Positive Traveling Wave Solutions Research Articles

Orbital stability of periodic traveling waves to some coupled BBM equations

<abstract><p>In this work, we show some results concerning the orbital stability of dnoidal wave solutions to some Benjamin-Bona-Mahony equations (BBM equations henceforth). First, by the standard argument, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type. Then, we show that this type of solutions are orbitally stable by perturbations with the same period L. The major tools to obtain these results are the Grillaks, Shatah and Strauss' general theory in the periodic case. The results in the present paper extend some previous stability results for the BBM equations.</p></abstract>

Wave propagation in a diffusive SAIV epidemic model with time delays

Based on the fact that the incubation periods of epidemic disease in asymptomatically infected and infected individuals are inevitable and different, we propose a diffusive susceptible, asymptomatically infected, symptomatically infected and vaccinated (SAIV) epidemic model with delays in this paper. To see whether epidemic disease can propagate spatially with a constant speed, we focus on the travelling wave solution for this model. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system is greater than one and the wave speed is greater than or equal to the critical speed, we prove that this model admits nontrivial positive travelling wave solutions. Our theoretical results are of benefit to the prevention and control of epidemic.

Traveling Waves for a Discrete Diffusion Epidemic Model with Delay

This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components

In this article, the authors consider the orbital stability of periodic traveling wave solutions for the coupled compound KdV and MKdV equations with two components \begin{document}$ \begin{equation*} \left\{ \begin{aligned} u'>Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period \begin{document}$ L $\end{document} for the coupled compound KdV and MKdV equations. Then, combining the orbital stability theory presented by Grillakis et al., and detailed spectral analysis given by using Lame equation and Floquet theory, we show that the dnoidal type periodic wave solution with period \begin{document}$ L $\end{document} is orbitally stable. As the modulus of the Jacobian elliptic function \begin{document}$ k\rightarrow 1 $\end{document} , we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the coupled compound KdV and MKdV equations from our work. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition \begin{document}$ v = 0 $\end{document} , called the compound KdV and MKdV equation.

Super-critical and critical traveling waves in a three-component delayed disease system with mixed diffusion

This paper deals with traveling waves of a three-component delayed disease system with mixed diffusion. When the basic reproduction number of the corresponding spatial-homogenous delayed differential system R0>1 and the wave velocity c≥c∗ (c∗ is critical velocity), we obtain that the system admits a positive traveling wave solution. When R0≤1 and c∈R or R0>1 and c<c∗, we prove that the system has no positive traveling wave solutions. Our theoretical results may be conducive to disease prevention and control.

On positive travelling wave solutions for a general class of KdV-Burger type equation

On positive travelling wave solutions for a general class of KdV-Burger type equation

Traveling waves in a diffusive epidemic model with criss‐cross mechanism

In this paper, we propose a reaction‐diffusion system to describe the spread of infectious diseases within two population groups by self and criss‐cross infection mechanism. Firstly, based on the eigenvalues, we give two methods for the calculation of the critical wave speed c∗. Secondly, by constructing a pair of upper‐lower solutions and using the Schauder fixed‐point theorem, we prove that the system admits positive traveling wave solutions, which connect the initial disease‐free equilibrium at t = −∞, but the traveling waves need not connect the final disease‐free equilibrium at t = +∞. Hence, we study the asymptotic behaviors of the traveling wave solutions to show that the traveling wave solutions converge to at t = +∞. Finally, by the two‐sided Laplace transform, we establish the nonexistence of traveling waves for the model. The approach in this paper provides an effective method to deal with the existence of traveling wave solutions for the nonmonotone reaction‐diffusion systems consisting of four equations.

ORBITAL STABILITY OF PERIODIC TRAVELING WAVE SOLUTIONS TO THE GENERALIZED LONG-SHORT WAVE EQUATIONS

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Long-Short wave equations $ \left\{ \begin{array}{l} \!\!\!i\varepsilon_{\!t}\!+\!\varepsilon_{\!xx}\!\! = \!\!n\varepsilon\!\!+\!\! \alpha|\varepsilon|^{2}\!\varepsilon,\\ \!\!n_{t}\!\! = \!\!(|\varepsilon|^{2})_{x}, x\!\in\! R. \end{array} \right. $ Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period $ L $ for the generalized Long-Short wave equations. Then, combining the classical method proposed by Benjamin, Bona et al., and detailed spectral analysis given by using Lamé equation and Floquet theory, we show that the dnoidal type periodic wave solution is orbitally stable by perturbations with period $ L $. As the modulus of the Jacobian elliptic function $ k\rightarrow 1 $, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the generalized Long-Short equations. In particular, as $ \alpha = 0 $, we can also obtain the orbital stability results of periodic wave solutions and solitary wave solutions for the long-short wave resonance equations. The results in the present paper improve and extend the previous stability results of long-shore wave equations and its extension equations.

Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models

This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. By using Schauder's fixed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modified Ikehara's Theorem. With the help of their asymptotic behavior, we provide a sufficient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modified Leslie-Gower predator-prey models with different kinds of functional responses are also discussed.

Spatial propagation for a parabolic system with multiple species competing for single resource

A model of \begin{document}$m$\end{document} species competing for a single growth-limiting resource is considered. We aim to use the dynamics of such a problem to describe the invasion and spread of \begin{document}$m$\end{document} species which are introduced localized in space \begin{document}$\mathbb{R}^N$\end{document} . The existence, uniqueness and uniform boundedness of the Cauchy problem are investigated by semigroup theory and local \begin{document}$L^p$\end{document} -estimates. The asymptotic speed of spread is achieved by uniform persistence ideas. The existence of traveling wave is obtained by upper-lower solutions and sliding techniques. Our result shows that the asymptotic speed of spread for \begin{document}$m$\end{document} species is characterized by the minimum wave speed of the positive traveling wave solutions associated with this system.

Stability and traveling waves of a vaccination model with nonlinear incidence

In this paper, we propose a spatial vaccination model with nonlinear incidence. First, we consider the well-posedness of solutions of the model. Second, in the case of the bounded spatial habitat Ω⊆Rn, we investigate the global stability of the model. More precisely, it is shown that, if the threshold value R0≤1, then the disease-free equilibrium E0 is globally asymptotically stable; if R0>1, then there exists a unique disease equilibrium E∗ which is globally asymptotically stable. Third, in the case of the unbounded spatial habitat Ω=Rn, we study the existence of traveling wave solutions of the model. Here we show that when the threshold value R0>1, then there exists c∗>0 such that there exist positive traveling wave solutions of the model connecting the two equilibria E0 and E∗ with speed c>c∗. And when R0>1, there is not such a traveling wave solution with speed c<c∗. Numerical simulations are performed to illustrate our analytic results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R0.

Orbital stability of periodic traveling wave solutions to the generalized zakharov equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations {iut+uxx=uv+|u|2u,vtt-vxx=(|u|2)xx.First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].

A spatial echinococcosis transmission model with time delays: Stability and traveling waves

In this paper, we derive a time-delayed and diffusive echinococcosis transmission model. We first address the well-posedness to the initial-value problem for the model and give the basic reproduction number [Formula: see text]. In the case of a bounded spatial domain, we establish the local stability as well as the global stability of the disease-free and disease equilibria of the model. The methods to prove the local and the global stability are to analyze the corresponding characteristic equations and construct Lyapunov functionals, respectively. In the case of an unbounded spatial domain, by applying Schauder’s fixed point theorem and the limiting arguments, we show that when [Formula: see text], there exists a constant [Formula: see text] such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for [Formula: see text], and when [Formula: see text] and [Formula: see text], the model has no positive traveling wave solutions connecting them.

Traveling waves for the nonlocal diffusive single species model with Allee effect

In this paper, we study a nonlocal diffusive single species model with Allee effect. We prove that the model admits positive traveling wave solutions connecting the equilibrium 0 to some unknown positive steady state if and only if the wave speed c≥2r, where r>0 is the intrinsic rate of increase of the species. For the sufficient large wave speed c, we show that the unknown steady state is the unique positive equilibrium. For two types of convolution kernel functions, we investigate the change of the wave profile as the non-locality increases, and illustrate that the unknown steady state may be a positive periodic solution.

Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays

This paper is concerned with the traveling wave solutions of delayed reaction–diffusion systems. By using Schauder’s fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka–Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka–Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.

Traveling waves for nonlocal and non-monotone delayed reaction-diffusion equations

We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-diffusion equation. Based on the construction of two associated auxiliary reaction diffusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations, the existence of the positive traveling wave solutions for c ≥ c* is obtained. Also, the exponential asymptotic behavior in the negative infinity was established. Moreover, we apply our results to some reactiondiffusion equations with spatio-temporal delay to obtain the existence of traveling waves. These results cover, complement and/or improve some existing ones in the literature.

Asymptotic behaviour for a class of delayed cooperative models with patch structure

For a class of cooperative population models with patch structure and multiple discrete delays, we give conditions for the absolute global asymptotic stability of both the trivial solution and -- when it exists -- a positive equilibrium. Under a sublinearity condition, sharper results are obtained. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. As a by-product, we obtain a criterion for the existence of positive traveling wave solutions for an associated reaction-diffusion model with patch structure. Our results improve and generalize criteria in the recent literature.

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.

Positive travelling fronts for reaction–diffusion systems with distributed delay

We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction–diffusion systems with distributed delay. To prove the existence of travelling waves, we give an abstract formulation of the equation for the wave profiles in some suitable Banach spaces and apply known results about the index of some associated Fredholm operators. After a Lyapunov–Schmidt reduction, these waves are obtained via the Banach contraction principle, as perturbations of a positive heteroclinic solution for the associated system without diffusion, whose existence is proven under some requirements. By a careful analysis of the exponential decay of the travelling wave profiles at −∞, their positiveness is deduced. The existence of positive travelling waves is important in terms of applications to biological models. Our method applies to systems of delayed reaction–diffusion equations whose nonlinearities are not required to satisfy a quasi-monotonicity condition. Applications are given, and include the delayed Fisher–KPP equation.

KPP type flame fronts in porous media

In this paper we study the model of pressure-driven flames in porous media proposed by Brailovsky et al (1997 Combust. Sci. Technol. 124 145–65). We show that, under the assumption of first order reaction with linear reaction kinetics (quadratic nonlinearity), the model admits a family of positive travelling wave solutions. Moreover, under the same assumption, we prove that propagation of disturbances in the system is fully determined by the rate of decay of the initial data at infinity. We also give an upper bound of the burning rate in the case of arbitrary chemical kinetics bounded by linear function.