Multispin-wave contributions to the excitation spectrum of one-dimensional easy-plane antiferromagnets, (CH3)4NMnCl3

(${\mathrm{C}{\mathrm{H}}_3)}_4$NMn${\mathrm{Cl}}_3$ (TMMC) is a one-dimensional easy-plane antiferromagnet which, when subjected to a transverse applied magnetic field, is a realization of a sine-Gordon field theory. Inelastic neutron scattering experiments and NMR relaxation measurements have shown that the magnetic excitation spectrum has a $\omega \sim 0$ central peak, which has been associated with the soliton excitations. In this paper we examine the spin-wave contributions to the excitation spectrum. We find multispin-wave excitations that contribute to the central peak. However, we find that these multispin-wave excitations are not in agreement with the experimental data, which require an additional soliton contribution to describe them. The multispin-wave excitations also give rise to inelastic excitation spectra. For most values of the applied field, the anharmonic interactions in TMMC modify the inelastic excitations so as to resemble the sine-Gordon excitations, such as breathers, etc. This resemblance ceases for special values of the applied field which produces a resonance between the inelastic two-spin-wave excitations and the zone-boundary spin wave. This purely anharmonic effect, which has been observed experimentally, is well described by our results.