  Motivated by field-theoretic predictions we investigate the stable
excitations that exist in two characteristic gapped phases of a spin-1 model
with Ising-like and single-ion anisotropies. The sine-Gordon theory indicates a
region close to the phase boundary where a stable breather exists besides the
stable particles, that form the Haldane triplet at the Heisenberg isotropic
point. The numerical data, obtained by means of the Density Matrix
Renormalization Group, confirm this picture in the so-called large-D phase for
which we give also a quantitative analysis of the bound states using standard
perturbation theory. However, the situation turns out to be considerably more
intricate in the Haldane phase where, to the best of our data, we do not
observe stable breathers contrarily to what could be expected from the
sine-Gordon model, but rather only the three modes predicted by a novel
anisotropic extension of the Non-Linear Sigma Model studied here by means of a
saddle-point approximation.
