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

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