Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems

This paper is concerned with nonlinear stability of traveling wave fronts for a delayed reaction diffusion system. We prove that the traveling wave front is exponentially stable to perturbation in some exponentially weighted L^∞ spaces, when the difference between initial data and traveling wave front decays exponentially as x→−∞, but the initial data can be suitable large in other locations. Moreover, the time decay rates are obtained by weighted energy estimates.     

The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay

In this paper, we study a reaction–diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step A+B→2B$A+B\to 2B$ and a decay step B→C$B\to C$, where A, B, and C are the reactant, the autocatalyst, and the inner product, respectively. Previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant A initially distributes uniformly in the whole space, then a pair of waves will be generated and will propagate outwards from the initial reaction zone. One crucial feature of this phenomenon is that for the strong decay case, the formation of waves is independent of the amount of the autocatalyst B introduced into the system. It is this phenomenon of KPP-type which we would like to address in this paper. To study the propagation of reactant and autocatalyst analytically, we first use the tail behavior of waves to construct a pair of generalized super-/sub-solutions for the approximate system of the autocatalytic reaction system. Note that the autocatalytic reaction system does not enjoy comparison principle. Together with a family of truncated problems, we can establish the existence of a family of traveling waves with the minimal speed. Second, we use this pair of generalized super-/sub-solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of Aronson–Weinberger formulation, which in turn confirms the aforementioned numerical and experimental results.     

Recurrent traveling waves in a two-dimensional saw-toothed cylinder and their average speed

We study a curvature-dependent motion of plane curves in a two-dimensional infinite cylinder with spatially undulating boundary. The law of motion is given by V=κ+A$V=\kappa +A$, where V is the normal velocity of the curve, κ is the curvature, and A is a positive constant. The boundary undulation is assumed to be almost periodic, or, more generally, recurrent in a certain sense. We first introduce the definition of recurrent traveling waves and establish a necessary and sufficient condition for the existence of such traveling waves. We then show that the traveling wave is asymptotically stable if it exists. Next we show that a regular traveling wave has a well-defined average speed if the boundary shape is strictly ergodic. Finally we study what we call “virtual pinning”, which means that the traveling wave propagates over the entire cylinder with zero average speed. Such a peculiar situation can occur only in non-periodic environments and never occurs if the boundary undulation is periodic.     

Turning points and traveling waves in FitzHugh-Nagumo type equations

Consider the following FitzHugh-Nagumo type equation

     

An application of traveling wave analysis in economic growth model

A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance.     

An approximation to the minimum traveling wave for the delayed diffusive Nicholson's blowflies equation

In this work, we study the approximation of traveling wave solutions propagated at minumum speeds c ₀(h ) of the delayed Nicholson's blowflies equation: In order to do that, we construct a subsolution and a super solution to (?). Also, through that construction, an alternative proof of the existence of traveling waves moving at minimum speed is given. Our basic hypothesis is that p /δ ∈(1,e ] and then, the monostability of the reaction term. Copyright © 2017 John Wiley & Sons, Ltd.     

Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities

We study the existence, uniqueness and asymptotic behavior, as well as the stability of a special kind of traveling wave solutions for competitive PDE systems involving intrinsic growth, competition, crowding effects and diffusion. The traveling waves are exclusive in the sense that as the variable goes to positive or negative infinity, different species are close to extinction or carrying capacity. We perform an appropriate affine transformation of the traveling wave equations into monotone form and construct appropriate upper and lower solutions. By this means, we reduce the existence proof to application of well-known theory about monotone traveling wave systems (cf. [A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, MIA, Kluwer, Boston, 1989; J. Wu, X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651-687] and [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994]). Then, by using spectral analysis of the linearization over the profile, we prove the orbital stability of the traveling wave in some Banach spaces with exponentially weighted norm. Furthermore, we show that the introduction of some weight is necessary in the sense that, in general, traveling wave solutions with initial perturbations in the (unweighted) space C₀ are unstable (cf. [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994] and [D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981]).     

Uniqueness of traveling wave solutions for a biological reaction-diffusion equation

The existence of traveling wave solutions for a reaction-diffusion, which serves as models for microbial growth in a flow reactor and for mathematical epidemiology, was previously confirmed. However, the problem on the uniqueness of traveling wave solutions remains open. In this paper we give a complete proof of the uniqueness of traveling wave solutions.     

A type of bounded traveling wave solutions for the Fornberg-Whitham equation

In this paper, by using bifurcation method, we successfully find the Fornberg-Whitham equation

ut−uxxt+ux=uuxxx−uux+3uxuxx,     

Asymptotic stability of traveling waves in a discrete convolution model for phase transitions

In a recent paper [P. Bates, A. Chmaj, A discrete convolution model for phase transition, Arch. Rational Mech. Anal. 150 (1999) 281-305], a discrete convolution model for Ising-like phase transition has been derived, and the existence, uniqueness of traveling waves and stability of stationary solution have been studied. This nonlocal model describes l₂-gradient flow for a Helmholts free energy functional with general range interaction. In this paper, by using the comparison principle and the squeezing technique, we prove that the traveling wavefronts with nonzero speed is globally asymptotic stable with phase shift.     

Solitary wave and chaotic behavior of traveling wave solutions for the coupled KdV equations

Using the method of dynamical systems to study the coupled KdV system, some exact explicit parametric representations of the solitary wave and periodic wave solutions are obtained in the given parameter regions. Chaotic behavior of traveling wave solutions is determined.     

Traveling Wave Fronts of a Degenerate Parabolic Equation with Non-divergence Form

We study the traveling wave solutions of a nonlinear degenerate parabolic equation with non-divergence form. Under some conditions on the source, we establish the existence, and then discuss the regularity of such solutions.     

Periodic traveling wave solution for non-homogeneous BBM and KdVB equations

In this paper the Green’s function method and results about fixed point are used to get existence results on periodic traveling wave solution for non-homogeneous problems of generalized versions of the BBM and KdVB equations. It is shown through the constructions of explicit Green’s functions that the periodic boundary value problems for the traveling wave solutions of the BBM and KdVB equations are equivalent to integral equations which generate compact operators in the space of periodic functions. These integral representations allowed us to prove that if the speed of the wave propagation is suitably chosen, then the BBM and KdVB equations will admit periodic traveling wave solution.     

Minimum wave speed for a diffusive competition model with time delay

In this paper we study mono-stable traveling wave solutions for a Lotka-Volterra reaction-diffusion competition model with time delay. By constructing upper and lower solutions, we obtain the precise minimum wave speed of traveling waves under certain conditions. Our results also extend the known results on the minimum wave speed for Lotka-Volterra competition model without delay.     

Research on traveling wave solutions for a class of (3+1)-dimensional nonlinear equation

Nonlinear wave phenomena are of great importance in the nature, and have became for a long time a challenging research topic for both pure and applied mathematicians. In this paper the solitary wave, kink (anti-kink) wave and periodic wave solutions for a class of (3+1)-dimensional nonlinear equation were obtained by some effective methods from the dynamical systems theory.