  Nonlinear phenomena are key for magnon-based information processing, but the
nonlinear interaction between two spin-wave signals requires their
spatio-temporal overlap which can be challenging for directional processing
devices. Our study focuses on a gallium-substituted yttrium iron garnet film,
which exhibits an exchange-dominated dispersion relation and thus provides a
particularly broad range of group velocities compared to pure YIG. Using time-
and space-resolved Brillouin light scattering spectroscopy, we demonstrate the
excitation of time-separated spin-wave pulses at different frequencies from the
same source, where the delayed pulse catches up with the previously excited
pulse and outruns it due to its higher group velocity. By varying the
excitation power of the faster pulse, the outcome can be finely tuned from a
linear superposition to a nonlinear interaction of both pulses, resulting in a
full attenuation of the slower pulse. Therefore, our findings demonstrate the
all-magnonic erasing process of a propagating magnonic signal, which enables
the realization of complex temporal logic operations with potential
application, e.g., in inhibitory neuromorphic functionalities.
