# WAVES IN LATTICES

This class of problems is motivated by the desire to understand the dynamical behavior of lattices of particles in a force field. The central model problem is a chain of coupled pendula with forcing and damping. The problem arises in studying the classical model of electrons in a crystal lattice, the motion of charge--density waves, the Josephson junction, the discretization in space of sine--Gordon equations and their analogues and in other applications.

Several numerical and experimental studies of the last two decades uncovered some interesting nonlinear wave analogues in the case of just two pendula -- one could callthese ``caterpillar waves'' because of the way they behave.

I gave a transparent geometrical explanation of the mechanism of these nonlinear ``waves" and gave a bifurcation diagram which predicts what happens for different values of parameters in the case of an arbitrary number of particles -- this was done in [18] for $ n=2 $ pendula and in [21] for any number n.

What makes the ``caterpillar waves'' ``crawl'' in the final analysis is the well--known fact that the velocity of the vectorfield near a degenerate equilibrium (i.e., the point of coalescence of two equilibria) is small compared to the velocity of the vectorfield near a nondegenerate equilibrium. The difficulty of this work was to connect the simple fact with the phenomenon of the waves.

The paper [21] predicts the coexistence of two stable traveling waves with different speeds. This phenomenon was observed experimentally several years later in newly discovered Josephson junction crystals.

The application of dynamical systems to studying PDE's, with an eye on turbulence, has not lived up to expectations, despite the appeal of this approach. The lattice of $n\geq 2$ pendula with weak torsional coupling is one of very few examples which are on the borderline, being rich enough to inherit the interesting behavior while still simple enough to be manageable analytically, and one of few cases when the geometric approach to discretized PDE viewed as a dynamical system gives a nearly complete insight into the qualitative dynamics of the waves in a lattice and into various bifurcations.