Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary.

This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.