Asymptotic speed of spread and traveling fronts for a nonlocal reaction–diffusion model with distributed delay

In this paper, a nonlocal reaction–diffusion model with distributed delay is studied. The asymptotic speed of spread is established for this model, and its coincidence with the minimal wave speed for traveling wave fronts is proved. Moreover, the dependence of the asymptotic speed of spread on delay and the nonlocal effect is considered. Our main finding here is that the delay can induce slow asymptotic speed of spread while the nonlocal effect can increase fast asymptotic speed of spread.     

Existence and qualitative features of entire solutions for delayed reaction diffusion system: the monostable case

The paper is concerned with the existence and qualitative features of entire solutions for delayed reaction diffusion monostable systems. Here the entire solutions mean solutions defined on the $ (x,t)\in\mathbb{R}^{N+1} $. We first establish the comparison principles, construct appropriate upper and lower solutions and some upper estimates for the systems with quasimonotone nonlinearities. Then, some new types of entire solutions are constructed by mixing any finite number of traveling wave fronts with different speeds $ c\geq c_* $ and propagation directions and a spatially independent solution, where $c_*>0$ is the critical wave speed. Furthermore, various qualitative properties of entire solutions are investigated. In particularly, the relationship between the entire solution, the traveling wave fronts and a spatially independent solution are considered, respectively. At last, for the nonquasimonotone nonlinearity case, some new types of entire solutions are also investigated by introducing two auxiliary quasimonotone controlled systems and establishing some comparison theorems for Cauchy problems of the three systems.     

Effect of stochastic perturbations for front propagation in Kolmogorov Petrovskii Piscunov equations

This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation:

where the stochastic differential is taken in the sense of Itô and $\zeta$ is a Gaussian random field satisfying $\mathbb{E}\left[\zeta \right]=0$ and $\mathbb{E}\left[\zeta (s,x)\zeta (t,y)\right]=(s\wedge t)\Gamma (x?y)$. Two situations are considered: firstly, $\zeta$ is simply a standard Wiener process (i.e. $\Gamma \equiv 1$): secondly, $\Gamma \in C^{\infty }\left(\mathbb{R}\right)$ with ${lim}_{\left|z\right|\to +\infty }\left|\Gamma \left(z\right)\right|=0$.The results are as follows: in the first situation (standard Wiener process: i.e. $\Gamma \left(x\right)\equiv 1$), there is a non-degenerate travelling wave front if and only if $\frac{{?}^2}{2}<1$, with asymptotic wave speed $max\left(\sqrt{2\kappa (1?\frac{{?}^2}{2})},\left(\frac{1}{N}(1?\frac{{?}^2}{2})+\frac{\kappa N}{2}\right){\mathbf{1}}_{\{N<\sqrt{\frac{2}{\kappa }(1?\frac{{?}^2}{2})}\}}\right)$; the noise slows the wave speed. If the stochastic integral is taken instead in the sense of Stratonovich, then the asymptotic wave speed is the classical McKean wave speed and does not depend on $?$.In the second situation (noise with spatial covariance which decays to 0 at $\pm \infty$, stochastic integral taken in the sense of Itô), a travelling front can be defined for all $?>0$. Its average asymptotic speed does not depend on $?$ and is the classical wave speed of the unperturbed KPP equation.     

Evans functions and bifurcations of standing wave fronts of a nonlinear system of reaction diffusion equations

Consider the following nonlinear system of reaction diffusion equations arising from mathematical neuroscience $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u]-w,~ \frac{\partial w}{\partial t}=\varepsilon(u-\gamma w).$ Also consider the nonlinear scalar reaction diffusion equation $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u].$ In these model equations, $\alpha>0$, $\beta>0$, $\gamma>0$, $\varepsilon>0$ and $\theta>0$ are positive constants, such that $0<2\theta<\beta$. In the model equations, $u=u(x,t)$ represents the membrane potential of a neuron at position $x$ and time $t$, $w=w(x,t)$ represents the leaking current, a slow process that controls the excitation.\\indent The main purpose of this paper is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing wave fronts) and Evans functions (complex analytic functions) to establish the existence, stability, instability and bifurcations of standing wave fronts of the nonlinear system of reaction diffusion equations and to establish the existence and stability of the standing wave fronts of the nonlinear scalar reaction diffusion equation.     

Persistence of travelling fronts of KdV-Burgers-Kuramoto equation

We study the travelling fronts of KdV-Burgers-Kuramoto equation from geometric singular perturbation point of view. Motivated by the analogue between travelling fronts and heteroclinic orbits of the corresponding ordinary differential equations, we prove the persistence of these waves for sufficiently small dissipation. The result of numerical investigation also establishes our analysis.     

Optimization of the wave front of high order harmonics

We present a full optimization of the high harmonics wave front owing to the use of a soft X-ray Hartmann sensor. The sensor was calibrated using a high harmonic source with an accuracy of λ/50 root-mean-square (rms) with λ around 30 nm. We observed a high harmonic wave front of λ/7 rms, which is two times the diffraction-limit, astigmatism being the dominant aberration for every condition of generation. By clipping slightly the unfocused high harmonic beam, it is possible to produce a diffraction-limited beam containing approximately 90% of the incident energy.     

On the holographic simulation of non-circular zone plates

New methods involving the manipulation of fundamental wavefronts (e.g., plane and spherical) with simple optical components such as pinholes and spherical lenses have been developed for the fabrication of elliptic, hyperbolic and conical holographic zone plates. Also parabolic zone plates by holographic techniques have been obtained for the first time. The performance behaviour of these zone plates has been studied. Further a phenomenological explanation is offered for the observed improved fringe contrast obtained with a spherical reference wave. An erratum to this article is available at .     

Turbulent bore in a supercritical flow over an irregular bed

Breaking waves in a free-surface homogeneous fluid flow in the neighborhood of a local variation in the channel depth are studied experimentally and theoretically. The structure of both a steady-state hydraulic jump generated by a local obstacle in the channel and an unsteady wave configuration consisting of two turbulent bores in the problem of lock failure is studied. Using the turbulent bore model [1], analytic profiles of breaking waves are obtained and the time-dependent problem is numerically investigated and compared with experimental data. It is shown that the model [1] with a hydrostatic pressure distribution over the depth adequately describes both the location and the structure of the steady-state and unsteady wave fronts.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 62–70. Original Russian Text Copyright © 2005 by Gusev and Lyapidevskii.     

Evaporation and Condensation Fronts in Porous Media

Flow of a fluid through a porous medium is considered with allowance for heat conduction. Both fronts at which the liquid is transformed into steam or a liquid-steam mixture and fronts with inverse transformations are studied. The evolutionarity conditions of these fronts are considered and a model of their structure is proposed.     

The true molecular weight distributions of acrylate polymers formed in propagating fronts

We have analyzed fractionated samples of poly(methacrylic acid) produced in a propagating front for the amount of anhydride that formed and determined that a large percentage of acid groups exist as anhydrides (>20%). By analyzing the samples after cleavage, we found that the molecular weight dropped significantly (from M_n = 1.4 × 10⁵ to M_n = 1.0 × 10⁴). We conclude that the high molecular weights observed previously were the result of intermolecular anhydride formation. Poly(butyl acrylate), which cannot form anhydride bonds, produced in fronts had broad (M_w/M_n = 1.7–2.0) but unimodal molecular weight distributions with M_u < 10⁵. The average molecular weight decreased with increasing initiator concentrations. © 1996 John Wiley & Sons, Inc.