Publication 23-CNA-013
On long waves and solitons in particle lattices with forces of infinite range
Benjamin Ingimarson
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
bwi@andrew.cmu.edu
Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@cmu.edu
Abstract: We study waves on infinite one-dimensional lattices of particles that each interact with all others through power-law forces
F ~ 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.
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