Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations

We study dynamical phenomena for a class of lattice differential equations, namely infinite systems of ordinary differential equations coordinatized by points on a spatial lattice. We examine in particular the dependence of traveling wave solutions on the direction of motion of the traveling wave. The phenomenon of crystallographic pinning occurs when there is a tendency for a wave to become pinned in selected directions. In previous work with J.W. Cahn and E.S. Van Vleck we demonstrated this phenomenon for a special class of systems with piecewise linear nonlinearities. In the present work we show how this phenomenon holds for a general class of systems with smooth nonlinearities and how it follows from general principles of dynamical systems.