Traveling Plane Wave Research Articles

Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

AbstractWe study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.

Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Stability of Planar Traveling Waves for a Class of Lotka–Volterra Competition Systems with Time Delay and Nonlocal Reaction Term

Pattern formation and front stability for a moving-boundary model of biological invasion and recession

We investigate pattern formation in a two-dimensional (2D) Fisher–Stefan model, which involves solving the Fisher–KPP equation on a compactly-supported region with a moving boundary. The boundary evolves analogously to the classical one-phase Stefan problem, such that boundary speed is proportional to the local population density gradient. By combining the Fisher–KPP and classical Stefan theory, the Fisher–Stefan model alleviates two limitations of the Fisher–KPP equation for biological populations. Unlike the Fisher–KPP equation, solutions to the Fisher–Stefan model have compact support, explicitly defining the region occupied by the population. Furthermore, the Fisher–Stefan model admits travelling wave solutions with non-negative density for all wave speeds c∈(−∞,∞), and can thus model both population invasion and population recession. In this work, we investigate whether the 2D Fisher–Stefan model predicts pattern formation, by analysing the linear stability of planar travelling wave solutions to sinusoidal transverse perturbations. Planar fronts of the Fisher–KPP equation are linearly stable. Similarly, we demonstrate that invading planar fronts (c>0) of the Fisher–Stefan model are linearly stable to perturbations of all wave numbers. However, our analysis demonstrates that receding planar fronts (c<0) of the Fisher–Stefan model are linearly unstable for all wave numbers. This is analogous to unstable solutions for planar solidification in the classical Stefan problem. Introducing a surface tension regularisation stabilises receding fronts for short-wavelength perturbations, giving rise to a range of unstable modes and a most unstable wave number. We supplement linear stability analysis with level-set numerical solutions that corroborate theoretical results. Overall, front instability in the Fisher–Stefan model suggests a new mechanism for pattern formation in receding biological populations.

Emerging Spiral Waves and Coexisting Attractors in Memductance-Based Tabu Learning Neurons

Understanding neuron function may aid in determining the complex collective behavior of brain systems. To delineate the collective behavior of the neural network, we consider modified tabu learning neurons (MTLN) with magnetic flux. Primarily, we explore the rest points and stability of the isolated MTLN, as well as its dynamical characteristics using maximal Lyapunov exponents. Surprisingly, we discover that for a given set of parameter values with distinct initial conditions, the periodic and the chaotic attractors may coexist. In addition, experimental analysis is carried out using a microcontroller-based implementation technique to support the observed complex behavior of the MTLN. We demonstrate that the observed numerical results are in good agreement with the experimental verification. Eventually, the collective behaviors of the considered MTLN are investigated by extending them to the network of the lattice array. We discover that when the magnetic flux coupling coefficient is varied in the presence of an external stimulus, the transition from spiral waves to traveling plane waves occurs. Finally, we manifest the formation of spiral waves in the absence of an external stimulus in contrast to previous observations.

Acoustic travelling wave separation in the time domain using electronic time delay circuits and leaky recursion

Acoustic travelling plane wave separation is implemented in the time domain from the sound pressure signals of two closely spaced microphones using analog electronic circuits based on a leaky recursion formulation. The sound pressure and acoustic particle velocity signals at the midpoint between the microphones are estimated from delayed recursive expressions modified for stability by introducing mild loss recursion. The forward and backward travelling wave components are obtained electronically by simple linear combinations of the sound pressure and particle velocity signals. Two travelling wave separation circuits (TWSC) were designed, built, and evaluated from measurements in an acoustic test tube apparatus in the operational frequency range from 710 to 5,680 Hz. The two designs differ in the use of first- or second-order time delay electronic circuits, with performance of the first-order circuit degrading with increasing frequency. Experimental results for the second-order circuit show a travelling wave separation error of -30.4 dB in the operational frequency range, a travelling wave amplitude accuracy within ±0.1 dB in most of the frequency range, with an implementation that offers operation in real-time with low latency, and resilience to unknown initial conditions or similar errors during operation, with typical transient relaxation times of 2 ms, that can be made lower at the expense of reduced accuracy.

Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

This paper is concerned with the spatial propagation of bistable nonlocal dispersal equations in exterior domains. We first obtain the existence and uniqueness of an entire solution which behaves like a planar traveling wave front for large negative time. Then, when the entire solution comes to the interior domain, the profile of the front will be disturbed. However, the disturbance is local in space for finite time, which means the disturbance disappears as its location is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile uniformly in space and continue to propagate in the same direction for large positive time provided that the interior domain is compact and convex. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010).

Acoustic Radiation Force and Torque Acting on Asymmetric Objects in Acoustic Bessel Beam of Zeroth Order Within Rayleigh Scattering Limit

Acoustic momentum exchange between objects and the surrounding fluid can be quantified in terms of acoustic radiation force and torque, and depends on several factors including the objects’ geometries. For a one-dimensional plane wave type, the induced torque on the objects with arbitrary shape becomes a function of both, direct polarization and Willis coupling, as a result of shape asymmetry, and has only in-plane components. Here, we investigate, in the Rayleigh scattering limit, the momentum transfer to objects in the non-planar pressure field of an acoustic Bessel beam with axisymmetric wave front. This type of beam is selected since it can be practically realized by an array of transducers that are cylindrically arranged and tilted at the cone angle β which is a proportionality index of the momentum distribution in the transverse and axial propagation directions. The analytical expressions of the radiation force and torque are derived for both symmetric and asymmetric objects. We show the dependence of radiation force and torque on the characteristic parameters β and radial distance from the beam axis. By comparing against the case of a plane travelling plane wave, zero β angle, we demonstrated that the non-planar wavefront of a zeroth order Bessel beam causes an additional radial force and axial torque. We also show that, due to Willis coupling, an asymmetric object experiences greater torques in the θ direction, by minimum of one order of magnitude compared to a plane travelling wave. Further, the components of the partial torques owing to direct polarization and Willis coupling act in the same direction, except for a certain range of cone angle β. Our findings show that a non-planar wavefront, which is quantified by β in the case of a zeroth-order Bessel beam, can be used to control the magnitude and direction of the acoustic radiation force and torque acting on arbitrarily shaped objects, implying that the wavefront should be adjusted according to the object’s shape to impart acoustic momentum in all directions and achieve a desired acoustophoretic response.

Traveling waves for reaction-diffusion PDE coupled to difference equation with nonlocal dispersal term and time delay

We consider a class of biological models represented by a system composed of reactiondiffusion PDE coupled with difference equations (renewal equations) in n-dimensional space, with nonlocal dispersal terms and implicit time delays. The difference equation generally arises, by means of the method of characteristics, from an age-structured partial differential system. Using upper and lower solutions, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.

Isolated periodic wave solutions arising from Hopf and Poincaré bifurcations in a class of single species model

In this paper, we consider the bifurcations of local and global isolated periodic traveling waves in a single species population model described by a reaction-diffusion equation. Based on the singular point quantity algorithm of conjugate symmetric complex systems, we investigate Hopf bifurcation from all equilibrium points for the corresponding planar traveling wave system. We obtain all center conditions and construct one perturbed Hamiltonian system to study Poincaré bifurcation. Further, using the Chebyshev criterion, we develop a utilized approach to prove the existence of at most two limit cycles in a piecewise continuous parameter interval. Finally, the existence of double isolated periodic traveling waves for the model is established, and the results are illustrated by numerical simulation. It is shown that in a population model with density-dependent migrations and Allee effect, two large amplitude oscillations (isolated periodic traveling waves) can exist simultaneously.

Acoustic radiation force and radiation torque beyond particles: Effects of nonspherical shape and Willis coupling.

Acoustophoresis mainly deals with the manipulation of subwavelength scatterers in an incident acoustic field. The geometric details of manipulated particles are often neglected by replacing them with equivalent symmetric geometries such as spheres, spheroids, cylinders, or disks. It has been demonstrated that geometric asymmetry, represented by Willis coupling terms, can strongly affect the scattering of a small object; hence neglecting these terms may miss important force contributions. In this work, we present a generalized formalism of acoustic radiation force and radiation torque based on the polarizability tensor, where Willis coupling terms are included to account for geometric asymmetry. Following Gorkov's approach, the effects of geometric asymmetry are explicitly formulated as additional terms in the radiation force and torque expressions. By breaking the symmetry of a sphere along one axis using intrusion and protrusion, we characterize the changes in the force and torque in terms of partial components, associated with the direct and Willis coupling coefficients of the polarizability tensor. We investigate the cases of standing and traveling plane waves and show how the equilibrium positions and angles are shifted by these additional terms. We show that while the contributions of asymmetry to the force are often negligible for small particles, these terms greatly affect the radiation torque. Our presented theory, providing a way of calculating radiation force and torque directly from polarizability coefficients, shows that it is essential to account for shape of objects undergoing acoustophoretic manipulation, with important implications for applications such as the manipulation of biological cells.

Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion

We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−Δ)α/2. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power α∈[1,2] but unstable for α∈(0,1). In the focusing case, they can be linearly unstable for any α∈(0,2]. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as α∈(1,2] decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For α≤1, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for α&lt;1, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities.

주변 유체와 연성된 수중 구조물의 진동분석을 위한 스펙트럼 요소법

To design a spacer grid for protecting underwater structures consisting of cylinders against sudden external loads, it is necessary to perform a numerical analysis on the coupling system considering the structural vibration and surrounding fluids. The structural vibration of cylinders coupled with viscous fluid is predicted by using the spectral element method (SEM). Scattering waves induced by traveling plane waves are calculated and utilized to investigate the pressure forces of structures, and the transfer function for the flexural vibration and pressure amplitudes of plane wave is calculated. The finite element method (FEM) is used to verify the numerical results obtained using the SEM. The amplitudes of the added mass of each cylinder are dependent on the gap between surrounding fluids and adjacent structures. When the input frequency of traveling waves is increased, the pressure fields become significantly more complicated. When the number of cylinders is increased, the transfer function at the specific frequency ranges show overlapping resonance modes. The proposed numerical method can be used to save the computing time and improve the accuracy of the numerical analysis technique for studying the vibration of underwater structures.

A Quantum Optical Approach to the effect of a Laser Mode on the Motion of Atomic Vapor by Varying the Field Coherence Angles

We follow theoretically the motion of the sodium atoms in vapor state under the influence of a laser mode in (1 + 1) D, which is achieved by introducing different optical filters. In the Dirac interaction representation, the equations of motion are represented via the Bloch form together with the Pauli operators to find the elements of the density matrix of the system. The emergence of the principle of coherence in varying the angles of the laser mode permits the evaluation of the average force affecting the atoms' acceleration or deceleration, and hence the corresponding velocities and temperatures are investigated. The atomic vapor is introduced in a region occupied by a heat bath presented by the laser field, such that the state of the atomic vapor is unstable inside the system due to the loss or gain of its kinetic energy to or from the laser field. This instability is studied by finding the eigenvalues of the system's entropy. Resorting to the assumption of Botin, Kazantsev, and Pusep, who issued in the presence of the weak and strong spontaneous emission, a coupling between the mean numbers of photons in terms of time, which allows the evaluation of the rate of entropy production of the system under study. No singularities are found throughout the process of equations solving and other calculations. Resorting to symbolic software, a set of figures illustrating the nonlinear behavior in the dynamics of the problem is present. In this paper, we introduce a theoretical study of the effect of two-counter propagation traveling plane waves on the motion of the sodium atoms in the vapor state by varying the coherence angles to investigate the atomic behavior. Good agreements are found with previous studies.

Non‐linear behaviour of electron acoustic wave dynamics in a magnetized plasma with non‐thermal hot electrons

AbstractNon‐linear dynamical behaviour of electron acoustic waves (EAWs) is studied in a magnetized non‐thermal plasma (containing inertial cold electrons, inertialess hot electrons following non‐thermal distribution function, and static ions) via a fluid dynamical approach. A linear dispersion relation is derived and the propagation of two possible modes and their evolution are studied through the different plasma configuration parameters, such as non‐thermality and external magnetic field strength. In a non‐linear perturbation regime, a reductive perturbation technique is employed to derive the non‐linear evolution equation and the analysis is executed for travelling plane waves in terms of a non‐linear dynamical system to enlighten the numerous aspects of the phase space dynamics. The results of numerical simulation predict the existence of a wide class of non‐linear structures, namely solitonic, periodic, quasiperiodic, and chaotic depending upon different controlling plasma parameters. Also, Poincaré return map analysis confirms these non‐linear structures of the EAWs.

A one-dimensional symmetry result for entire solutions to the Fisher-KPP equation

We consider the Fisher-KPP reaction-diffusion equation in the whole space. We prove that if a solution has, to main order and for all times (positive and negative), the same exponential decay as a planar traveling wave with speed larger than the minimal one at its leading edge, then it has to coincide with the aforementioned traveling wave.

Development of exact mathematic models for planar slotted traveling wave antennas using the dyadic Green function and their experiment foundation

Exact mathematic models were developed for traveling wave antennas with a constant slot width based on the dyadic Green function, taking into account both the main and cross-polarization components of the field in the far radiation zone. The possibility of using these models as basic models for studying the electrodynamic properties of traveling wave antennas with a different slot configuration (linearly, exponentially tapering, etc.) is justified. The influence of the geometric parameters of the investigated traveling wave antennas on their characteristics was studied by means of using the established mathematic models. It has been found that in the electrodynamic E plane, with an increase of the length and width of the slot, the main beam of the radiation pattern narrows, which leads to an increase the directed properties of the antennas. In the electrodynamic H plane, the width of the main beam of radiation pattern decreases only as the length of the slot of traveling wave antennas increases and does not depend on its width in practice. It was shown that the established theory results of the study are well consistent with the results of experiments, which also revealed the limitations of the applicability of the developed models. Analysis of traveling wave antennas characteristics was carried out at central frequency of 10 GHz in the frequency range of 8-12 GHz.

Propagation Phenomena with Nonlocal Diffusion in Presence of an Obstacle

We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form \({\mathbb {R}}^N\setminus K\), where \(K\subset {\mathbb {R}}^N\) is a compact “obstacle”. The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as \(t\rightarrow -\infty \) and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front.

Vanishing diffusion limits for planar fronts in bistable models with saturation

We deal with planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like u t = ε d i v ( ∇ u 1 + | ∇ u | 2 ) + f ( u ) , u = u ( x , t ) , x ∈ R n , t ∈ R , \begin{equation*}u_t=\varepsilon \, \mathrm {div}\, \left (\frac {\nabla u}{\sqrt {1+\vert \nabla u \vert ^2}}\right ) + f(u), \quad u=u(x, t), \; x \in \mathbb {R}^n, \, t \in \mathbb {R}, \end{equation*} analyzing in particular their behavior for ε → 0 \varepsilon \to 0 . First, we construct monotone and non-monotone planar traveling waves, using a change of variables allowing to analyze a two-point problem for a suitable first-order reduction of the equation above; then, we investigate the asymptotic behavior of the monotone fronts for ε → 0 \varepsilon \to 0 , showing their convergence to suitable step functions. A remarkable feature of the considered diffusive term is that the fronts connecting 0 0 and 1 1 are necessarily discontinuous (and steady, namely with 0 0 -speed) for small ε \varepsilon , so that in this case the study of the convergence concerns discontinuous steady states, differently from the linear diffusion case.

Radar system with dedicated planar traveling wave antennas for elderly people monitoring

Abstract. Radar systems for contactless vital sign monitoring are well known and an actual object of research. These radar-based sensors could be used for monitoring of elderly people in their homes but also for detecting the activity of prisoners and to control electrical devices (light, audio, etc.) in smart living environments. Mostly these sensors are foreseen to be mounted on the ceiling in the middle of a room. In retirement homes the rooms are mostly rectangular and of standardized size. Furniture like beds and seating are found at the borders or the corners of the room. As the propagation path from the center of the room ceiling to the borders and corners of a room is 1.4 and 1.7 time longer the power reflected by people located there is 6 or even 10 dB lower than if located in the center of the room. Furthermore classical antennas in microstrip technology are strengthening radiation in broadside direction. Radar systems with only one single planar antenna must be mounted horizontally aligned when measuring in all directions. Thus an antenna pattern which is increasing radiation in the room corners and borders for compensation of free space loss is needed. In this contribution a specification of classical room sizes in retirement homes are given. A method for shaping the antenna gain in the E-plane by an one-dimensional series-fed traveling wave patch array and in the H-plane by an antenna feeding network for improvement of people detection in the room borders and corners is presented for a 24 GHz digital beamforming (DBF) radar system. The feeding network is a parallel-fed power divider for microstrip patch antennas at 24 GHz. Both approaches are explained in theory. The design parameters and the layout of the antennas are given. The simulation of the antenna arrays are executed with CST MWS. Simulations and measurements of the proposed antennas are compared to each other. Both antennas are used for the transmit and the receive channel either. The sensor topology of the radar system is explained. Furthermore the measurement results of the protoype are presented and discussed.

Existence and stability traveling wave solutions for a system of social outbursts

We study the existence and stability of planar traveling wave solutions to the system of reaction-diffusion equations:{ut=∂2u∂x2+Γu(1−u)1+e−β(v−1)−u,x∈Rn,t>0,vt=∂2v∂x2−v(1+u)p+1,x∈Rn,t>0, which was introduced in [8] as a model for the spatiotemporal dynamics of rioting activity and social tension. The “wave-like” spread of rioting activity observed in the 2005 French Riots [12] motivates the study of these type of solutions, which were numerically analyzed in [46]. Of particular interest is the qualitative difference in traveling wave solutions between the p<0 and p≥0 case. We present results for the existence and stability of these type of solutions for p≥0 and provide the asymptotic approach rates of the traveling wave solutions to the steady states as |x|→∞. For the case of p<0 we study the stability of such solutions using Evans Functions and a combination of analytical and numerical methods to show irrefutable evidence of spectral stability.