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

Thawatchai Mayteevarunyoo,Boris A Malomed,Athikom Reoksabutr

doi:10.1103/physreve.88.022919

Abstract

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 (δ-functional) gain-loss dipole ~δ'(x), combined with the usual attractive potential ~δ(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 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 exact solutions are compared with numerical counterparts, which 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. 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.

Highlights

A great deal of interest has arisen in physical systems featuring parity-time (PT ) symmetry [1,2,3], i.e., dissipative quantum or wave systems with antisymmetry between spatially separated gain and loss
The optical realizations of PT symmetry suggest additional interest in nonlinearity in these systems [8], where stable solitons can be supported by a combination of the Kerr nonlinearity and a spatially periodic complex potential, whose odd imaginary part accounts for the balanced gain and loss, as mentioned above
The objective of this work is to introduce a solvable model of a nonlinear PT -symmetric medium, in which the gain-loss combination is represented by a pointlike dipole ∼δ (x), which is embedded into a uniform Kerr-nonlinear SF or SDF medium, in combination with a linear and/or nonlinear potential pinning the wave field to the PT dipole

Summary

Introduction

A great deal of interest has arisen in physical systems featuring parity-time (PT ) symmetry [1,2,3], i.e., dissipative quantum or wave systems with antisymmetry between spatially separated gain and loss. The optical realizations of PT symmetry suggest additional interest in nonlinearity in these systems [8], where stable solitons can be supported by a combination of the Kerr (cubic) nonlinearity and a spatially periodic complex potential, whose odd (antisymmetric) imaginary part accounts for the balanced gain and loss, as mentioned above. The stability of such solitons was rigorously analyzed in Ref. Bright solitons were predicted too in PT -symmetric systems with quadratic (second-harmonicgenerating) nonlinearity [11]

Objectives

Results

Conclusion