Traveling Front Solutions Research Articles

Exact Analytic Solution of the Simplified Telegraph Model of Propagation and Dissipation of Excitation Fronts

The telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in several branches of sciences (E. Ahmed, H. A. Abdusalam, and E. S. Fahmy, 2001, Int. J. Mod. Phys. C 12(5), 717; E. Ahmed and H. A. Abdusalam, 2004, Chaos, Solitons and Fractals 22, 583; H. A. Abdusalam and E. S. Fahmy, 2003, Chaos, Solitons and Fractals 18, 259; Abdusalam, 2004 Appl. Math. Comp. 157, 515). An excitation wave in cardiac tissue fails to propagate if the transmembrane voltage at its front rises too slow and does not excite the tissue ahead of it. Then the sharp voltage profile of the front will dissipate, and subsequent spread of voltage will be purely diffusive. This mechanism is impossible in FitzHugh–Nagumo type system (V. N. Biktashev, 2003, Int. J. Bifarcation and Chaos 13(12), 3605). Biktashev suggested a simplified mathematical model for this mechanism and in the present work we generalize this model to telegraph system. Our generalized telegraph model has exact traveling front solutions and we show the effect of the time delay on the velocity and we show that, the post-front voltage depends on two parameters in which one of them is the time delay.

SOME ENTIRE SOLUTIONS OF THE ALLEN–CAHN EQUATION

This paper is dealing with entire solutions of a bistable reaction-diffusion equation with Nagumo type nonlinearity, so called the Allen--Cahn equation. Here the entire solutions are meant by the solutions defined for all $(x,t)\in \mathbb{R}\times\mathbb{R}$. In this article we first show the existence of an entire solution which behaves as two traveling front solutions coming from both sides of $x$-axis and annihilating in a finite time, using the explicit expression of the traveling front and the comparison theorem. We also show the existence of an entire solution emanating from the unstable standing pulse solution and converges to the pair of diverging traveling fronts as the time tends to infinity. Then in terms of the comparison principle we prove a rather general result on the existence of an unstable set of an unstable equilibrium to apply to the present case.

A SIMPLIFIED MODEL OF PROPAGATION AND DISSIPATION OF EXCITATION FRONTS

An excitation wave in nerve or cardiac tissue may fail to propagate if the temporal gradient of the transmembrane voltage at the front becomes too small to excite the tissue ahead of it. A simplified mathematical model is suggested, that reproduces this phenomenon and has exact traveling front solutions. The spectrum of possible propagation speeds is bounded from below. This causes a front to dissipate if it is not allowed to propagate quickly enough. A crucial role is played by the Na inactivation gates, even if their dynamics are by an order of magnitude slower than the dynamics of the voltage.

Traveling front solutions to directed diffusion-limited aggregation, digital search trees, and the Lempel-Ziv data compression algorithm.

We use the traveling front approach to derive exact asymptotic results for the statistics of the number of particles in a class of directed diffusion-limited aggregation models on a Cayley tree. We point out that some aspects of these models are closely connected to two different problems in computer science, namely, the digital search tree problem in data structures and the Lempel-Ziv algorithm for data compression. The statistics of the number of particles studied here is related to the statistics of height in digital search trees which, in turn, is related to the statistics of the length of the longest word formed by the Lempel-Ziv algorithm. Implications of our results to these computer science problems are pointed out.

Travelling waves in a singularly perturbed sine-Gordon equation

We determine the linearised stability of travelling front solutions of a perturbed sine-Gordon equation. This equation models the long Josephson junction using the RCSJ model for currents across the junction and includes surface resistance for currents along the junction. The travelling waves correspond to the so-called fluxons and their linear stability is determined by calculating the Evans function. Surface resistance corresponds to a singular perturbation term in the governing equation, which specifically complicates the computation of the corresponding Evans function. Both the flow of quasi-particles across and along the junction stabilise the waves.

Existence of traveling wave solutions in a diffusive predator-prey model.

We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R(4) and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R(4). The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem.

Stability and Traveling Fronts in Lotka-Volterra Competition Models with Stage Structure

This paper is concerned with a delay differential equation model for the interaction between two species, the adult members of which are in competition. The competitive effects are of the Lotka--Volterra kind, and in the absence of competition it is assumed that each species evolves according to the predictions of a simple age-structured model which reduces to a single equation for the total adult population. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. Thus, the time delays appear in the adult recruitment terms. The dynamics of the model are determined, and global stability results are established for each equilibrium. The equilibria of the model involve the maturation delays. The criteria for global convergence to each equilibrium are sharp and involve these delays. A reaction-diffusion extension of the model is also studied for the case when only the adult members of each species can diffuse. We prove the existence of a traveling front solution connecting the two boundary equilibria for the case when there is no coexistence equilibrium. This represents invasion by the stronger species of territory previously inhabited only by the weaker. The proof of the existence of such a front uses Wu and Zou's theory for traveling front solutions of delayed reaction-diffusion systems.

Convergence and Travelling Fronts in Functional Differential Equations with Nonlocal Terms: A Competition Model

In this paper we consider a two-species competition model described by a reaction- diffusion system with nonlocal delays. In the case of a general domain, we study the stability of the equilibria of the system by using the energy function method. When the domain is one-dimensional and infinite, by employing linear chain techniques and geometric singular perturbation theory, we investigate the existence of travelling front solutions of the system.

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species.

We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay.

Dissipation of the excitation wave fronts.

An excitation wave in cardiac tissue will fail to propagate if the transmembrane voltage at its front rises too slowly and does not excite the tissue ahead of it. Then the sharp voltage profile of the front will dissipate, and the subsequent spread of voltage will be purely diffusive. This mechanism is impossible in FitzHugh-Nagumo type systems. Here a simplified mathematical model for this mechanism is suggested. The model has exact traveling front solutions, and gives conditions for the front dissipation. In particular, a front will dissipate if it is not allowed to propagate faster than a certain nonzero speed. This critical speed depends only on the properties of the fast sodium current.

Travelling fronts in a food-limited population model with time delay

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.

Multiplication invariant subspaces of the Bergman space

We establish sharp pointwise Green's function bounds and consequent linearized stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([32, 108, 110]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, hyperbolic stability of the corresponding ideal shock of the associated equilibrium system, and transversality of the connecting profile, with no additional assumptions on the structure or strength of the shock. Restricting to Lax type shocks, we establish the further result of nonlinear stability with respect to small L 1 n H 2 perturbations, with sharp rates of decay in L p , 2 ≤ p ≤ ∞, for weak shocks of general simultaneously symmetrizable systems; for discrete kinetic models, and initial perturbation small in W 3,1 n W 3, ∞ , we obtain sharp rates of decay in L p , 1 ≤ p < ∞, for (Lax type) shocks of arbitrary strength. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin-Xin and Broad-well models, for which spectral stability has been established in [61, 43], and in [52], respectively. Our analysis follows the basic pointwise semigroup approach introduced by Zumbrun and Howard [107] for the study of traveling waves of parabolic systems; however, significant extensions are required to deal with the nonsectorial generator and more singular short-time behavior of the associated (hyperbolic) linearized equations. Our main technical innovation is a systematic method for refining large-frequency (short-time) estimates on the resolvent kernel, suitable in the absence of parabolic smoothing.

Wave front solutions of a diffusive delay model for populations of Daphnia magna

In this paper, we consider a diffusional food-limited population model incorporating a discrete time-delay. For the case when the delay is small, we show that monotone travelling front solutions exist connecting the two uniform steady states of the model. The effect of a larger delay is studied numerically, suggesting that in this case travelling fronts still exist but lose their monotonicity.

Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays

We consider the diffusive Nicholson's blowflies equation where the time delay is of the distributed kind, incorporated as an integral convolution in time. Of interest is the question of the existence of travelling front solutions and their qualitative form. For small delay, existence of such fronts is proved when the convolution kernel assumes a special form, enabling the use of linear chain techniques. The resulting higher-dimensional system is studied using geometric singular perturbation theory. The method should be applicable to other such kernels as well. For larger delays, numerical simulations show that the main effect is a loss of monotonicity of the wave front.

Stability of travelling-wave solutions for reaction-diffusion-convection systems

We are concerned with the asymptotic behaviour of classical solutions of systems of the form $$ \cases u_{t} = A u_{xx} + f(u, u_{x}) &\text{for } x \in {\mathbb R}, t> 0, u(x,t) \in {\mathbb R}^N,\\ u(x,0) = \varphi (x),& \endcases\tag{1} $$ where $A$ is a positive-definite diagonal matrix and $f$ is a bistable nonlinearity satisfying conditions which guarantee the existence of a comparison principle for (1). Suppose that (1) has a travelling-front solution $w$ with velocity $c$, that connects two stable equilibria of $f$. (There are hypotheses on $f$ under which such a front is known to exist [E. C. M. Crooks and J. F. Toland, Travelling waves for reaction-diffusion-convection systems , Topol. Methods Nonlinear Anal. 11 (1998), 19–43].) We show that if $\varphi$ is bounded, uniformly continuously differentiable and such that $\Vert w(x) - \varphi (x) \Vert $ is small when $|x|$ is large, then there exists $\chi \in {\mathbb R}$ such that $$ \Vert u(\cdot, t) - w(\cdot + \chi - ct) \Vert _{BUC^{1}} \rightarrow 0 \quad\text{as } t \rightarrow \infty.\tag{2} $$ Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where $f$ is independent of $u_{x}$. First $\varphi$ is assumed to be increasing in $x$, and (2) proved via a homotopy argument. Then we deduce the result for arbitrary $\varphi$ by showing that there is an increasing function in the $\omega$-limit set of $\varphi$.

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.

Non-existence of travelling front solutions of some bistable reaction-diffusion equations

This work deals with travelling fronts solutions of some reaction-diffusion equations in an infinite cylinder in dimension $\ge 2$. The problem is set in $\Sigma=\{(x_1,y)\in{\mathbb R}\times\omega\}$ where $\omega\subset{\mathbb R}^{N-1}$ is a bounded and smooth domain with outward normal $\nu$. The equations, with unknowns $c\in {\mathbb R}$ and $u\in C^2(\overline{\Sigma})$, are $$ (P) \qquad\qquad \left\{\begin{array}{rl} \Delta u-(c+\alpha(y))\ \partial_{1}u+f(u)=0 & \hbox{ in }\Sigma={\mathbb R}\times\omega\\ \displaystyle{\frac{\partial u }{\partial\nu}}=0 & \hbox{ on }\partial\Sigma= {\mathbb R}\times\partial\omega\\ u(-\infty,\cdot)=0\hbox{ and }u(+\infty,\cdot)=1 & \end{array} \right. $$ The function $\alpha \in C^0(\overline{\omega})$ is given. The nonlinearity $f$ is assumed to be of the ``bistable type": it changes sign once in $(0,1)$. Berestycki and Nirenberg [8] proved that if $\omega$ is convex then the problem has a solution. Here, by using the invariance by translation and the sliding method, we construct an example of a non-convex domain $\omega$ and of a function $\alpha$ for which we prove that $(P)$ has no solutions. This is in sharp contrast with other types of nonlinearities for which solutions exist whatever $\omega$ may be.

Travelling front solution for a class of competition-diffusion system with high-order singular point

In this paper, the existence of travelling front solution for a class of competition-diffusion system with high-order singular point $$w_{it} = d_i w_{ixx} - w_{i^i }^a f_i \left( w \right),x \in R,t > 0,i = 1,2$$ (I) is studied, where di,ai>0 (i=1,2) and w=(w1(x,t),w2(x,t)). Under the certain assumptions on f, it is showed that if ai 0, where c* is called the minimal wave speed of (I), such that if c≥c0 or c=c*, then (I) has a travelling front solution, if c<c*, then (I) has no travelling front solution by using the shooting method in combination with a compactness argument.