A solvable model for solitons pinned to a PT-symmetric dipole

TL;DR

This paper introduces an analytically solvable one-dimensional PT-symmetric nonlinear model with a point-like gain-loss dipole, revealing exact localized mode solutions and their stability properties relevant for optical waveguides.

Contribution

It presents the first exact analytical solutions for PT-symmetric localized modes with a delta-function dipole, including stability analysis and comparison with regularized numerical solutions.

Findings

Exact solutions for pinned modes are derived in both models.

Mode shape transitions occur with increasing dipole strength, affecting stability.

Pinned modes with SDF nonlinearity are stable, while SF modes can become unstable.

Abstract

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-functional) 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 dipole's 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.