  In recent years, there has been considerable interest in the study of wave
propagation in nonlinear photonic lattices. The interplay between nonlinearity
and periodicity has led researchers to manipulate light and discover new and
interesting phenomena such as new classes of localized modes, usually referred
to as solitons, and novel surface states that propagate robustly. A field where
both nonlinearity and periodicity arises naturally is nonlinear optics. But
there are other areas where waves propagating on background lattices play an
important role, including photonic crystal fibers and Bose-Einstein
condensation. In this review article the propagation of wave envelopes in one
and two-dimensional periodic lattices associated with additional potential in
the nonlinear Schrodinger (NLS) equation, termed lattice NLS equations, are
studied. A discrete reduction, known as the tight-binding approximation, is
employed to find the linear dispersion relation and the equations governing
nonlinear discrete envelopes for two-dimensional simple periodic lattices and
two-dimensional non-simple honeycomb lattices. In the limit under which the
envelopes vary slowly, continuous envelope equations are derived from the
discrete system. The coefficients of the linear evolution system are related to
the dispersion relation in both the discrete and continuous cases. For simple
lattices, the continuous systems are NLS type equations. In honeycomb lattices,
in certain cases, the continuous system is found to be nonlinear Dirac
equations. Finally, it is possible to realize so-called topological insulator
systems in an optical waveguide setting. The modes supported by these systems
are associated with spectral topological invariants and, remarkably, can
propagate without backscatter from lattice defects.
