title: Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials.
abstract: We consider a class of fully nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of orderα>1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyse the propagation of localized waves whenαis close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic Korteweg-de Vries (KdV) equation and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When [Formula: see text], we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.
authors.affiliation: INRIA Grenoble Rhône-Alpes and Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, Grenoble Cedex 9 38041, France.
authors.forename: Guillaume
authors.initials: G
authors.lastname: James
authors.affiliation: Department of Mathematics , McMaster University , Hamilton, Ontario, Canada L8S 4K1.
authors.forename: Dmitry
authors.initials: D
authors.lastname: Pelinovsky
keywords: Fermi–Pasta–Ulam lattices; Gaussian solitary waves; Hertzian interactions; compactons; granular chains; logarithmic Korteweg–de Vries equation