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.     

Blow-up profiles for positive solutions of nonlocal dispersal equation

In this paper, we study the blow-up profiles of the nonlocal dispersal equation. More precisely, we prove that the positive solution of nonlocal dispersal equation has different blow-up profiles, depending on the refuge domain.     

Multi-type forced waves in nonlocal dispersal KPP equations with shifting habitats

The current paper is concerned with the forced waves to nonlocal dispersal KPP equations with shifting habitats. Without asking the sign of intrinsic growth rate at negative infinity, we derive the conditions for the existence of multi-type forced waves with different profiles corresponding to different shifting speeds. Our results show that (i) there exist not only monotone but also non-monotone forced waves, and (ii) even at the same wave speed as the shifting speed of habitats, more than one forced waves can be found.     

Traveling Waves for Nonlocal and Non-monotone Delayed Reaction-difusion Equations

We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-difusion equation.Based on the construction of two associated auxiliary reaction difusion 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 infnity was established.Moreover,we apply our results to some reactiondifusion 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.     

Traveling waves in a nonlocal dispersal SIR epidemic model

This paper is concerned with traveling wave solutions of a nonlocal dispersal SIR epidemic model. The existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. This threshold dynamics are proved by Schauder’s fixed point theorem and the Laplace transform. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.     

On the stability of a population model with nonlocal dispersal

This paper is concerned with a nonlocal dispersal population model with spatial competition and aggregation. We establish the existence and uniqueness of positive solutions by the method of coupled upper-lower solutions. We obtain the global stability of the stationary solutions.     

Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model

This paper is devoted to developing a nonlocal dispersal HIV infection dynamical model. The existence of travelling wave solutions is investigated by employing Schauder's fixed point theorem. That is, we study the existence of travelling wave solutions for $R_0>1$ and each wave speed $c>c^{?}$. In addition, the boundary asymptotic behaviour of travelling wave solutions at +∞ is obtained by constructing suitable Lyapunov functions and employing Lebesgue dominated convergence theorem. By employing a limiting argument, we investigate the existence of travelling wave solutions for $R_0>1$ and $c=c^{?}$. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.     

A finite difference scheme for traveling wave solutions to Burgers equation

We construct a finite difference scheme for the ordinary differential equation describing the traveling wave solutions to the Burgers equation. This difference equation has the property that its solution can be calculated. Our procedure for determining this solution follows closely the analysis used to obtain the traveling wave solutions to the original ordinary differential equation. The finite difference scheme follows directly from application of the nonstandard rules proposed by Mickens. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 815–820, 1998     

Monotonicity,uniqueness, and stability of traveling waves in a nonlocal reaction‐diffusion system with delay

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction‐diffusion system with delay arising from the spread of an epidemic by oral‐faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c ?c _?. Then we show that the traveling wave fronts with speed c >c _? are exponentially asymptotically stable.     

Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation

We report the analytical one- and two-rogon-like solutions for the two-dimensional nonlocal nonlinear Schrödinger equation by means of the similarity transformation. These obtained solutions can be used to describe the possible physical mechanisms for rogue-like wave phenomenon. Moreover, the free function of space y involved in the obtained solutions excites the abundant structures of rogue-like wave propagations. The Hermite-Gaussian function of space y (normalized function) is, in particular, chosen to depict the dynamical behaviors for rogue-like wave phenomenon.     

Nonlocal anisotropic dispersal with monostable nonlinearity

We study the travelling wave problem

     

Symmetry reductions and traveling wave solutions for the Krichever–Novikov equation

In this paper, we study the Krichever–Novikov equation from the point of view of the theory of symmetry reductions in PDEs. By using this theory, we find that for the Krichever–Novikov equation some similarity solutions are solutions with physical interest: solitons, kinks, antikinks, and compactons. Copyright © 2012 John Wiley & Sons, Ltd.     

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.     

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.     

On the nonlocal Boussinesq equation: Multiple-soliton solutions

In this work we investigate the first-order nonlocal Boussinesq system. We use the simplified form of Hirota’s direct method to determine multiple-soliton solutions for this system.     

Spreading speed for a nonlocal dispersal vaccination model with general incidence

This paper is concerned with the spreading speed for a nonlocal dispersal vaccination model with general incidence. We first prove the existence and uniform boundedness of solutions for this model by using the Schauder’s fixed point theorem. Then, applying comparison principle, we establish the existence of spreading speed for the infective individuals. According to our result, one can see that the spreading speed coincides with the critical speed of traveling wave solution connecting the disease-free and endemic equilibria. In addition, the diffusion rate of the infected individuals can increase the spread of infectious diseases, while the vaccination rate reduces the spread of infectious diseases.     

Pulsating waves and entire solutions for a spatially periodic nonlocal dispersal system with a quiescent stage

Science China Mathematics - This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and...