Periodic permanent waves in an anharmonic chain with nearest-neighbour interaction

The existence of longitudinal periodic permanent waves in a one-dimensional translationally invariant anharmonic chain with nearest-neighbour interaction is established by means of variational methods. A general expression for the energy is given in terms of the dispersion relation. The interaction potential is not specified in detail. Fundamental assumptions are: the potential has a vertical asymptote for small distance and a horizontal one for large distance between two neighbouring particles. In the limit for high energy the wave looks like an infinite series of 'spikes' with finite amplitude and 'infinite' velocity