Macroscopic Lattice Dynamics
M. H. Hays, C. D. Levermore, and P. D. Miller
Program in Applied Mathematics
The University of Arizona, Tucson AZ 85721 USA
Abstract:
Fully nonlinear modulation equations are presented that govern the slow evolution of single-phase harmonic wavetrains under a large family of spatially discrete nonintegrable flows, among which is the discrete nonlinear Schrödinger equation (DNLS). These modulation equations are a pair of partial differential equations in conservation form that are hyperbolic for some data and elliptic for other data. In some cases these equations are capable of dynamically changing type from hyperbolic to elliptic, a phenomenon that has been associated with the modulational instability of the underlying wavetrain. By putting the modulation equations in Riemann invariant form, one can select initial data that avoid this dynamic change of type. Numerical experiments demonstrating the theoretical results are presented.