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. © 1992.