  We perform a systematic DMRG investigation of the two natural spin-1
generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 $J_1-J_2$
Heisenberg chain, where $J_2$ is a next-nearest neighbor Heisenberg coupling,
and the spin-1 $J_1-J_3$ model, where $J_3$ refers to a three-site interaction
defined by $J_3\left[({\bf S}_{i-1}\cdot {\bf S}_i)({\bf S}_i\cdot {\bf
S}_{i+1})+\mathrm {H.c.}\right]$. Although both models are rigorously
equivalent to the Majumdar-Ghosh chain for spin-1/2, their physics appears to
be quite different for spin 1. Indeed, when all couplings are
antiferromagnetic, the spin-1 $J_1-J_2$ model undergoes an effective decoupling
into two next-nearest neighbour (NNN) Haldane chains upon increasing $J_2$,
while the $J_1-J_3$ chain undergoes a spontaneous dimerization similar to the
spin-1/2 Majumdar-Ghosh chain upon increasing $J_3$. By extending the phase
diagram to all signs of the couplings, we show that both the dimerized and the
NNN-Haldane phase are actually present in the $J_1-J_3$ model, the former one
adjacent to the Haldane one, the latter one to the ferromagneric one, with an
Ising transition between them. By contrast, the $J_1-J_2$ chain only has a
NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for
positive $J_2$. In both cases, our DMRG data are consistent with a continuous
Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic
phases.
