Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials

We investigate the generalized (n+1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity in PT-symmetric potentials, and derive two families of analytical sech-type and Gaussian-type localized modes (soliton solutions). Based on these analytical solutions, the powers, power-flow densities and the phase jumps are analyzed. The linear stability analysis and the direct numerical simulation for these exact solutions indicate that sech-type and Gaussian-type solutions are both stable below some thresholds for the imaginary part of PT-symmetric potentials (except for the extended Rosen-Morse potential) in the focusing power-law nonlinear medium, while they are always unstable in the defocusing power-law nonlinear medium. The gain (loss) related to the values of the imaginary part of the potential (Wn) should be enough small compared with the fixed value of the real part of the potential (V0) in order to ensure the stability of exact solutions.