We introduce the simplest one-dimensional nonlinear model with the 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 point-like (delta-fu
nctional) gain-loss dipole {delta}^{prime}(x), combined with the usual attractive potential {delta}(x). The nonlinearity is represented by self-focusing (SF) or self-defocusing (SDF) 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 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}^{prime}- and {delta}- functions replaced by their Lorentzian regularization. With the increase of the dipoles strength, {gamma}, 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. In the case of the SDF uniform nonlinearity, the pinned modes are stable, keeping the single-peak shape.
We introduce the two-dimensional Gross-Pitaevskii/nonlinear-Schrodinger (GP/NLS) equation with the self-focusing nonlinearity confined to two identical circles, separated or overlapped. The model can be realized in terms of Bose-Einstein condensates
(BECs) and photonic-crystal fibers. Following the recent analysis of the spontaneous symmetry breaking (SSB) of localized modes trapped in 1D and 2D double-well nonlinear potentials (also known as pseudopotentials), we aim to find 2D solitons in the two-circle setting, using numerical methods and the variational approximation (VA). Well-separated circles support stable symmetric and antisymmetric solitons. The decrease of separation L between the circles leads to destabilization of the solitons. The symmetric modes undergo two SSB transitions. First, they are transformed into weakly asymmetric breathers, which is followed by a transition to single-peak modes trapped in one circle. The antisymmetric solitons perform a direct transition to the single-peak mode. The symmetric solitons are described reasonably well by the VA. For touching (L=0) and overlapping (L<0) circles, single-peak solitons are found-asymmetric ones, trapped in either circle, and symmetric solitons centered at the midpoint of the bi-circle configuration. If the overlap is weak, the symmetric soliton is unstable. It may spontaneously leap into either circle and perform shuttle motion in it. A region of stability of the symmetric solitons appears with the increase of the overlap degree. In the case of a moderately strong overlap, another SSB effect is found, in the form of a pair of symmetry-breaking and restoring bifurcations which link families of the symmetric and asymmetric solitons.
