Title: Few-magnon excitations in a frustrated spin-$S$ ferromagnetic chain

Abstract: We study few-magnon excitations in a finite-size spin-$S$ ferromagnetic nearest-neighbor (NN) XXZ chain with additional antiferromagnetic next-nearest-neighbor (NNN) interaction $J'$ and single-ion (SI) anisotropy $D$. Using a set of exact two-magnon Bloch states, the two-magnon problem is mapped to a single-particle one on an effective open chain with both NN and NNN hoppings. For the commensurate momentum $k=-\pi$, the effective chain is decoupled into two NN open chains that can be exactly solved via a plane-wave ansatz. Based on this, we identify in the $\Delta'-D/|J'|$ plane (with $\Delta'$ the anisotropy parameter for the NNN coupling) the regions supporting the SI or NNN exchange two-magnon bound states near the edge of the band. We prove that there always exists a lower-energy NN exchange two-magnon bound state near the band edge. For $S=1/2$, we numerically calculate the $n$-magnon spectra for $n\leq5$ 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.