Particle-in-Cell Simulation of the Parametric Decay Instability of Alfv\'en Waves with Absorbing Boundary Conditions

The Alfv\'en wave parametric decay instability (PDI) facilitates energy transfer, plasma heating, and turbulence generation in space, astrophysical, and fusion plasmas. Most simulation studies of Alfv\'en wave PDI have focused on kinetic ions under periodic boundary conditions. Here, we present fully kinetic one-dimensional simulations (perpendicular wave-vector $k_\perp=0$) of the Alfv\'en wave PDI at low plasma beta using absorbing boundary conditions for the waves to understand the energy partition in an open system. For $\beta=5\times 10^{-4}$ and a normalized wave amplitude $\frac{\delta B}{B_0}=0.01$, nearly 92\% of the pump wave energy is transferred to the backward-propagating child Alfv\'en wave, and the remaining energy is partitioned between electrons ($\sim 1$-$2\%$) and ions ($\sim 6$-$7\%$). In the parameter regime considered, the ion and electron heating appears only when the PDI has sufficiently developed, and their rates are approximately twice the linear PDI growth rate, which roughly corresponds to the quadratic dependence of energy on the fluctuation amplitude. Furthermore, we find a qualitative agreement between theoretical and numerical growth rates over a range of plasma and wave parameters. This work establishes critical steps for future extension to finite $k_\perp$ waves in high dimensions, where stronger electron heating may be induced.