Direction reversing travelling waves in the Fermi-Pasta-Ulam chain

Abstract

This paper considers the famous Fermi-Pasta-Ulam chain with periodic boundary con- ditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is completely integrable, as was shown in [16]. We study how the level sets of the integrals foliate the phase space. Our study reveals all the integrable structure in the low energy domain of the chain. If the number of particles in the chain is even, then this foliation is singular. The method of singular reduction shows that the system has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle vari- ables. The pinched tori are interpreted as homoclinic and heteroclinic connections between travelling waves. Thus we discover a new class of solutions which can be de- scribed as direction reversing travelling waves. They remarkably show an interesting interaction of normal modes in the absense of energy transfer. These solutions can easily be observed numerically.