  We investigate the nonlinear scattering dynamics in interacting atomic
Bose-Einstein condensates under non-Hermitian dissipative conditions. We show
that by carefully engineering a momentum-dependent atomic loss profile one can
achieve matter-wave amplification through four wave mixing in a one-dimensional
quasi free-space setup - a process that is forbidden in the counterpart
Hermitian systems due to energy mismatch. Additionally, we show that similar
effects lead to rich nonlinear dynamics in higher dimensions. Finally, we
propose a physical realization for selectively tailoring the momentum-dependent
atomic dissipation. Our strategy is based on a two step process: (i) exciting
atoms to narrow Rydberg- or metastable excited states, and (ii) introducing
loss through recoil; all while leaving the bulk condensate intact due to
protection by quantum interference.
