  We study the non-resonant (non-$\phi$) production of $K^+K^-$ pairs by
protons of 2.83 GeV kinetic energy on C, Cu, Ag, and Au targets within the
collision model, based on the nuclear spectral function, for incoherent primary
proton--nucleon and secondary pion--nucleon creation processes. The model takes
into account the initial proton and final kaon absorption, target nucleon
binding and Fermi motion as well as nuclear mean-field potential effects on
these processes. We calculate the antikaon momentum dependences of the
exclusive absolute and relative $K^+K^-$ pair yields in the acceptance window
of the ANKE magnetic spectrometer, used in a recent experiment performed at
COSY, within the different scenarios for the antikaon-nucleus optical
potential. We demonstrate that the above observables are strongly sensitive to
this potential. Therefore, they can be useful to help determine the $K^-$
optical potential from the direct comparison of the results of our calculations
with the data from the respective ANKE-at-COSY experiment. We also show that
the pion--nucleon production channels dominate in the low-momentum $K^-$, $K^+$
production in the considered kinematics and, hence, they have to be accounted
for in the analysis of these data.
