  A quantum system in complex potentials obeying parity-time (PT ) symmetry
could exhibit all real spectra, starting out in non-Hermitian quantum
mechanics. The key physics behind a PT-symmetric system consists of the
balanced gain and loss of the complex potential. In this work, we plan to
include the nonequilibrium nature, i.e. the intrinsic kinds of gain and loss of
a system, to a PT-symmetric many-body quantum system, with the emphasis on the
combined effects of non-Hermitian due to nonequilibrium nature and PT symmetry
in determining the properties of a system. In this end, we investigate the
static and dynamical properties of a dark soliton of a polariton Bose-Einstein
condensate under the PT-symmetric non-resonant pumping by solving the
driven-dissipative GrossPitaevskii equation both analytically and numerically.
We derive the equation of motion for the center of mass of the dark solitons
center analytically with the help of the Hamiltonian approach. The resulting
equation captures how the combination of the open-dissipative character and
PT-symmetry affects the properties of dark soliton, i.e. the soliton relaxes by
blending with the background at a finite time. Further numerical solutions are
in excellent agreement with the analytical results.
