Variational approximation for wave propagation in continuum and discrete media

We Develop a variational approximation for wave propagation in continuum and discrete media based on the modulation of wave profiles Described by appropriate trial functions. We illustrate the method by consiDering an application to the theory of dislocation of materials. We first consiDer the continuum approximation of the moDel and reproduce the exact traveling known solution. We then consiDer the fully discrete non integrable moDel and obtain an approximate solution based on trial functions with functional form similar to the exact solution of the continuum. The Description of this discrete approximate solution is in terms of a discrete nonlinear dispersion relation between the wave parameters. In this last situation we compare the numerical and variational solutions at the stationary case. We thus illustrate the usage of a variational asymptotic approximation to study nonlinear problems and we contrast the differences and difficulties between continuum and discrete problems.