  We experimentally study nonlinear propagation of spin waves in microscopic
yttrium iron garnet waveguides, where the dispersion spectrum is engineered to
enable efficient four-magnon interactions over a wide range of wavelengths. We
show that under these conditions, the initial monochromatic spin wave
nonlinearly generates co-propagating spin waves with well-defined, discrete
frequencies. This process is characterized by a low energy threshold and can be
observed in a wide range of frequencies and excitation powers. Thanks to the
engineered dispersion, the process allows the generation of waves with short
wavelengths that cannot be excited directly by a linear excitation mechanism.
The nonlinearly generated short-wavelength spin waves continuously acquire the
energy from the initial pump wave during co-propagation, which results in
compensation of their propagation losses over significant distances. The
observed phenomena can be used to implement frequency- and
wavelength-conversion operations in magnonic nanodevices and circuits.
