  Following up on a proposal to use four-wave mixing in an underdense plasma at
mildly relativistic laser intensities to produce vastly more energetic x-ray
pulses [V. M. Malkin and N. J. Fisch, Phys. Rev. E, 101, 023211 (2020)], we
perform the first numerical simulations in one dimension to demonstrate
amplification of a short high frequency seed through four-wave mixing. We find
that parasitic processes including phase modulation and spatial pulse slippage
limit the amplification efficiency. We numerically explore the previously
proposed "dual seed" configuration as a countermeasure against phase
modulation. We show how this approach tends to be thwarted by longitudinal
slippage. In the examples we considered, the best performance was in fact
achieved through optimization of signal and pump parameters in a "single seed"
configuration.
