  In this paper, weak turbulence theory is used to investigate the nonlinear
evolution of the parametric instability in 3D low-$\beta$ 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 Alfv\'en wave quanta, and
several exact solutions to the wave kinetic equations are presented. The main
results of the paper concern the parametric decay of Alfv\'en waves that
initially satisfy $e^+ \gg e^-$, where $e^+$ and $e^-$ are the frequency
($\,f$) spectra of Alfv\'en waves propagating in opposite directions along the
magnetic field lines. If $e^+$ initially has a peak frequency $f_0$ (at which
$f e^+$ is maximized) and an "infrared" scaling $f^p$ at smaller $f$ with $-1 <
p < 1$, then $e^+$ acquires an $f^{-1}$ scaling throughout a range of
frequencies that spreads out in both directions from $f_0$. At the same time,
$e^-$ acquires an $f^{-2}$ scaling within this same frequency range. If the
plasma parameters and infrared $e^+$ spectrum are chosen to match conditions in
the fast solar wind at a heliocentric distance of 0.3 astronomical units (AU),
then the nonlinear evolution of the parametric instability leads to an $e^+$
spectrum that matches fast-wind measurements from the Helios spacecraft at 0.3
AU, including the observed $f^{-1}$ scaling at $f \gtrsim 3 \times 10^{-4}
\mbox{ Hz}$. The results of this paper suggest that the $f^{-1}$ spectrum seen
by Helios in the fast solar wind at $f \gtrsim 3\times 10^{-4} \mbox{ Hz}$ is
produced in situ by parametric decay and that the $f^{-1}$ range of $e^+$
extends over an increasingly narrow range of frequencies as $r$ decreases below
0.3 AU. This prediction will be tested by measurements from the Parker Solar
Probe.
