In this paper, weak-turbulence theory is used to investigate the nonlinear evolution of the parametric instability in three-dimensional low-
plasmas at wavelengths much greater than the ion inertial length under the assumption that slow magnetosonic waves are strongly damped. It is shown analytically that the parametric instability leads to an inverse cascade of Alfvén wave quanta, and several exact solutions to the wave kinetic equations are presented. The main results of the paper concern the parametric decay of Alfvén waves that initially satisfy
are the frequency (
) spectra of Alfvén waves propagating in opposite directions along the magnetic field lines. If
initially has a peak frequency
is maximized) and an ‘infrared’ scaling
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