具有非局部反应的时滞扩散Nicholson方程的行波解

对具有扩散项的时滞Nicholson方程的行波解进行了研究．特别是考虑到生物个体在空间位置上的迁移,研究了具有非局部反应的时滞扩散模型．对于弱生成时滞核,运用几何奇异摄动理论,在时滞充分小的情况下,证明了行波解的存在性．     

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c* > 0, and prove the existence of waves when c ≥ c* and the nonexistence when 0 ≤ c < c*.     

行波的稳定性

文［１］引入一种扩充系统用来计算带Ｏ（２）对称性的非线性常微分方程的行波（或回转波）．本文证明，这些行波的稳定性由扩充系统的特征值来决定，其机理与通常的静态解的稳定性的判别相同．     

Finite Travelling Waves for a Semilinear Degenerate Reaction-Diffusion System

In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system is studied, where 0 < α _i < 1, m _ij ≥ 0 and ∑ ^N _{j =1} m _ij > 0, i, j = 1,...,N. Necessary and sufficient conditions on existence and large time behaviours of FTWs of (I) are obtained by using the matrix theory, Schauder's fixed point theorem, and upper and lower solutions method. Received July 3, 1997, Revised June 23, 1999, Accepted March 29, 2000     

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.     

一类半线性退化反应扩散方程组的有限行波解

本文考虑一类半线性退化反应扩散方程组有限行波解的存在性与不存在性,以及有限行波解存在的充要条件与大时间行为被获得.     

Travelling Waves and Numerical Approximations in a Reaction Advection Diffusion Equation with Nonlocal Delayed Effects

Summary. In this paper, we consider the growth dynamics of a single-species population with two age classes and a fixed maturation period living in a spatial transport field. A Reaction Advection Diffusion Equation (RADE) model with time delay and nonlocal effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling waves for the delay model with three birth functions which appeared in the well-known Nicholson's blowflies equation, and we consider and analyze numerical solutions of the travelling wavefronts from the wave equations for the problems with nonlocal temporally delayed effects. In particular, we report our numerical observations about the change of the monotonicity and the possible occurrence of multihump waves. The stability of the travelling wavefront is numerically considered by computing the full time-dependent partial differential equations with nonlocal delay.     

Travelling waves for reaction-diffusion equations with time depending nonlinearities

The existence of travelling wave with given end points for parabolic system of nonlinear equations is proven. The nonlinear term depends also on a·x−ct where x is the multidimensional space variable, t—time, c—the speed of the wave and a—the direction of travel.     

Travelling wave fronts in a vector disease model with delay

In this paper, we study the diffusive vector disease model with delay. This problem with strong biological background has attracted much research attention. We focus on the existence of traveling wave fronts, and find that there is a moving zone for the transition from the disease-free state to the infective state. To complete the theoretical analysis, we employ the mathematical tools including the monotone iteration technique as well as the upper and lower solution method.     

Watershed initial conditions for propagating waves in a system with bistability: A weight-function approach

We consider non-adiabatic combustion waves arising from a two-step exothermic system. Our previous work showed that in certain parameter regions, the combustion wave can evolve to the “fast” solution branch, the “slow” solution branch or diffuse to the ambient temperature (extinction wave). Here, we are interested to find critical initial temperature profiles which evolve to these three types of steady solutions. For a particular family of temperature profiles, we construct a weight function which can be used to predict which of these three types of waves an initial temperature profile will evolve to.     

时滞合作扩散系统的波前解

本文建立了两种群时滞合作系统波前解的存在性定理，扩展了单种群生物模型的结论．     

Bifurcation of equilibrium forms of a gas column rotating with constant speed around its axis of symmetry

We will be concerned with the problem of deformation of the lateral surface of a column that rotates with constant speed around its axis of symmetry. The column is filled by a gas and our goal is to investigate the deformation of the lateral surface depending on the pressure of the gas.     

Travelling waves for a biological reaction diffusion model with spatio-temporal delay

In this paper, we consider the reaction diffusion equations with spatio-temporal delay, which models the microbial growth in a flow reactor. Nonlocal spatial term, a weighted average in space, arises when the individuals have not necessarily been at the same point in space at previous time. By employing linear chain technique, geometric singular perturbation, and the center manifold theorem, we prove that the steady travelling wave does not only persist, but also it looks qualitatively the same as it do with no delay at all, under the introduction of delays, at least for small delay.     

A normalized solitary wave solution of the Maxwell-Dirac equations

We prove the existence of a $L^2$-normalized solitary wave solution for the Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, for the Coulomb-Dirac model, describing fermions with attractive Coulomb interactions in the mean-field limit, we prove the existence of the (positive) energy minimizer.     

Modelling Axi-symmetric Travelling Waves in a Dielectric with Nonlinear Refractive Index

We consider an isotropic dielectric with a nonlinear refractive index. The medium may be inhomogeneous but its spatial variation has an axial symmetry. We characterize all monochromatic axi-symmetric travelling waves as solution of a system of six second order differential equations on (0, ). Boundary conditions at 0 ensure the regularity of the fields on the axis. Guided waves satisfy additional conditions at . Special solutions of this system correspond to what are normally referred to as TE and TM modes.Lecture held in the Seminario Matematico e Fisico on June 13, 2003Received: February, 2004     

Modulational Stability of Travelling Waves in 2D Anisotropic Systems

The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion system and to the weak electrolyte model for electroconvection in nematic liquid crystals.      

Travelling wave solutions of some classes of nonlinear evolution equations in (1+1) and (2+1) dimensions

The tanh method is proposed to find travelling wave solutions in (1+1) and (2+1) dimensional wave equations. It can be extended to solve a whole family of modified Korteweg–de Vries type of equations, higher dimensional wave equations and nonlinear evolution equations.     

Computation of Travelling Combustion Waves in a Porous Medium

Numerical solutions for travelling combustion waves in a porous medium are sought. The algorithm of computation is based on a shooting method used in an existence proof. The numerical result suggests that there is a limit for the inlet gas velocity below which no travelling wave solution can be constructed.     

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.