In this paper, new travelling wave solutions to the Fornberg-Whitham equation ut − uxxt + ux + uux = uuxxx + 3uxuxx are investigated. They are characterized by two parameters. The expresssions of the periodic and solitary wave solutions are obtained.

A non-monotone time-delayed lattice system with global interaction is considered. The spreading speed, including the upward convergence, is established by comparison arguments and a fluctuation method. The existence of travelling waves is obtained by Schauder’s fixed-point theorem and a limiting process. It turns out that the minimal wave speed of travelling waves coincides with the spreading s...

We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed 2D lattice equations with non-monotone birth functions. First, with the help of Ikehara’s Theorem, the a priori asymptotic behavior of travelling wave is exactly derived. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. These results complement earlier res...

The recent theoretical discovery of families of travelling wave solutions in pipe flow (Faisst & Eckhardt 2003; Wedin & Kerswell 2004; Hof et al. 2004) at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series of numerical experiments are conducted in which we look for the spatial signature of these ...

By using the method of dynamical systems, this paper researches the bifurcation and the exact travelling wave solutions for (2 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Exact parametric representations of all wave solutions are given.

Periodic travelling waves are an important solution form in oscillatory reaction–diffusion equations. I have shown previously that such waves arise naturally near a boundary at which a Dirichlet condition is applied. This result has applications in ecology, providing a potential explanation for the periodic waves seen in a number of natural populations. However, in ecological applications the D...

An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green’s functions) for the linear ope...