Traveling Waves and Monodromy in Anharmonic Lattices

The study of anharmonic lattices can be considerably simplified by imposing a spatial periodicity condition on solutions. This reduces the infinite dimensional lattice equations to a finite dimensional system of ordinary differential equations for which we have at our disposal Birkhoff normal forms, invariant theory, singular reduction and the Kolmogorov Arnol’d Moser theorem. As an example we study the famous Fermi Pasta Ulam lattice for which we find traveling wave solutions. These traveling waves can become unstable and reverse their directions. Moreover, although the lattice is nearly integrable, the integrable approximation has monodromy and hence does not admit global action angle coordinates.