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

TL;DR

This paper investigates wave phenomena in infinite one-dimensional particle lattices with long-range power-law interactions, deriving explicit formulas for solitary and periodic waves, and exploring the connection to integrable systems and nonlocal PDEs.

Contribution

It provides explicit formulas for solitary and periodic waves in infinite lattices with long-range interactions, extending understanding of wave dynamics in such systems.

Findings

Explicit formulas for solitary waves in Calogero-Moser lattices

Connection between long-wave limits and nonlocal PDEs like Benjamin-Ono

Identification of regimes where KdV or nonlocal PDEs describe wave behavior

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.