Nonlinear interaction between three kinetic Alfv\'{e}n waves

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.