On Long Waves and Solitons in Particle Lattices with Forces of Infinite Range

.We study waves on infinite one-dimensional lattices of particles that each interact with all others through power-law forces \(F ~ r-β\). 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 \(β > 4\), but with \(2< β < 4\) it is a nonlocal dispersive PDE that reduces to the Benjamin–Ono equation for \(β=3\). For the infinite Calogero–Moser lattice, we find explicit formulas that describe solitary and periodic traveling waves.KeywordsKdV limitCalogero–Sutherland systemsBäcklund transformMSC codes37K6037K4070F4535Q51