Bifurcation to locked fronts in two component reaction–diffusion systems

We study invasion fronts and spreading speeds in two component reaction–diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.     

Diffusion of a test electron beam in a discrete spectrum of waves propagating along a traveling wave tube

The velocity diffusion of particles in a field of randomly phased waves is experimentally investigated. An arbitrary waveform generator is used to launch a prescribed discrete spectrum of waves along the helix of a traveling wave tube. A cold test electron beam propagates along the axis of the tube and interacts with the waves without self-consistently perturbing their amplitudes. A trochoidal energy analyzer records the beam energy distribution at the output of the tube. The energy spread of the beam is measured as the position of the emitter probe is varied. A velocity diffusion coefficient can thus be measured. Two different situations are compared: one with a large overlap parameter between neighboring modes where standard quasilinear diffusion theory is valid; the other one with an intermediate overlap parameter where numerical simulations have shown that the diffusion coefficient exceeds the quasilinear value by a factor of about 2.3.     

Traveling waves in a coarse-grained model of volume-filling cell invasion: Simulations and comparisons

Many reaction–diffusion models produce traveling wave solutions that can be interpreted as waves of invasion in biological scenarios such as wound healing or tumor growth. These partial differential equation models have since been adapted to describe the interactions between cells and extracellular matrix (ECM), using a variety of different underlying assumptions. In this work, we derive a system of reaction–diffusion equations, with cross-species density-dependent diffusion, by coarse-graining an agent-based, volume-filling model of cell invasion into ECM. We study the resulting traveling wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic comparison between the behaviors observed in this model and those predicted by simpler models in the literature that do not take into account volume-filling effects in the same way. Our study justifies the use of some of these simpler, more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes, but it also reveals some interesting properties arising from the introduction of cell and ECM volume-filling effects, where standard model simplifications might not be appropriate.     

Traveling wave solutions of the n-dimensional coupled Yukawa equations

We discuss traveling wave solutions to the Yukawa equations, a system of nonlinear partial differential equations which has applications to meson–nucleon interactions. The Yukawa equations are converted to a six-dimensional dynamical system, which is then studied for various values of the wave speed and mass parameter. The stability of the solutions is discussed, and the methods of competitive modes is used to describe parameter regimes for which chaotic behaviors may appear. Numerical solutions are employed to better demonstrate the dependence of traveling wave solutions on the physical parameters in the Yukawa model. We find a variety of interesting behaviors in the system, a few of which we demonstrate graphically, which depend upon the relative strength of the mass parameter to the wave speed as well as the initial data.     

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

This paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained.     

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.     

Traveling waves for delayed non-local diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed non-local diffusion equations with crossing-monostability. Based on constructing two associated auxiliary delayed non-local diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space using the traveling wave fronts of the auxiliary equations, the existence of traveling waves is proved by Schauder’s fixed point theorem. The result implies that the traveling waves of the delayed non-local diffusion equations with crossing-monostability are persistent for all values of the delay τ?0.     

Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity

This paper is concerned with the existence of traveling wave fronts for delayed non-local diffusion systems without quasimonotonicity, which can not be answered by the known results. By using exponential order, upper-lower solutions and Schauder's fixed point theorem, we reduce the existence of monotone traveling wave fronts to the existence of upper-lower solutions without the requirement of monotonicity. To illustrate our results, we establish the existence of traveling wave fronts for two examples which are the delayed non-local diffusion version of the Nicholson's blowflies equation and the Belousov-Zhabotinskii model. These results imply that the traveling wave fronts of the delayed non-local diffusion systems without quasimonotonicity are persistent if the delay is small.     

一类具有非线性发生率与时滞的离散扩散SIR模型临界行波解的存在性

研究了一类具有非线性发生率的离散扩散时滞SIR模型的临界行波解的存在性.在人口总数非恒定的条件下,首先,应用上下解法与Schauder不动点定理证明了解在有限闭区间上的存在性;其次,通过极限讨论了临界行波解在整个实数域上存在;最后,通过反证法与波动引理得到了行波解在无穷远处的渐近行为.     

Existence and Stability of Traveling Waves for Degenerate Reaction–Diffusion Equation with Time Delay

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of \(c\ge c^*\) for the degenerate reaction–diffusion equation without delay, where \(c^*>0\) is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay \(\tau >0\). Furthermore, we prove the global existence and uniqueness of \(C^{\alpha ,\beta }\)-solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted \(L^1\)-space. The exponential convergence rate is also derived.     

Existence of traveling wave solutions for influenza model with treatment

To investigate the spreading speed of influenza and the influence of treatment on the spreading speed, a reaction–diffusion influenza model with treatment is established. The existence of traveling wave solutions is shown by introducing an auxiliary system and applying the Schauder fixed point theorem. The non-existence of traveling wave solutions is proved by a two-sided Laplace transform, which needs a new approach for the prior estimate of exponential decay of traveling wave solutions.     

Existence of traveling waves to any Lax shock satisfying Oleinik’s criterion in conservation laws

Given any shock wave of a conservation law where the flux function may not be convex, we want to know whether it is admissible under the criterion of vanishing viscosity/capillarity effects. In this work, we show that if the shock satisfies the Oleinik’s criterion and the Lax shock inequalities, then for an arbitrary diffusion coefficient, we can always find suitable dispersion coefficients such that the diffusive-dispersive model admits traveling waves approximating the given shock. The paper develops the method of estimating attraction domain for traveling waves we have studied before.     

Monotonicity,uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c ?c _?. Then we show that the traveling wave fronts with speed c >c _? are exponentially asymptotically stable.     

具有接种和非局部扩散的流行病模型的行波解（英文）

An epidemic model with vaccination and nonlocal diffusion is proposed, and the existence of traveling wave solutions of this model is studied. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder’s fixed point theorem,sufficient conditions are obtained for the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.     

Bifurcations and Exact Solutions of the Raman Soliton Model in Nanoscale Optical Waveguides with Metamaterials

In this paper, we study Raman soliton model in nanoscale optical waveguides with metamaterials, having polynomial law non-linearity. By using the bifurcation theory method of dynamical systems to the equations of $\phi(\xi)$, under 24 different parameter conditions, we obtain bifurcations of phase portraits and different traveling wave solutions including periodic solutions, homoclinic and heteroclinic solutions for planar dynamical system of the Raman soliton model. Under different parameter conditions, 24 exact explicit parametric representations of the traveling wave solutions are derived. The dynamic behavior of these traveling wave solutions are meaningful and helpful for us to understand the physical structures of the model.     

Traveling wave solutions for an autocatalytic reaction–diffusion model

In this paper we consider an autocatalytic reaction–diffusion model which has many applications. We extend previous results using qualitative analysis and show the existence of an exponentially decaying traveling wave front for a minimum speed and algebraically decaying wave fronts for large speeds. Further, the wave front profiles are calculated and the minimum speed is accurately determined using different numerical methods.     

Traveling waves for chemotaxis–systems

In this paper we study the existence of traveling wave solutions to the Keller‐Segel model, a general model of chemotaxis, where the species do not reproduce. In the case of logarithmic sensitivity we show that various functionals modeling the reactive feedback on the chemo‐attractant do allow for traveling waves and a wide range of qualitatively different behavior is possible. We can find monotone fronts as well as pulse solutions in the densities of the population and the chemical. In particular, a new kind of solution exists, where both densities travel as pulses.     

A Numerical Method for Constructing a Self-Similar Isolated Wave Solution

An efficient numerical algorithm is developed for constructing self-similar isolated wave or switching wave solutions. The algorithm is developed for the well-known Kolmogorov–Petrovskii–Piskunov (KPP) problem, which has a switching-wave analytical solution, and is applied to construct an isolated traveling pulse in the four-component reaction–diffusion model.     

An application of traveling wave analysis in economic growth model

A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance.     

Minimal wave speed and spread speed of competing pioneer and climax species

In this article, for a diffusive population model describing interaction of pioneer-climax species, we explore the issues of spreading speed, linear determinacy and traveling wave fronts. Applying the theory developed by Weinberger et al. [J. Math. Biol. 2002;45:183–218], we identify some ranges of model parameters within which, the model is shown to have a single spreading speed which is linearly determinate and coincides with the corresponding minimal speed for the traveling wave fronts connecting two relevant equilibria, one being a boundary equilibrium and the other being a coexistence equilibrium.