Localized modes in a nonlinear medium with a PT-symmetric dipole

We introduce the simplest one-dimensional model of a nonlinear system with the parity-time (PT ) symmetry, which makes it possible to find numerical solutions for localized modes (“solitons”). The PT -symmetric element is represented by a point-like (delta-functional) gain-loss dipole ~ δ'(x), combined with the usual attractive potential ~ δ(x). The nonlinearity is represented by self-focusing (SF) Kerr terms, both spatially uniform and localized ones. The system can be implemented in planar optical waveguides. For the sake of comparison, also introduced is the model with separated δ-functional gain and loss, embedded into the linear medium and combined with the δ-localized Kerr nonlinearity and attractive potential. Full analytical solutions for pinned modes are found in both models. The numerical solutions are obtained in the gain-loss-dipole model with the δ'- and δ- functions replaced by their Lorentzian regularization. With the increase of the dipole's strength, γ, the single-peak shape of the numerically found mode, supported by the uniform SF nonlinearity, transforms into a double-peak one. This transition coincides with the onset of the escape instability of the pinned soliton.