  We study few-magnon excitations in a finite-size spin-$S$ chain with
ferromagnetic nearest-neighbor (NN) interaction $J>0$ and antiferromagnetic
next-nearest-neighbor (NNN) interaction $J'<0$, in the presence of the
single-ion (SI) anisotropy $D$. We first reveal the condition for the emergence
of zero-excitation-energy states. In the isotropic case with $\Delta=\Delta'=1$
($\Delta$ and $\Delta'$ are the corresponding anisotropy parameters), a
threshold of $J/|J'|$ above which the ground state is ferromagnetic is
determined by exact diagonalization for short chains up to $12$ sites. Using a
set of exact two-magnon Bloch states, we then map the two-magnon problem to a
single-particle one on an effective open chain with both NN and NNN hoppings.
The whole two-magnon excitation spectrum is calculated for large systems and
the commensurate-incommensurate transition in the lowest-lying mode is found to
exhibit different behaviors between $S=1/2$ and higher spins due to the
interplay of the SI anisotropy and the NNN interaction. For the commensurate
momentum $k=-\pi$, the effective lattice is decoupled into two NN open chains
that can be exactly solved via a plane-wave ansatz. Based on this, we
analytically identify in the $\Delta'-D/|J'|$ plane the regions supporting the
SI or NNN exchange two-magnon bound states near the edge of the band. In
particular, we prove that there always exists a lower-lying NN exchange
two-magnon bound state near the band edge for arbitrary $S\geq 1/2$. Finally,
we numerically calculate the $n$-magnon spectra for $S=1/2$ with $n\leq5$ by
using a spin-operator matrix element method. The corresponding $n$-magnon
commensurate instability regions are determined for finite chains and
consistent results with prior literature are observed.
