Range Of Wave Speeds Research Articles

Travelling wave solutions of the beam equation with jumping nonlinearity

We study the existence of travelling wave solutions of a nonlinear fourth–order partial differential equation which can be considered as a model of an asymmetrically supported beam. We use a variational approach and show that there exist infinitely many travelling wave solutions under considerably weakened assumptions than those previously used in the literature. On the other hand, allowing the presence of sign preserving nonlinearities results in a limitation of the possible values of the wave speed. We pose several questions that still remain open, particularly regarding the range of wave speeds and the number of solutions. Moreover, some forms of classical solutions are presented together with some numerical experiments

Control of diffusion-driven pattern formation behind a wave of competency

In certain biological contexts, such as the plumage patterns of birds and stripes on certain species of fishes, pattern formation takes place behind a so-called “wave of competency”. Currently, the effects of a wave of competency on the patterning outcome are not well-understood. In this study, we use Turing’s diffusion-driven instability model to study pattern formation behind a wave of competency, under a range of wave speeds. Numerical simulations show that in one spatial dimension a slower wave speed drives a sequence of peak splittings in the pattern, whereas a higher wave speed leads to peak insertions. In two spatial dimensions, we observe stripes that are either perpendicular or parallel to the moving boundary under slow or fast wave speeds, respectively. We argue that there is a correspondence between the one- and two-dimensional phenomena, and hypothesize that (as others have) pattern formation behind a wave of competency can account for the pattern organization observed in many biological systems.

Semi-infinite travelling waves arising in a general reaction–diffusion Stefan model

We examine travelling wave solutions of the reaction–diffusion equation, , with a Stefan-like condition at the edge of the moving front. With only a few assumptions on R(u) and D(u), a variety of new semi-infinite travelling waves arise in this reaction–diffusion Stefan model. While other reaction–diffusion models can admit semi-infinite travelling waves for a unique wavespeed, we show that semi-infinite travelling waves in the reaction–diffusion Stefan model exist over a range of wavespeeds. Furthermore, we determine the necessary conditions on R(u) and D(u) for which semi-infinite travelling waves exist for all wavespeeds. Using asymptotic analysis in various distinguished limits of the wavespeed, we obtain approximate solutions of these travelling waves, agreeing with numerical simulations with high accuracy.

Approximations of solitary waves on lattices using weak and variational formulations

We examine two different ways to approximate the width and height of travelling wave solutions in infinite dimensional systems of ordinary different equations describing the time evolution of quantities such as displacement and charge at the node points of a regular lattice. We consider examples of one dimensional lattice systems similar to the Toda lattice equations and using a suitable ansatz, derive differential-difference equations for the shape of the travelling waves. These equations involve terms with both delay and advance in the independent variable and, in general, little is known about their analytic properties. We present two approximation methods based on the use of a test solution which contains parameters controlling its height, width and, in some cases, shape. In the first method these parameters are chosen using a weak formulation of the problem and in the second method a variational formulation is used. These methods are easy to use and produce accurate and simple explicit algebraic relationships between the height, width and speed of the travelling waves over the whole range of wave speeds.