  Two different Perfectly Matched Layer (PML) formulations with efficient
pseudo-spectral numerical schemes are derived for the standard and
non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral
explicit exponential integrator scheme for a first-order formulation and a
linearly implicit preconditioned finite-difference scheme for a second-order
formulation are proposed and analyzed. To obtain a high spatial accuracy, new
regularized Berm\'udez type absorption profiles are introduced for the PML. It
is shown that the two schemes are efficient, but the linearly implicit scheme
should be preferred for accuracy purpose when used within the framework of
pseudo-spectral methods combined with the regularized Berm\'udez type
functions. In addition, in the non-relativistic regime, numerical examples lead
to the conclusion that the error related to regularized Berm\'udez type
absorption functions is insensitive to the small parameter $\varepsilon$
involved in the NKGE. The paper ends by a two-dimensional example showing that
the strategy extends to the rotating NKGE where the vortex dynamics is very
well-reproduced.
