G. James and D. Pelinovsky
Gaussian solitary waves and compactons in FPU lattices with Hertzian potentials
Proceedings of Royal Society A 470, 20130465 (2014) (20 pages)
We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices,
consisting of a chain of particles coupled by fractional power nonlinearities
of order a > 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 analyze the propagation of localized waves
when a is 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 KdV equation, and possesses linearly orbitally stable
Gaussian solitary wave solutions. The second model consists of
a generalized KdV equation with Holder-continuous
fractional power nonlinearity and admits compacton solutions, i.e.
solitary waves with compact support. When a approaches 1, 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.
Discrete FPU lattice, granular crystals, log-KdV equation, Gaussian solitary waves, linear stability.