  We consider a dimer lattice of the Fermi-Pasta-Ulam-Tsingou (FPUT) type,
where alternating linear couplings have a controllably small difference, and
the cubic nonlinearity ($\beta$-FPUT) is the same for all interaction pairs. We
use a weakly nonlinear formal reduction within the lattice bandgap to obtain a
continuum, nonlinear Dirac-type system. We derive the Dirac soliton profiles
and the model's conservation laws analytically. We then examine the cases of
the semi-infinite and the finite domains and illustrate how the soliton
solutions of the bulk problem can be ``glued'' to the boundaries for different
types of boundary conditions. We thus explain the existence of various kinds of
nonlinear edge states in the system, of which only one leads to the standard
topological edge states observed in the linear limit. We finally examine the
stability of bulk and edge states and verify them through direct numerical
simulations, in which we observe a solitary wave setting into motion due to the
instability.
