Solvable model for solitons pinned to a parity-time-symmetric dipole

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