Traveling Front Research Articles

On dynamics of gasless combustion in slowly varying periodic media

In this paper we consider a classical model of gasless combustion in a one dimensional formulation under the assumption of ignition temperature kinetics. We study the propagation of flame fronts in this model when the initial distribution of the solid fuel is a spatially periodic function that varies on a large scale. It is shown that in certain parametric regimes the model supports periodic traveling fronts. An accurate asymptotic formula for the velocity of the flame front is derived and studied. The stability of periodic fronts is also explored, and a critical condition in terms of parameters of the problem is derived. It is also shown that the instability of periodic fronts, in certain parametric regimes, results in a propagation-extinction-conduction-reignition pattern which is studied numerically.Novelty and significance statement: This work provides a closed form asymptotic description of periodic traveling fronts in a gasless combustion model with step-wise ignition temperature kinetics with a slowly varying concentration field. The stability analysis is performed, and the range of applicability of asymptotic formulas is given. A new propagation-extinction-conduction-reignition regime is identified. This regime emerges exclusively due to periodicity of the concentration field.

Entire solutions with and without radial symmetry in balanced bistable reaction–diffusion equations

AbstractLet $$n\ge 2$$ n ≥ 2 be a given integer. In this paper, we assert that an n-dimensional traveling front converges to an $$(n-1)$$ ( n - 1 ) -dimensional entire solution as the speed goes to infinity in a balanced bistable reaction–diffusion equation. As the speed of an n-dimensional axially symmetric or asymmetric traveling front goes to infinity, it converges to an $$(n-1)$$ ( n - 1 ) -dimensional radially symmetric or asymmetric entire solution in a balanced bistable reaction–diffusion equation, respectively. We conjecture that the radially asymmetric entire solutions obtained in this paper are associated with the ancient solutions called the Angenent ovals in the mean curvature flows.

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.

Large Amplitude Radially Symmetric Spots and Gaps in a Dryland Ecosystem Model

We construct far-from-onset radially symmetric spot and gap solutions in a two-component dryland ecosystem model of vegetation pattern formation on flat terrain, using spatial dynamics and geometric singular perturbation theory. We draw connections between the geometry of the spot and gap solutions with that of traveling and stationary front solutions in the same model. In particular, we demonstrate the instability of spots of large radius by deriving an asymptotic relationship between a critical eigenvalue associated with the spot and a coefficient which encodes the sideband instability of a nearby stationary front. Furthermore, we demonstrate that spots are unstable to a range of perturbations of intermediate wavelength in the angular direction, provided the spot radius is not too small. Our results are accompanied by numerical simulations and spectral computations.

The logarithmic Bramson correction for Fisher-KPP equations on the lattice ℤ

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice Z \mathbb {Z} . The level sets of solutions with step-like initial conditions are located at position c ∗ t − 3 2 λ ∗ ln ⁡ t + O ( 1 ) c_*t-\frac {3}{2\lambda _*}\ln t+O(1) as t → + ∞ t\rightarrow +\infty for some explicit positive constants c ∗ c_* and λ ∗ \lambda _* . This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed c ∗ c_* uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times.

Propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats

This paper is concerned with the propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats. We first show that such an equation admits a single spreading speed in every direction under certain conditions and then give several spreading properties in terms of spreading speeds such as asymptotic spreading ray speeds and asymptotic spreading sets. Furthermore, we consider the dependence of the spreading speed on the dispersal rate and reaction term and prove that taking the temporal average or spatial average can decrease the spreading speed. Finally, we employ the viscosity vanishing method to establish the existence of time-space periodic traveling fronts with the critical speed in every direction under the partially temporally homogeneous case and partially nearly flat case, which solves partially the open problem raised by Rawal, Shen, and Zhang (Discrete Contin. Dyn. Syst.35 (2015) 1609–1640).

Conical traveling fronts for nonlocal bistable equations in [formula omitted

We investigate the existence and qualitative properties of conical traveling fronts for the bistable nonlocal equations. Based on the existence result of pyramidal traveling fronts (Li et al., 2021), we construct a sequence of pyramidal traveling fronts and take its limit. However, it is not easy to show that the limit function is a real conical traveling front, and the proofs of its qualitative properties are quite different from those of classical equations, although some of the ideas used are similar.

Stability of periodic traveling fronts for a time–space periodic dengue transmission model

In this paper, we consider the periodic traveling waves for a time–space periodic and degenerate reaction–diffusion dengue model. By establishing two comparison theorems and using comparison methods, we prove the globally exponential stability of time–space periodic traveling fronts. Our result is a continuation of Fang et al. (2020).

Pyramidal traveling fronts of a time periodic diffusion equation with degenerate monostable nonlinearity

This article focuses on the nonplanar traveling fronts of degenerate monostable time periodic reaction-diffusion equations in Rn with n≥3. By constructing a couple of proper supersolution and subsolution, we prove the existence of periodic pyramidal traveling front in R3 and then in Rn with n>3.

Nonplanar traveling fronts of the diffusion system with Belousov-Zhabotinskii reaction in [formula omitted

This paper studies nonplanar traveling fronts for a diffusion system with Belousov-Zhabotinskii (BZ for short) reaction in R3. First, we show the existence and stability results of pyramidal traveling fronts, and investigate their qualitative properties. By taking the limit of a pyramidal traveling front sequence, we obtain the conical traveling front and study its qualitative properties. Although both the pyramidal fronts and the conical fronts have larger traveling speeds than the planar front, their global mean speed equals that of the planar front in R3.

Time-periodic traveling waves for a periodic Lotka–Volterra competition system with nonlocal dispersal

This paper is concerned with the time-periodic traveling waves for a periodic Lotka–Volterra competition system with nonlocal dispersal. Under the strong competition assumption, we first prove the existence of periodic traveling fronts connecting two stable semi-trivial periodic solutions. Then we establish the monotonicity and global exponential stability of smooth periodic traveling fronts. Since the standard strong comparison theorem does not hold for the Lotka–Volterra competition system, the monotonicity and global exponential stability are proved by establishing some new strong comparison theorems and constructing some auxiliary monotone systems. The Liapunov stability of the periodic traveling fronts is also proved.

Thermal hysteresis and front propagation in dense planetary rings

ABSTRACT Saturn’s rings are composed of icy grains, most in the mm to m size ranges, undergoing several collisions per orbit. Their collective behaviour generates a remarkable array of structures over many orders of magnitude, much of it not well understood. On the other hand, the collisional properties and parameters of individual ring particles are poorly constrained; usually, N-body simulations and kinetic theory employ hard-sphere models with a coefficient of restitution ϵ that is constant or a decreasing function of impact speed. Due to the plastic deformation of surface regolith, however, it is likely that ϵ will be more complicated, at the very least a non-monotonic function. We undertake N-body simulations with the REBOUND code with non-monotonic ϵ laws to approximate surfaces that are friable but not sticking. Our simulations reveal that such ring models can support two thermally stable steady states for the same (dynamical) optical depth: a cold and a warm state. If the ring breaks up into radial bands of one or the other, we find that warmer states tend to migrate into the colder states via a coherent travelling front. We also find stationary ‘viscous’ fronts, which connect states of different optical depths, but the same angular momentum flux. We discuss these preliminary results and speculate on their implications for structure formation in Saturn’s B and C-rings, especially with respect to structures that appear in Cassini images but not in occultations.

Propagation Dynamics for Time-Periodic and Partially Degenerate Reaction-Diffusion Systems

This paper is concerned with the propagation dynamics for partially degenerate reaction-diffusion systems with monostable and time-periodic nonlinearity. In the cooperative case, we prove the existence of periodic traveling fronts and the exponential stability of continuous periodic traveling fronts. In the noncooperative case, we establish the existence of the minimal wave speed of periodic traveling waves and its coincidence with the spreading speed. More specifically, when the system is nondegenerate, the existence of periodic traveling waves is proved by using Schauder's fixed point theorem and the regularity of analytic semigroup, while in the partially degenerate case, due to the lack of compactness and standard parabolic estimates, the existence result is obtained by appealing to the asymptotic fixed point theorem with the help of some properties of the Kuratowski measure of noncompactness. It may be the first work to study periodic traveling waves of partially degenerate reaction-diffusion systems with noncooperative and time-periodic nonlinearity.

Propagation dynamics for a periodic delayed lattice differential equation without quasi-monotonicity

This paper is concerned with the propagation dynamics for a periodic delayed lattice differential equation with age structures. In the quasi-monotone case, the existence of the spreading speed and its coincidence with the minimal wave speed of periodic traveling fronts has been established in Wang and Weng (2014). In this paper, we consider the spatial dynamics of the equation without quasi-monotonicity. We first establish the existence of the spreading speed c∗ and the non-existence of the periodic traveling waves with speed c∈(0,c∗). Then, we prove the existence of the periodic traveling waves by using the asymptotic fixed point theorem. Our result indicates that the spreading speed still equals to the minimal wave speed in the non-quasi-monotone case. It may be the first work to study the spatial dynamics of the time-periodic and delayed lattice differential systems without quasi-monotonicity.

Propagation phenomena in a diffusion system with the Belousov-Zhabotinskii chemical reaction

This paper is concerned with the propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction in [Formula: see text] under the bistability assumption. We prove that there is a new type of entire solution originated from three moving planar traveling fronts, and evolved to a V-shaped traveling front as time changes, which means that the profile of this solution is not invariant at all. Here an entire solution is referred to a solution that is defined for all time [Formula: see text] and in the whole space [Formula: see text]. Furthermore, we show that not only the entire solution but also all transition fronts share the same global mean speed by constructing suitable radially symmetric expanding and retracting super- and subsolutions.

Traveling Fronts in a Reaction–Diffusion Equation with a Memory Term

Based on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.

On the existence of cylindrically symmetric traveling fronts of fractional Allen-Cahn equation in $\mathbb{R}^{3}$

This paper is concerned with the existence of cylindrically symmetric traveling fronts for bistable fractional Allen-Cahn equation in $\mathbb{R}^{3}$. By constructing a sequence of pyramidal traveling fronts and taking its limit, we obtain a traveling front with cylindrically symmetric type.

Wavelike propagation of quorum activation through a spatially distributed bacterial population under natural regulation

Many bacteria communicate using diffusible pheromone signals known as autoinducers. When the autoinducer concentration reaches a threshold, which requires a minimum population density or ‘quorum’, the bacteria activate specific gene regulatory pathways. Simple diffusion of autoinducer can activate quorum-dependent pathways in cells that are located at substantial distances from the secreting source. However, modeling has predicted that autoinducer diffusion, coupled with positive feedback regulation in autoinducer synthesis, could also allow a quorum-regulated behavior to spread more rapidly through a population by moving as a self-sustaining front at constant speed. Here we show that such propagation can occur in a population of bacteria whose quorum pathway operates under its own natural regulation. We find that in unstirred populations of Vibrio fischeri, introduction of autoinducer at one location triggers a wavelike traveling front of natural bioluminescence. The front moves with a well-defined speed ∼2.5 mm h−1, eventually outrunning the slower diffusional spreading of the initial stimulus. Consistent with predictions from modeling, the wave travels until late in growth, when population-wide activation occurs due to basal autoinducer production. Subsequent rounds of waves, including waves propagating in the reverse direction, can also be observed late in the growth of V. fischeri under natural regulation. Using an engineered, lac-dependent strain, we show that local stimuli other than autoinducers can also elicit a self-sustaining, propagating response. Our data show that the wavelike dynamics predicted by simple mathematical models of quorum signaling are readily detected in bacterial populations functioning under their own natural regulation, and that other, more complex traveling phenomena are also present. Because a traveling wave can substantially increase the efficiency of intercellular communication over macroscopic distances, our data indicate that very efficient modes of communication over distance are available to unmixed populations of V. fischeri and other microbes.

Global stability of noncritical traveling front solutions of Fisher-type equations with degenerate nonlinearity

In this paper, for degenerate n-degree Fisher-type equations, we discuss the stability of their traveling front solutions with noncritical speeds. In fact, when the initial perturbations around these noncritical traveling front solutions are in some weighted Banach spaces, we have proved that these solutions are globally exponentially stable in the form of (1+t)13e−νt for ν ∈ (0, 1) via L1-energy estimates, L2-energy estimates, and the weighted energy method. Furthermore, by Fourier transform and the weighted energy method, we will prove that traveling front solutions with noncritical speeds are also globally exponentially stable in the form of t−12e−νt for some positive constant ν when the initial perturbations around these solutions are in some weighted Sobolev spaces. Our conclusions extend the local stability of noncritical traveling front solutions into the global case and also give some novel forms of exponential stability of these solutions.

Entire solutions of time periodic bistable Lotka–Volterra competition-diffusion systems in $${\mathbb {R}}^N$$

This paper is concerned with entire solutions of a two species time periodic bistable Lotka–Volterra competition-diffusion systems in $${\mathbb {R}}^N$$ . Here an entire solution refers to a solution that is defined for all time and in the whole space. It is known that “annihilating-front” type entire solutions have been obtained recently in $${\mathbb {R}}$$ . In the present paper, we first prove that there is a new type of entire solution which behaves as three time periodic moving planar traveling fronts as time goes to $$-\infty $$ and as a time periodic V-shaped traveling front as time goes to $$+\infty $$ in $${\mathbb {R}}^N$$ . Furthermore, we show that the propagating speed of such entire solutions coincides with the unique speed of the time periodic planar front, regardless of the shape of the level sets of the fronts.