  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 $\alpha >1$. 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
$\alpha$ 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
H\"older-continuous fractional power nonlinearity and admits compacton
solutions, i.e. solitary waves with compact support. When $\alpha \rightarrow
1^+$, 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.
