop in these cases can consist of dilatational and rotational direct, edge and diffracted, and head wavefronts. Approximations to the field near the wavefronts are obtained and discussed.     

The Equivalent of Fourier Holography at the Nanoscale

The paper deals with optical holographic interferometry at the nanometric scale. The observed objects are sodium chloride nanocrystals. The object illumination is done through the use of evanescent wavefronts. The observed crystals become self-luminous objects producing pseudo-non-diffracting wavefronts. The wavefronts emerging from the crystals are the result of electromagnetic resonances of the crystals. A microscope is utilized to register the wavefronts generated by the crystals. A 6 μm spherical particle made of polystyrene acts as a relay lens to collect the wavefronts that are recorded by a monochromatic CCD and a color camera attached to the microscope. The structure of the recorded images is determined through Fourier transform analysis. It is shown that the recorded images are lens holograms formed by the interference of the wavefronts generated by the crystals. Fourier transform algorithms and edge detection algorithms are utilized to obtain the dimensions of the crystals. The power of Gabor’s idea when he invented holography is again proven in this study. If the problem of super-resolution is viewed from the point of view of the Theory of Communications, the fact that one can register both amplitude and phase of a signal of a self-luminous object provides the means of reaching spatial resolutions with average standard deviation of ±3 nm using helium–neon laser illumination of λ = 632.8 nm. The resolution that can be achieved depends on the structure of the observed material, in the present case ±5d, where d is the distance of the atomic planes of the NaCl. With improvements in the hardware and software higher resolutions may be feasible.     

Response analysis of the piezoelectric energy harvester under correlated white noise

Nonlinear Dynamics - Energy harvesting of a monostable duffing-type harvester with piezoelectric coupling under correlated multiplicative and additive white noise is investigated in this paper. The...     

Experimental investigation on splashing and nonlinear fingerlike instability of large water drops

The fluid physics of the splashing and spreading of a large-scale water drop is experimentally observed and investigated. New phenomena of drop impact that differ from the conventional Rayleigh–Taylor instability theory are reported. Our experimental data shows good agreement with previous work at low Weber number but the number of fingers or instabilities begins to deviate from the R–T equation of Allen at high Weber numbers. Also observed were multiple waves (or rings) on the spreading liquid surface induced from pressure bouncing (or pulsation) within the impacting liquid. The first ring is transformed into a radially ejecting spray whose initial speed is accelerated to a velocity of 4–5 times that of the impacting drop. This first ring is said to be “splashing,” and its structure is somewhat chaotic and turbulent, similar to a columnar liquid jet surrounded by neighboring gas jets at relatively high impact speed. At lower impact speeds, splashing occurs as a crown-shaped cylindrical sheet. A second spreading ring is observed that transforms into fingers in the circumferential direction during spreading. At higher Weber number, the spreading of a third ring follows that of the second. This third ring, induced by the pressure pulsation, overruns and has fewer fingers than the second, which is still in a transitional spreading stage. Several important relationships between the drop impact speed, the spray ejection speed of the first ring, and the number of fingers of the second and third rings are presented, based on data acquired during a set of drop impact experiments. Issues related to the traditional use of the R–T instability are also addressed.     

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.     

Robust design optimization of a nonlinear monostable energy harvester with uncertainties

Meccanica - Based on the improved interval extension, a robust optimization method for nonlinear monostable energy harvesters with uncertainties is developed. In this method, the 2nd order terms in...     

Coarsening Fronts

We characterize the spatial spreading of the coarsening process in the Allen–Cahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the Allen–Cahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneous, stable equilibrium. During one cycle of the oscillatory propagation, two layers of the periodic pattern are annihilated. Galerkin approximations and the Conley index for ill-posed spatial dynamics are used to show existence of modulated fronts for all parameter values. In the limit of small amplitude patterns or large wave speeds, we establish uniqueness and asymptotic stability of the modulated fronts. We show that the minimal speed of propagation can be characterized by a dichotomy which depends on the existence of pulled fronts. The main tools here are an Evans function type construction for the infinite-dimensional ill-posed dynamics and an analysis of the complex dispersion relation based on Sturm–Liouville theory.     

Design,Manufacture, and Quasi-Static Testing of Metallic Negative Stiffness Structures within a Polymer Matrix

A composite material system comprised of a monostable negative stiffness (NS) structure within a polymer matrix was designed, fabricated, and experimentally evaluated. The monostable negative stiffness (NS) structure was designed using a combination of analytical and numerical models and manufactured in stainless steel. The NS structure was arranged in parallel with different polymer matrices to experimentally evaluate the effects of the matrix properties on the overall stiffness and energy dissipation of the composite NS-matrix system when loaded in uniaxial compression. A strong influence of the matrix properties on the stiffness and energy absorption capacity of the composite system was observed. Unlike conventional composites for which there is a natural tradeoff between stiffness and energy absorption capacity, the composite NS-matrix system enhanced stiffness while simultaneously improving energy absorption relative to a neat matrix, but only when the stiffness of the matrix was carefully matched to the stiffness of the NS structure.     

Reflector Problem in $${\mathbb R^n}$$ Endowed with Non-Euclidean Norm

In this paper, we establish existence and uniqueness up to dilations for the reflector problem in a nonisotropic medium in for which light wavefronts are described by non-Euclidean norm spheres, through approaches of paraboloid approximation and optimal mass transport. Research of L. A. Caffarelli supported in part byNSF grant No. DMS-0140338; research of Q. Huang supported in part by NSF grants No. DMS-0201599 and No. DMS-0502045.