Higher-dimensional localized mode families in parity-time-symmetric potentials with competing nonlinearities

Both two-dimensional and three-dimensional localized mode families in different parity-time (PT)-symmetric potentials with competing nonlinearities are investigated. We show that localized mode families described by a (2 + 1)-dimensional nonlinear Schrödinger equation in the extended complex PT-symmetric Rosen-Morse potential wells are unstable for all parameters due to the residue of gain (loss) in the system from the nonvanishing imaginary part in the extended Rosen-Morse potentials. In the extended hyperbolic Scarf II potentials, spatial localized modes are stable only for the defocusing cubic and focusing quintic nonlinearities. In this case, the gain (loss) should also be small enough for a certain real part of the PT-symmetric potential; otherwise, localized modes eventually lead to instability. These results have been verified by linear stability analysis from analytical solutions and direct numerical simulation of the governing equation. The phase switch, power, and power-flow density associated with these fundamental localized modes have also been examined. Moreover, the spatial and spatiotemporal localized mode families are presented, and the corresponding stability analysis for these solutions is also carried out.