  We present a theoretical study of the scattering of spin waves by a domain
wall (DW) in a ferrimagnetic (FiM) spin chain in which two sublattices carry
spins of unequal magnitudes. We find that a narrow, but atomically smooth FiM
DW exhibits a different behavior in comparison with similarly smooth
ferromagnetic and antiferromagnetic DWs due to the inequivalence of the two
sublattices. Specifically, for sufficiently weak anisotropy, the smaller spin
at the center of the DW is found to become precisely normal to the easy-axis,
selecting an arbitrary direction in the $xy$-plane and thereby breaking the
U(1) spin-rotational symmetry spontaneously. This particular form of a FiM DW
does not occur in antiferromagnetic systems and is shown to lead to a strong
dependence of spin wave scattering pattern on the state of polarization of the
spin wave, which can be either right-handed or left-handed, suggesting the
utilization of such a narrow DW as a spin-wave filter. Moreover, we find that
in the case of an atomically sharp DW, where all the spins point either up or
down due to strong easy-axis anisotropy and therefore the polarization of the
spin wave is conserved upon transmission, the wave vector of the spin wave
changes after passing through the DW leading to a change in the group velocity
of the spin wave. This change of the wave vector indicates the acceleration or
deceleration of the spin waves and thus a sharp FiM DW could serve as a spin
wave accelerator or decelerator in spintronics devices, offering a
functionality absent in a ferromagnetic and an antiferromagnetic counterpart.
Our results indicate that FiM spin textures can interact with spin waves
distinctly from ferromagnetic and antiferromagnetic counterparts, suggesting
that they may offer spin-wave functionalities that are absent in more
conventional magnets.
