Many models in biology and ecology can be described by reaction-diffusion equations wit time delay. One of important solutions for these type of equations is the traveling wave solution that shows the phenomenon of wave propagation. The existence of traveling wave fronts has been proved for large class of equations, in particular, the monotone systems, such as the cooperative systems and some c...

We investigate the qualitative behavior of solutions of a Burgers-Boussinesq system – a reaction-diffusion equation coupled via gravity to a Burgers equation – by a combination of numerical, asymptotic and mathematical techniques. Numerical simulations suggest that when the gravity ρ is small the solutions decompose into a traveling wave and an accelerated shock wave moving in opposite directio...

In this paper, the (G′/G)-expansion method is applied to seek traveling wave solutions of the combined and the double combined sinh-cosh-Gordon equations. This traveling wave solutions are expressed by the hyperbolic functions and the trigonometric functions. It is shown that the proposed method is direct, effective and more general. 2000 Mathematics Subject Classification: 35K01; 35J05.

We explore numerically different aspects of periodic traveling-wave solutions of the Camassa–Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied. 2005 Elsevier Ltd. All rights reserved.

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt − Luxx = B(±|u|u)xx, p > 1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operatorsL and B. Members of the class arise as mathematical models ...

We study the traveling wave solutions for a system of coupled KdV equations derived by Lou et al [11]. In that paper, they found 5 types of Painlevé integrable systems for the coupled KdV system. We show that each of them can be reduced to a partially or completely uncoupled system, through which the dynamical behavior of traveling wave solutions can be determined. In some parameter regions, ex...

We establish the existence of traveling wave solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator–prey model with Holling type-III functional response. The analysis is in the three-dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator– prey system in the traveling wave variable. ...

This paper investigates traveling wave solutions of the spatially discrete reaction-diffusion systems whose kinetics are modeled by the McKean caricature of the FitzHugh-Nagumo model. In the limit of a weak coupling strength, we construct the traveling wave solutions and obtain the critical coupling constant below which propagation failure occurs. We report the existence of two different pulse ...

In this paper, the modified exp-function method is used to seek generalized wave solutions of Klein-Gordon equation. As a result, some new types of exact traveling wave solutions are obtained which include kink wave solutions, periodic wave solution, and solitary wave solutions. Obtained results clearly indicate the reliability and efficiency of the proposed modified exp-function method.

Traveling wave solutions to a class of dispersive models, ut − utxx + uux = θuuxxx + (1− θ)uxuxx, are investigated in terms of the parameter θ, including two integrable equations, the Camassa-Holm equation, θ = 1/3, and the Degasperis-Procesi equation, θ = 1/4, as special models. It was proved in [H. Liu and Z. Yin, Contemporary Mathematics, 2011, 526, pp273–294] that when 1/2 < θ ≤ 1 smooth so...