Wave Front Solutions Research Articles

Comparison theorems and variable speed waves for a scalar reaction–diffusion equation

This paper concerns the reaction-diffusion equation ut = uxx + u2(1 − u). Previous numerical solutions of this equation have demonstrated various different types of wave front solutions, generated by different initial conditions. In this paper, the authors use a phase-plane form of comparison theorems for partial differential equations (PDEs) to confirm analytically these numerical results. In particular, they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave, while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave, or to the exponentially decaying wave, or to a perpetually accelerating wave, dependent upon the exact form of the decay of the initial conditions. We then focus on the case of accelerating waves and investigate their form in more detail, by approximating the full equation in this case with a hyperbolic PDE, which we solve using the method of characteristics. We use this approximate solution to derive a leading-order approximation to the wave speed.

Traveling waves for the diffusive Nicholson's blowflies equation

We consider traveling wave front solutions for the diffusive Nicholson's blowflies equation on the real line. The existence of such solutions is proved using the technique developed by J. Wu and X. Zou (J. Dyn. Differ. Equations 13 (3) (2001)). Some numerical simulation using the iteration formula of Wu and Zou [7] is also provided.

Speed of reaction-transport processes.

We present an approach to determining the speed of wave-front solutions to reaction-transport processes. This method is more accurate than previous ones. This is explicitly shown for several cases of practical interest: (i) the anomalous diffusion reaction, (ii) reaction diffusion in an advective field, and (iii) time-delayed reaction diffusion. There is good agreement with the results of numerical simulations.

Critical wave speeds for a family of scalar reaction-diffusion equations

We study the set of traveling waves in a class of reaction-diffusion equations for the family of potentials f m ( U) = 2 U m (1 − U). We use perturbation methods and matched asymptotics to derive expansions for the critical wave speed that separates algebraic and exponential traveling wave front solutions for m → 2 and m → ∞. Also, an integral formulation of the problem shows that nonuniform convergence of the generalized equal area rule occurs at the critical wave speed.

Traveling Wave Fronts of Reaction-Diffusion Systems with Delay

This paper deals with the existence of traveling wave front solutions of reaction-diffusion systems with delay. A monotone iteration scheme is established for the corresponding wave system. If the reaction term satisfies the so-called quasimonotonicity condition, it is shown that the iteration converges to a solution of the wave system, provided that the initial function for the iteration is chosen to be an upper solution and is from the profile set. For systems with certain nonquasimonotone reaction terms, a convergence result is also obtained by further restricting the initial functions of the iteration and using a non-standard ordering of the profile set. Applications are made to the delayed Fishery–KPP equation with a nonmonotone delayed reaction term and to the delayed system of the Belousov–Zhabotinskii reaction model.

Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations

Linear diffusion is an established model for spatial spread in biological systems, including movement of cell populations. However, for interacting, closely packed cell populations, simple diffusion is inappropriate, because different cell populations will not move through one another: rather, a cell will stop moving when it encounters another cell. In this paper, I introduce a nonlinear diffusion term that reflects this phenomenon, known as contact inhibition of migration. I study this term in the context of two competing cell populations, one of which has a proliferative advantage over the other; this is motivated by the very early stages of solid tumour growth. I focus in particular on travelling–wave solutions, corresponding to moving interfaces between the two cell populations. Numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value, and I present linear analysis that explains the selection of wave speeds by initial conditions. I obtain an approximation to the shape of these waves for high speeds, and show that the minimum speed arises via quite new behaviour in the travelling–wave equations, with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum. Exploiting this structure, I use singular perturbation theory to investigate the wave shape for speeds close to the minimum.

Travelling front solutions of a nonlocal Fisher equation.

We consider a scalar reaction-diffusion equation containing a nonlocal term (an integral convolution in space) of which Fisher's equation is a particular case. We consider travelling wavefront solutions connecting the two uniform states of the equation. We show that if the nonlocality is sufficiently weak in a certain sense then such travelling fronts exist. We also construct expressions for the front and its evolution from initial data, showing that the main difference between our front and that of Fisher's equation is that for sufficiently strong nonlocality our front is non-monotone and has a very prominent hump.

Existence and construction of travelling wavefront solutions of Fisher equations with fourth-order perturbations

We consider a generalized Fisher equation containing a fourth-order spatial derivative term. Of interest is the question of the existence of travelling front solutions of the equation and their qualitative form. When the fourth-order terms has a sufficiently small coefficient existence of such fronts is shown using invariant manifold theory. Asymptotic analysis is employed to construct an expression for the travelling front when its speed is large.

Curvature dynamics and stability of topological solitons in the optical parametric oscillator

The first analytic verification of the stability of topological solitons corresponding to wave-front solutions of the optical parametric oscillator is given. A translational invariance allows perturbations to the system to shift the front position without affecting the underlying exponential stability of the fronts themselves. For the two-dimensional problem with a positive O(1) signal detuning, the front curvature is shown to be governed by a heat equation, so that the only stable topological solitons supported must be stripes.

Role of fibroblast migration in collagen fiber formation during fetal and adult dermal wound healing.

Adult dermal wounds, in contrast to fetal wounds, heal with the formation of scar tissue. A crucial factor in determining the degree of scarring is the ratio of types I and III collagen, which regulates the diameter of the combined fibers. We developed a reaction-diffusion model which focuses on the control of collagen synthesis by different isoforms of the polypeptide transforming growth factor-beta (TGF beta). We used the model to investigate the current controversy as to whether the fibroblasts migrate into the wound from the surrounding unwounded dermis or from the underlying subcutaneous tissue. Numerical simulations of a spatially independent, temporal model led to a value of the collagen ratio consistent with that of healthy tissue for the fetus, but corresponding to scarring in the adult. We investigated the effect of topical application of TGF beta and show that addition of isoform 3 reduces scar tissue formation, in agreement with the experiment. However, numerical solutions of the reaction-diffusion system do not exhibit this sensitivity to growth factor application. Mathematically, this corresponds to the observation that behind healing wave-front solutions, a particular healed state is always selected independent of transients, even though there is a continuum of possible positive steady states. We explain this phenomenon using a caricature system of equations, which reflects the key qualitative features of the full model but has a much simpler mathematical form. Biologically, our results suggest that the migration into a wound of fibroblasts and TGF beta from the surrounding dermis alone cannot account for the essential features of the healing process, and that fibroblasts entering from the underlying subcutaneous tissue are crucial to the healing process.

Role of fibroblast migration in collagen fiber formation during fetal and adult dermal wound healing

Adult dermal wounds, in contrast to fetal wounds, heal with the formation of scar tissue. A crucial factor in determining the degree of scarring is the ratio of types I and III collagen, which regulates the diameter of the combined fibers. We developed a reaction-diffusion model which focuses on the control of collagen synthesis by different isoforms of the polypeptide transforming growth factor-β (TGFβ). We used the model to investigate the current controversy as to whether the fibroblasts migrate into the wound from the surrounding unwounded dermis or from the underlying subcutaneous tissue. Numerical simulations of a spatially independent, temporal model led to a value of the collagen ratio consistent with that of healthy tissue for the fetus, but corresponding to scarring in the adult. We investigated the effect of topical application of TGFβ and show that addition of isoform 3 reduces scar tissue formation, in agreement with the experiment. However, numerical solutions of the reaction-diffusion system do not exhibit this sensitivity to growth factor application. Mathematically, this corresponds to the observation that behind healing wavefront solutions, a particular healed state is always selected independent of transients, even though there is a continuum of possible positive steady states. We explain this phenomenon using a caricature system of equations, which reflects the key qualitative features of the full model but has a much simpler mathematical form. Biologically, our results suggest that the migration into a wound of fibroblasts and TGFβ from the surrounding dermis alone cannot account for the essential features of the healing process, and that fibroblasts entering from the underlying subcutaneous tissue are crucial to the healing process.

Dynamics and thermodynamics of delayed population growth

Grup de Fi ´sica Estadistica, Departament de Fisica, Universitat Autonoma de Barcelona, E-08193 Bellaterra, Spain~Received 31 January 1997!The dynamic and thermodynamic properties of delayed nonlinear reaction-diffusion equations describingpopulation growth with memory are analyzed. In the dynamic study we first apply the speed selection mecha-nisms for wave fronts connecting two steady states obtaining, on one hand, a decrease in the lower boundspeed, and also an upper bound velocity; we also calculate an exact wave front solution. In the thermodynamicstudy, we show an agreement between the stochastic description and extended irreversible thermodynamics inthe presence of a source of particles. @S1063-651X~97!10205-7#PACS number~s!: 05.70.2a, 05.40.1jI. INTRODUCTION

Thermal-activation effects in reaction-diffusion models: A two-component bistable system. I.

We study a two-component bistable reaction-diffusion system arising in a model for a simple reversible chemical reaction subject to the effect of thermal activation and energy production and dissipation. Some mathematical simplications enables the analytical resolution of the corresponding reaction-diffusion equation set in a variety of situations. As a first step, we analyze the evolution of the system in spatially homogeneous conditions. This makes it possible to detect the parameter regions where bistability occurs, identifying the physical conditions under which two stable states exist. Then, we study shape-preserving wave-front solutions. These fronts should stand for the motion of domain walls, formed during the self-organizing stage of the evolution.

Existence of wave front solutions and estimates of wave speed for a competition-diffusion system

Existence of wave front solutions and estimates of wave speed for a competition-diffusion system

Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Wavefront solutions of scalar reaction-diffusion equations have been intensively studied for many years. There are two basic cases, typified by quadratic and cubic kinetics. An intermediate case is considered in this paper, namely, u l = u xx + u 2 (1 − u). It is shown that there is a unique travelling-wave solution, with a speed 1/√2, for which the decay to zero ahead of the wave is exponential with x. Moreover, numerical evidence is presented which suggests that initial conditions with such exponential decay evolve to this travelling-wave solution, independently of the half-life of the initial decay. It is then shown that for all speeds greater than 1/√2 there is also a travelling-wave solution, but that these faster waves decay to zero algebraically, in proportion to 1/x. The numerical evidence suggests that initial conditions with such a decay rate evolve to one of these faster waves; the particular speed depends in a simple way on the details of the initial condition. Finally, initial conditions decaying algebraically for a power law other than 1/x are considered. It is shown numerically that such initial conditions evolve either to an algebraically decaying travelling wave, or in some cases to a wavefront whose shape and speed vary as a function of time. This variation is monotonic and can be quite pronounced, and the speed is a function of u as well as of time. Using a simple linearization argument, an approximate formula is derived for the wave speed which compares extremely well with the numerical results. Finally, the extension of the results to the more general case of u l = u xx + u m (1 − u), with m > 1, is discussed.

Wave fronts in a bistable reaction-diffusion system with density-dependent diffusivity

We obtain wave-front solutions for a one-dimensional bistable reaction-diffusion model with density-dependent diffusivity. These solutions — which are expected to stand for the asymptotic behaviour of a wide class of initial conditions — should describe the evolution of the walls of constant density domains, spontaneously formed in this system. The piecewise linearized form of the reaction terms and of the diffusivity makes it possible to obtain analytical results for a situation of interest in many real applications — namely, a diffusivity that changes abruptly at a critical value of the density. We pay particular attention to the dependence of the wave-front velocity on the relevant parameters, and are able to outline some physical arguments that explain its features.

Exact wave front solutions to two generalized coupled nonlinear physical equations

We find the analytical wave front solutions to two coupled physical models by presenting various ansatze for the two unknowns in the equations of interest.

Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Exact solutions for Belousov-Zhabotinskii reaction-diffusion system

We consider the following simplified model for the Belousov-Zhabotinskii (B-Z) reaction:\(\left\{ {\mathop {}\limits_{\frac{{\partial u}}{{\partial t}} = - suv + d\frac{{\partial ^2 u}}{{\partial x^2 }}, }^{\frac{{\partial u}}{{\partial t}} = - u(1 - u - rv) + d\frac{{\partial ^2 u}}{{\partial x^2 }},} } \right.\) wherer ands are positive parameters, andd is the diffusing constant for the concentrationu. Seeking travelling wave front solution and making an ansatz for the solution, we obtain a nonlinear system of algebraic equations. The system is solved using Wu Elimination and then we are able to find several exact solutions which are of interest.

Evolution of reaction-diffusion patterns in infinite and bounded domains

Abstract We introduce a semi-analytical method to study the evolution of spatial structures in reaction-diffusion systems. It consists in writing an integral equation for the relevant densities, from the propagator of the linear part of the evolution operator. In order to test the method, we perform an exhaustive study of a one-dimensional reaction-diffusion model associated to an electrical device - the ballast resistor. We consider the evolution of step and bubble-shaped initial density profiles in free space as well as in a semi-infinite domain with Dirichlet and Neumann boundary conditions. The piecewise-linear form of the reaction term, which preserves the basic ingredients of more complex nonlinear models, makes it possible to obtain exact wave-front solutions in free space and stationary solutions in the bounded domain. Short and long-time behaviour can also be analytically studied, whereas the evolution at intermediate times is analyzed by numerical techniques. We paid particular attention to the features introduced in the evolution by boundary conditions.