Traveling Wave Fronts Research Articles

Two-dimensional visualisation of a travelling sound wave over porous medium

Prior research has demonstrated a variation in sound speed and a wave bending in the air when an acoustic wavefront approaches a porous medium at a grazing angle. The impact is found to be explicitly dependent on the frequency of the sound and the nature of the adjacent absorber. The current study expands the investigations by using a distributed array of eight microphones mounted on a single-axis motorised mover. The measurement setup allowed for an extensive collection of impulse responses within a confined area near a horizontally placed porous panel. The sensors captured a frequency sweep every 10 millimetres on multiple horizontal planes above the air-material boundary. Moreover, the array configuration allowed for simultaneous capture of wave propagation both above and outside the projection area of the absorbing material. The progression of a travelling wavefront is reconstructed as an image-sequence-based 2D animation on four discrete vertical planes, 60 cm by 50 cm each, traversing or passing outside the porous material.

Spatial propagation in a delayed spruce budworm diffusive model

We investigate the spatial propagation in a delayed spruce budworm diffusive model where and represent, respectively, the incubation and the maturation delays for the spruce budworm. We find the minimal wave speed to determine the existence of traveling wave fronts of the model. More specifically, the model admits traveling wave fronts when ; the model has no traveling wave solutions when . The proofs are based on combining the upper and lower solutions with the approach of Wu and Zou's theorems, the limit arguments, and Laplace transform. The obtained results help us to understand the spreading patterns and the spreading speed of spruce budworm population.

Modeling and Simulating an Epidemic in Two Dimensions with an Application Regarding COVID-19

We derive a reaction–diffusion model with time-delayed nonlocal effects to study an epidemic’s spatial spread numerically. The model describes infected individuals in the latent period using a structured model with diffusion. The epidemic model assumes that infectious individuals are subject to containment measures. To simulate the model in two-dimensional space, we use the continuous Runge–Kutta method of the fourth order and the discrete Runge–Kutta method of the third order with six stages. The numerical results admit the existence of traveling wave solutions for the proposed model. We use the COVID-19 epidemic to conduct numerical experiments and investigate the minimal speed of spread of the traveling wave front. The minimal spreading speeds of COVID-19 are found and discussed. Also, we assess the power of containment measures to contain the epidemic. The results depict a clear drop in the spreading speed of the traveling wave front after applying containment measures to at-risk populations.

Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

<p>In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.</p>

Speed determinacy of traveling waves for a lattice stream-population model with Allee effect

<abstract><p>This paper investigates the speed selection mechanism for traveling wave fronts of a reaction-diffusion-advection lattice stream-population model with the Allee effect. First, the asymptotic behaviors of the traveling wave solutions are given. Then, sufficient conditions for the speed determinacy of the traveling wave are successfully obtained by constructing appropriate upper and lower solutions. We examine the model with the reaction term $ f (\psi) = \psi(1-\psi)(1+\rho\psi) $, with $ \rho $ being a nonnegative constant, as a specific example. We give a novel conjecture that there exists a critical value $ \rho_c > 1 $, such that the minimal wave speed is linearly selected if and only if $ \rho\leq\rho_c $. Finally, our speculation is verified by numerical calculations.</p></abstract>

Existence of traveling wave fronts for a diffusive Mackey–Glass model with two delays

Based on the diapause effect, a diffusive Mackey–Glass model with distinct diapause and developmental delays is proposed. By constructing appropriate upper and lower solutions and employing the inequality techniques, some sufficient conditions on the existence of traveling wave fronts are obtained. These results not only improve but also supplement some existing ones. Moreover, two numerical examples are provided to demonstrate their reliability and feasibility.

Stability and uniqueness of generalized traveling front for non-autonomous Fisher–KPP equations with nonlocal diffusion

This paper is concerned with the stability and uniqueness of generalized traveling front of non-autonomous Fisher–KPP equations with nonlocal diffusion. In such non-autonomous nonlocal dispersal equations, both the diffusion kernel and the reaction term generally depend on time. The existence of generalized traveling wave fronts of such equations has been studied in Ducrot and Jin (2022). By using comparison principle and part metric, we show the generalized traveling front is asymptotically stable under well-fitted perturbation and it is also unique.

Semi-waves and spreading speeds of stage-structured diffusive competition model with a free boundary

A stage-structured diffusive competition model with a free boundary is considered. The main aim of this paper is to study the evolutions of mature invasive species with competitive ability and determine whether it disrupts the stable ecosystem provided by native species. According to the long-time behaviors under the case of weak-strong competition, strong-weak competition, and weak competition, we obtain respectively the exact asymptotic spreading speed by delay-induced semi-wave problems if the spreading of invasive species occurs. And we demonstrate that, as the moving coefficient goes to infinite, the corresponding Cauchy problem can be viewed as the limiting profile of the model, and the asymptotic spreading speed approaches the minimal speed of the corresponding traveling wavefront solutions. Moreover, under the strong-weak competition situation, the asymptotic spreading speed of invasive species will decrease when the maturation time is longer.

The stability of traveling wave fronts for Belousov-Zhabotinskii system with small delay

This paper deals with the stability of traveling wave fronts for Belousov-Zhabotinskii system with small delay. First, the precise spatial decay rates of traveling wave fronts with small delay are given. Then by using Theorem 4.1 of Chapter 5 in [22], each traveling front solution with wave speed $ c>2\sqrt{1-r} $ is proved to be nonlinearly exponentially stable in some appropriate exponentially weighted spaces.

Traveling wave fronts of a diffusive Nicholson’s Blowflies equation with two delays

In this paper, we consider a diffusive Nicholson’s Blowflies equation with two delays. We construct a new upper solution for the wave equation and then show the equation admits the traveling wave fronts connecting the two equilibria. And also, we obtain the non-existence for traveling waves of the equation. This paper can be as a corrigendum for the paper (Huang and Liu, 2022).

MODAL TRANSFORM FEATURES OF CABLE LINE ELECTRICAL VALUES IN A TRAVELING WAVE FAULT LOCATOR

It is known that during a short circuit on an overhead line, the traveling waves in the phases contain components of two aerial and one ground modes. Due to the different propagation velocity along the line of these components at the locator installation place they arrive at different times. This leads to the fact that the traveling wave front in the phases becomes less pronounced, and, consequently, the accuracy of determining their arrival time and accuracy of the fault location decrease. To divide the phase values ​​of an overhead line into independent components of modes, invariant transformations – Clark transformation, Karrenbauer transformation, or Wedepohl transformation – are used. However, these transformations in their classical form cannot be applied to the electrical values of a cable line, since its design differs from that of an overhead power line. The purpose of the article is to illustrate the use of modal transformation on a cable line. As a result of the study, the following conclusions were obtained: currents in the cable line cores do not affect each other, therefore, in the wave fault locator that measures current waves, it is necessary to use directly the phase currents of the cores without modal transformation; if the wave fault locator measures the currents or voltages of the cable screens, the modal transformation should be performed, and it must be taken into account that the three modes formed by the screens are similar to the three modes of the overhead line; if a power transformer is adjacent to the cable line, the current traveling waves in the cores will be completely reflected from the measurement place, in this case in the wave fault locator it is necessary to use voltage traveling waves in the mode between the core and the screen of each cable of the three-phase group.

Estimation of the Traveling Wave Speed in Pancreatic Islets Using High Period Periodic Orbits

In response to an increase in blood glucose levels, insulin is released into the bloodstream by the pancreatic islets of Langerhans. As a result of this influx of glucose, the islets start what are called bursting oscillations of the membrane potential and the intracellular calcium concentration. Time delays of several seconds in the activity of distant cells in the islets have been observed, indicating the presence of traveling waves through the islets. By considering a robust model of a pancreatic islet in one dimension, we study the relationship between the wave speed and the model parameters for the existence of traveling wave fronts and traveling wave pulses. After a systematic reduction of the model equations, the wave fronts (or heteroclinic connection) are studied. Using the bi-stable equation, for which an exact expression of the heteroclinic connection can be computed, we use a homotopy parameter to move from this equation to an islet model. A relationship between the wave speed and the conductance of the ATP-modulated potassium channel is constructed. Upon the inclusion of the slow gating variable back into the model equations, we observe the presence of a traveling wave pulse (or homoclinic connection). Using a high period periodic orbit to approximate the homoclinic orbit, a similar relationship between these two parameters is constructed. We observe that the heteroclinic connection is a good approximation for a portion of the homoclinic connection. Comparisons of the speed of the wave traveling through the islet in the partial differential equation model and the model in traveling coordinates is carried out.

Global stability of bistable traveling wavefronts for a three-species Lotka–Volterra competition system with nonlocal dispersal

This paper is devoted to the study of traveling wavefronts for a three-component Lotka–Volterra system with nonlocal dispersal. This system arises in the study of three-species competition model in which there is no competition between two of these three species. It has been shown that this system admits a bistable traveling wavefront. In this paper, we further investigate the stability of bistable traveling wavefronts. By constructing suitable super- and sub-solutions and using a dynamical system approach, we obtain the globally asymptotic stability of the bistable traveling wavefronts.

Existence of Traveling Wave Fronts for a Generalized Nonlinear Schrodinger Equation

In the presented paper, a generalized nonlinear Schr o dinger equation without delay convolution kernel and with special delay convolution kernel is investigated. By using the geometric singular perturbation theory, the existence of traveling wave fronts is proved. Firstly, we show that such traveling wave fronts exist without delay by non-Hamiltonian qualitative analysis. Then, for the generalized nonlinear Schr o dinger equation with a special local strong delay convolution kernel, the desired heteroclinic orbit is obtained by using the Fredholm theory.

Asymptotics of transverse momentum broadening in dense QCD media

We study the asymptotic behaviour of the transverse momentum broadening distribution of an energetic quark or gluon propagating through dense QCD matter, in the large system size LL limit, taking into account radiative corrections in the double logarithmic approximation. Thanks to a connection between the evolution of the jet quenching parameter \hat{q}q̂ and the formation of traveling wave fronts in nonlinear physics, we obtain a formula for the LL dependence of the characteristic transverse momentum scale Q_sQs of the distribution valid up to terms of order 1/\ln(L)1/ln(L). We briefly discuss the physical implications of this formula for jet quenching and small-xx phenomenology.

Stability of traveling wave fronts for a modified vector disease model

Stability of traveling wave fronts for a modified vector disease model

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).

Traveling wave fronts for a diffusive Nicholson’s Blowflies equation accompanying mature delay and feedback delay

For the diffusive Nicholson’s Blowflies equation accompanying disparate mature delay and feedback delay, the traveling wave front analysis has not been touched. By using the inequality technique and constructing suitable upper and lower solutions, we successfully built up a novel criterion to ensure the existence of traveling wave front solutions of the proposed model, which is essentially new and supplement some existing literatures. In particular, we afford an explanatory example to substantiate the validness and efficacy of the theoretical results.

The Traveling Wave Solutions in a Mixed-Diffusion Epidemic Model

In this paper, we study the traveling wave solution of an epidemic model with mixed diffusion. First, we give two definitions of the minimum wave speeds and prove that they are equivalent. Second, the existence, decaying behavior, and uniqueness of traveling wave fronts are obtained. Third, the signs of minimum wave speeds are studied, and further, in two specific cases of the dispersal kernel, we show how to identify the signs of minimum wave speeds.

Dynamic peculiarities of interphase boundaries and domain walls for non-spin domains

The analysis of magnetization oscillations (de Haas – van Alphen effect; dHvA effect) and features of the Fermi surface shape in three (3D)- and two-dimensional (2D) metals shows that the dynamic phenomena — the dynamics of domain walls and interphase boundaries during diamagnetic phase transitions — are described by a new nonlinear partial differential equation. The equation is a result of the inclusion of the case of multiple extremal cross sections of the Fermi surface in these metals. Our consideration indicates the possibility of the appearance of metastable non-spin domains (Condon domains) and first-order phase transitions to the ordered phase in the regime of dHvA oscillations for the two-frequency case. Sine–Gordon, Klein–Gordon, double sine–Gordon, time-dependent Ginzburg–Landau equations, and the telegraph equation are limiting cases of the derived equation. We show that particular moving kink-soliton solutions of the equation describe traveling wave fronts being moving domain walls and interphase boundaries in the Condon domain phase.