一类时滞格种群模型波前解的存在性

This paper considers the travelling wave fronts to a delayed lattice differential equation.The existence of the travelling wave solutions is proved by making use of the technique of the upper and lower solutions developed by J Wu and X Zou in[6].This work extends that of [4]in a general class of nonlinear terms.     

Travelling Wave Solutions in Delayed Reaction Diffusion Systems with Partial Monotonicity

This paper deals with the existence of travelling wave fronts of delayed reaction diffusion systems with partial quasi-monotonicity. We propose a concept of ＂desirable pair of upper-lower solutions＂, through which a subset can be constructed. We then apply the Schauder＇s fixed point theorem to some appropriate operator in this subset to obtain the existence of the travelling wave fronts.     

Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications

This paper is concerned with the travelling wave fronts of nonlocal reaction-diffusion systems with delays. The existence of travelling wave fronts for nonlocal reaction-diffusion systems with delays is established by using Schauder’s fixed point theorem and upper-lower solution technique. Then these results are applied to the nonlocal delayed Logistic model and the delayed Belousov-Zhabotinskii reaction-diffusion system. Our results show that the time delay can reduce the minimal wave speed while the nonlocality can increase the minimal wave speed. Wan-Tong Li: Supported by NNSF of China (10571078) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.     

Traveling wave fronts for generalized Fisher equations with spatio-temporal delays

We study the existence of traveling wave fronts for a reaction-diffusion equation with spatio-temporal delays and small parameters. The equation reduces to a generalized Fisher equation if small parameters are zero. We present two results. In the first one, we deal with the equation with very general kernels and show the persistence of Fisher wave fronts for all sufficiently small parameters. In the second one, we deal with some particular kernels, with which the nonlocal equation can be reduced to a system of singularly perturbed ODEs, and we are then able to apply the geometric singular perturbation theory and phase plane arguments to this system to show the existence of the minimal wave speed, the existence of a continuum of wave fronts, and the global uniqueness of the physical wave front with each wave speed.     

非线性发展方程行波解的符号计算

Abstract. A Riccati equation involving a parameter and symbolic computation are used to uni-formly construct the different forms of travelling wave solutions for nonlinear evolution equa-tions. It is shown that the sign of the parameter can be applied in judging the existence of vari-ous forms of travelling wave solutions. An efficiency of this method is demonstrated on some e-quations,which include Burgers-Huxley equation,Caudrey-Dodd-Gibbon-Kawada equation,gen-eralized Benjamin-Bona-Mahony equation and generalized Fisher equation.     

一类Fisher方程的行波解

本文讨论了一类广义Fisher方程,得到了它的多个显式行波解.     

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.     

Characterizing and minimizing the operator split error for Fisher’s equation

Operator splitting (OS) is a popular and convenient technique used to numerically solve reactive transport problems such as Fisher’s equation. Although OS has been widely used to solve Fisher’s equation, no characterization of the innate OS error has been presented. Here the exact characteristics of the OS error for travelling wave solutions of Fisher’s equation are revealed and explored. The analysis shows that the OS error behaves differently to previously studied linear problems by smoothing or steepening the wave front depending on the sequential order of splitting. Further analysis confirms that the OS error is reduced by implementing an alternating OS scheme.     

关于指数率Logistic方程在无限柱体内的波前解的存在性

与一维空间中研究连接两个常数的波前解的存在性不同的是,本文建立了在多维无限长的柱体内连接两个曲面的单调行波解的存在性.相应的模型是一种具有指数率的Logistic方程.所用的方法是一种改进了的单调性方法.本文的研究结果对自然界中波的实际传播行为给出了有益的启示.     

Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays

This paper deals with the existence of travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. Our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system. New iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions. Following this, the existence of travelling wave fronts for reaction-diffusion systems with spatio-temporal delays is established. Finally, we apply the main results to a single-species diffusive model with spatio-temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels.     

Fisher方程和Burgers-Fisher方程新的精确行波解

本文研究了Fisher方程和Burgers-Fisher方程.运用一种辅助微分方程方法,得到了这两种非线性偏微分方程新的精确行波解.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.     

A modified tanh–coth method for solving the general Burgers–Fisher and the Kuramoto–Sivashinsky equations

In this work we use a modified tanh–coth method to solve the general Burgers–Fisher and the Kuramoto–Sivashinsky equations. The main idea is to take full advantage of the Riccati equation that the tanh-function satisfies. New multiple travelling wave solutions are obtained for the general Burgers–Fisher and the Kuramoto–Sivashinsky equations.     

New travelling wave solutions for the Fisher–KPP equation with general exponents

The Fisher–KPP equation with general nonlinear diffusion and arbitrary kinetic orders in the reaction terms is considered. The existence of oscillatory travelling wave solutions is proved for this model. Conditions for the existence of such solutions are provided.     

一类退化-奇异抛物型方程的行波解Ⅰ

多孔介质动力学方程吸引了众多人的注意,引起了数学家们的极大兴趣.退化-奇异抛物型方程的行波解也成为令人瞩目的问题.Aronson,Hosono,Atkinson,Engler,Grindron 和 Sleeman,Wang 和 Ye 等在这一领域中都有出色的工作.Aronson 对 m=k=1,u~nf(u)∈C~1[0,1],讨论了单调行波解的存在性.Hosono 对m=1,k≥2,n=0,f∈C~2[0,1],f′(0)<0,f″(0)(?)0,f′(1)<0,且存在α∈     

Entire solution of the degenerate Fisher equation

This paper deals with the solutions defined for all time of the degenerate Fisher equation. Some solutions are obtained by considering two traveling fronts with critical speed that come from both sides of the X-axis and mix. Unfortunately, the entire solutions which behave as two opposite wave fronts with non-critical speed approaching each other from both sides of the X-axis can not be obtained, because the essential difficulty originates from the algebraic decay rate of the fronts with non-critical speed.     

Tropical discretization: ultradiscrete Fisher–KPP equation and ultradiscrete Allen–Cahn equation

A systematic approach to the construction of ultradiscrete analogues for differential systems is presented. This method is tailored to first-order differential equations and reaction–diffusion systems. The discretizing method is applied to Fisher–KPP equation and Allen–Cahn equation. Stationary solutions, travelling wave solutions and entire solutions of the resulting ultradiscrete systems are constructed.     

LOCAL STABILITY OF TRAVELLING FRONTS FOR A DAMPED WAVE EQUATION

The paper is concerned with the long-time behaviour of the travelling fronts of the damped wave equation αutt+ut=uxx V’(u) on R.The long-time asymptotics of the solutions of this equation are quite similar to those of the corresponding reaction-diffusion equation ut=uxxV’(u).Whereas a lot is known about the local stability of travelling fronts in parabolic systems,for the hyperbolic equations it is only briefly discussed when the potential V is of bistable type.However,for the combustion or monostable type of V,the problem is much more complicated.In this paper,a local stability result for travelling fronts of this equation with combustion type of nonlinearity is established.And then,the result is extended to the damped wave equation with a case of monostable pushed front.     

On the Structure of Spectra of Modulated Travelling Waves

Modulated travelling waves are solutions to reaction-diffusion equations that are time-periodic in an appropriate moving coordinate frame. They may arise through Hopf bifurcations or essential instabilities from pulses o fronts. In this article, a framework for the stability analysis of such solutions is presented: the reaction-diffusion equation is cast as an ill-posed elliptic dynamical system in the spatial variable acting upon time-periodic functions. Using this formulation, points in the esolvent set, the point spectrum, and the essential spectrum of the linearization about a modulated travelling wave are related to the existence of exponential dichotomies on appopriate intervals for the associated spatial elliptic eigenvalue problem. Fredholm properties of the linearized operator are characterized by a relative Morse-Floe index of the elliptic equation. These results are proved without assumptions on the asymptotic shape of the wave. Analogous results are true for the spectra of travelling waves to parabolic equations on unbounded cylinders. As an application, we study the existence and stability of modulated spatially-periodic patterns with long-wavelength that accompany modulated pulses.     

Modified extended tanh-function method for solving nonlinear partial differential equations

Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers–Fisher equation and Drinfeld–Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.