On long waves and solitons in particle lattices with forces of infinite range

Abstract

We study waves on infinite one-dimensional lattices of particles that each interact with all others through power-law forces $F \sim r^{-\beta}$. The inverse-cube case corresponds to Calogero-Moser systems, which are well known to be completely integrable for any finite number of particles. The formal long-wave limit for unidirectional waves in these lattices is the Korteweg-de Vries equation if $\beta >4$, but with $2<\beta <4$ it is a nonlocal dispersive PDE that reduces to the Benjamin-Ono equation for $\beta=3$. For the infinite Calogero-Moser lattice, we find explicit formulas that describe solitary and periodic traveling waves.