  The existence and stability of the nonlinear spatial localized modes are
investigated in parity-time symmetric optical media characterized by a generic
complex hyperbolic refractive index distribution with competing gain and loss
profile. The exact analytical expressions of the localized modes are found for
all values of the competing parameter and in the presence of both the
self-focusing and self-defocusing Kerr nonlinearity. The effect of competing
gain/loss profile on the stability structure of these localized modes are
discussed with the help of linear stability analysis followed by the direct
numerical simulation of the governing equation. The spatial localized modes in
two-dimensional geometry as well as the transverse power-flow density
associated with these localized modes are also examined.
