  Solitons are of the important significant in many fields of nonlinear science
such as nonlinear optics, Bose-Einstein condensates, plamas physics, biology,
fluid mechanics, and etc.. The stable solitons have been captured not only
theoretically and experimentally in both linear and nonlinear Schrodinger (NLS)
equations in the presence of non-Hermitian potentials since the concept of the
parity-time (PT)-symmetry was introduced in 1998. In this paper, we present
novel bright solitons of the NLS equation with third-order dispersion in some
complex PT-symmetric potentials (e.g., physically relevant PT-symmetric
Scarff-II-like and harmonic-Gaussian potentials). We find stable nonlinear
modes even if the respective linear PT-symmetric phases are broken. Moreover,
we also use the adiabatic changes of the control parameters to excite the
initial modes related to exact solitons to reach stable nonlinear modes. The
elastic interactions of two solitons are exhibited in the third-order NLS
equation with PT-symmetric potentials. Our results predict the dynamical
phenomena of soliton equations in the presence of third-order dispersion and
PT-symmetric potentials arising in nonlinear fiber optics and other physically
relevant fields.
