Cnoidal Waves on Fermi-Pasta-Ulam Lattices

We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion with generic nearest-neighbour potential . We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of Friesecke and Pego (Nonlinearity 12:1601-1627, 1999) to a periodic setting and the spectral theory of the periodic Schrodinger operator with KdV cnoidal wave potential.