  Using Hall-MHD theory, we focus on the resonant interaction between three
kinetic Alfv\'{e}n waves. We thus derive three coupled equations which govern
this process. It turns out that these equations contain the same coupling
coefficient, directly showing that they satisfy the Manley-Rowe relations. The
coupling coefficient can be written in a comparatively very simple form, that
has not been deduced before. The decay rate, when a pump kinetic Alfv\'{e}n
wave decays into two similar Alfv\'{e}n waves, is therefore easily estimated
for plasmas of astrophysical interest.
