Properties of traveling waves for integrodifference equations with nonmonotone growth functions

In this paper, we will establish some new properties of traveling waves for integrodifference equations with the nonmonotone growth functions. More precisely, for c ≥ c _*, we show that either lim_x?+￥ f(x)=u*{\lim\limits_{\xi\rightarrow+\infty} \phi(\xi)=u*} or 0 < liminf_{x? + ￥} f(x) < u* < limsup_x?+￥f(x) ￡ b,{0 < \liminf\limits_{\xi \rightarrow + \infty} \phi(\xi) < u* < \limsup \limits_{\xi\rightarrow+\infty}\phi(\xi)\leq b,} that is, the wave converges to the positive equilibrium or oscillates about it at +∞. Sufficient conditions can assure that both results will arise. We can also obtain that any traveling wave with wave speed c > c* possesses exponential decay at −∞. These results can be well applied to three types of growth functions arising from population biology. By choosing suitable parameter numbers, we can obtain the existence of oscillating waves. Our analytic results are consistent with some numerical simulations in Kot (J Math Biol 30:413–436, 1992), Li et al. (J Math Biol 58:323–338, 2009) and complement some known ones.     

Spreading speeds and traveling wavefronts for second order integrodifference equations

This paper is concerned with the spreading speeds and traveling wavefronts for second order integrodifference equations. By introducing an auxiliary integrodifference system, the spreading speed is established for the integrodifference equation. It is shown that the spreading speed coincides with the minimal wave speed for monotonic traveling wavefronts. Furthermore, we prove that the traveling wavefronts are stable by applying the squeezing technique. Finally, we analyze the different effects of the delay term appearing in the integrodifference equation from the viewpoint of ecology.     

Stability of traveling waves for Hamilton-Jacobi equations with finite speed perturbations

We prove nonlinear stability of planar shock for general Hamilton-Jacobi equations with finite speed perturbation. Here we use energy estimates. It is shown that the solution connecting a weak shock is asymptotically stable under small perturbations.     

Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossing-monostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson??s blowflies equation in population dynamics and Mackey?CGlass model in physiology.     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

Existence and uniqueness of traveling waves for non-monotone integral equations with applications

A class of integral equations without monotonicity is investigated. It is shown that there is a spreading speed c^∗>0 for such an integral equation, and that its limiting integral equation admits a unique traveling wave (up to translation) with speed c?c^∗ and no traveling wave with c<c^∗. These results are also applied to some nonlocal reaction-diffusion population models.     

Entire solutions of integrodifference equations

This paper is concerned with the existence of nontrivial entire solutions of integrodifference equations. By constructing proper auxiliary integrodifference equations and functions, we present the existence of nontrivial entire solutions even if the birth function is not monotone, which are different from the well-studied travelling wave solutions. The convergence of entire solutions is also given.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

Global existence of diffusive-dispersive traveling waves for general flux functions

We establish a global existence of traveling waves for diffusive-dispersive conservation laws for locally Lipschitz flux functions. Using Lyapunov stability techniques, we reduce the global problem of finding traveling waves to considering local behaviors of a stable trajectory of the saddle point.     

Existence for implicit differential equations with nonmonotone perturbations

This paper is devoted to the existence of solutions for a class of implicit Cauchy problems. The main tools in our study will be a convergent approximation procedure and the theory of pseudomonotone perturbations of maximal monotone mappings. This research is supported by the National Natural Science Foundation of China (No. 10171008).     

Bounded traveling waves of the oceanic currents motion equations

The bifurcation methods of differential equations are employed to investigate traveling waves of the oceanic currents motion equations. The sufficient conditions to guarantee the existence of different kinds of bounded traveling wave solutions are rigorously determined. Further, due to the existence of a singular line in the corresponding traveling wave system, the smooth periodic traveling wave solutions gradually lose their smoothness and evolve to periodic cusp waves. The results of numerical simulation accord with theoretical analysis.     

Propagation of second order integrodifference equations with local monotonicity

This paper is concerned with the spreading speeds and traveling wavefronts of second order integrodifference equations with local monotonicity. By introducing two auxiliary integrodifference equations, the spreading speed and traveling wave solutions are studied. In particular, we obtain the nonexistence of monotone traveling wave solutions for an example if it is local monotone. These results are applied to a model which is obtained by introducing the spatial variable to a difference equation used by the International Whaling Commission.     

Existence of traveling waves in compressible Euler equations with viscosity and capillarity

Given any Lax shock of the compressible Euler dynamics equations, we show that there exists the corresponding traveling wave of the system when viscosity and capillarity are suitably added. For a traveling wave corresponding to a given Lax shock, the governing viscous–capillary system is reduced to a system of two differential equations of first-order, which admits an asymptotically stable equilibrium point and a saddle point. We then develop the method of estimating attraction domain of the asymptotically stable equilibrium point for the compressible Euler equations and show that the saddle point in fact lies on the boundary of this set. Then, we establish a saddle-to-stable connection by pointing out that there is a stable trajectory leaving the saddle point and entering the attraction domain of the asymptotically stable equilibrium point. This gives us a traveling wave of the viscous–capillary compressible Euler equations.     

Turning points and traveling waves in FitzHugh-Nagumo type equations

Consider the following FitzHugh-Nagumo type equation

     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models

The theory of asymptotic speeds of spread and monotone traveling waves is generalized to a large class of scalar nonlinear integral equations and is applied to some time-delayed reaction and diffusion population models.     

Interactions of delta shock waves for the transport equations with split delta functions

This paper is concerned with the interactions of δ-shock waves and the vacuum states between the two contact discontinuities for the transport equations. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter ε tends to zero.