  In this work, we study the dynamics of an infinite array of nonlinear dimer
oscillators which are linearly coupled as in the classical model of Su,
Schrieffer and Heeger (SSH). The ratio of in-cell and out-of-cell couplings of
the SSH model defines distinct $\textit{phases}$: topologically trivial and
topologically non-trivial. We first consider the case of weak out-of-cell
coupling, corresponding to the topologically trivial regime for linear SSH; for
any prescribed isolated dimer frequency, $\omega_b$, which satisfies
non-resonance and non-degeneracy assumptions, we prove that there are discrete
breather solutions for sufficiently small values of the out-of-cell coupling
parameter. These states are $2\pi/\omega_b$- periodic in time and exponentially
localized in space. We then study the global continuation with respect to this
coupling parameter. We first consider the case where $\omega_b$, the seeding
discrete breather frequency, is in the (coupling dependent) phonon gap of the
underlying linear infinite array. As the coupling is increased, the phonon gap
decreases in width and tends to a point (at which the topological transition
for linear SSH occurs). In this limit, the spatial scale of the discrete
breather grows and its amplitude decreases, indicating the weakly nonlinear
long wave regime. Asymptotic analysis shows that in this regime the discrete
breather envelope is determined by a vector gap soliton of the limiting
envelope equations. We use the envelope theory to describe discrete breathers
for SSH- coupling parameters corresponding to topologically trivial and, by
exploiting an emergent symmetry, topologically nontrivial regimes, when the
spectral gap is small. Our asymptotic theory shows excellent agreement with
extensive numerical simulations over a wide range of parameters.
