  We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam (FPU)
problem with a trilinear force-strain relation of soft-hard-soft type that is
in general non-symmetric. In addition to the classical spatially localized
solitary waves, such hardening-softening model also exhibits supersonic kinks
and finite-amplitude, spatially delocalized flat-top solitary waves that
acquire the structure of a kink-antikink bundle when their velocity approaches
the kink limit. Exploiting the fact that traveling waves are periodic modulo
shift by a lattice spacing, we compute these solutions as fixed points of the
corresponding nonlinear map and investigate how their properties depend on the
parameter measuring the asymmetry of the problem. In a particularly interesting
case when one of the soft regimes has zero elastic modulus, we obtain explicit
solutions for sufficiently slow solitary waves. In contrast to conventional
delocalization in the sonic limit, these compact structures mounted on a
constant background become localized at the lattice scale as their velocity
tends to zero. Numerical simulations of Riemann-type initial value problem in
this degenerate model show the emergence of Whitham shocks that involve
periodic trains of solitary waves. We investigate stability of the obtained
solutions using direct numerical simulations and Floquet analysis. We also
obtain explicit solutions for a quasicontinuum model that captures some
important features of the discrete problem.
