  The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine
thermalization in non-metallic solids and develop ``experimental'' techniques
for studying nonlinear problems, continues to yield a wealth of results in the
theory and applications of nonlinear Hamiltonian systems with many degrees of
freedom. Inspired by the studies of this seminal model, solitary-wave dynamics
in lattice dynamical systems have proven vitally important in a diverse range
of physical problems--including energy relaxation in solids, denaturation of
the DNA double strand, self-trapping of light in arrays of optical waveguides,
and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular,
due to their widely ranging and easily manipulated dynamical apparatuses--with
one to three spatial dimensions, positive-to-negative tuning of the
nonlinearity, one to multiple components, and numerous experimentally
accessible external trapping potentials--provide one of the most fertile
grounds for the analysis of solitary waves and their interactions. In this
paper, we review recent research on BECs in the presence of deep periodic
potentials, which can be reduced to nonlinear chains in appropriate
circumstances. These reductions, in turn, exhibit many of the remarkable
nonlinear structures (including solitons, intrinsic localized modes, and
vortices) that lie at the heart of the nonlinear science research seeded by the
FPU paradigm.
