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

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 δ^{\prime}(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 ones. 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^{\prime}- 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 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.

21 pages, 11 figures, Physical Review E, in press

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Cited by in corpus (21)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Nonlinear switching and solitons in PT-symmetric photonic systems
• Families of stationary modes in complex potentials
• Discrete solitons and vortices on two-dimensional lattices of $\mathcal{P}\mathcal{T}$-symmetric couplers
• Nonlinear quantum dynamics in a PT-symmetric double well
• Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear $\mathcal{P}\mathcal{T}$-symmetric defect
• Cusp bifurcation in the eigenvalue spectrum of PT-symmetric Bose-Einstein condensates
• Analytical stable Gaussian soliton supported by a parity-time-symmetric potential with power-law nonlinearity
• Discrete solitons in self-defocusing systems with $\mathcal{P}\mathcal{T}$-symmetric defects
• Attraction centers and PT-symmetric delta-functional dipoles in critical and supercritical self-focusing media
• Interactions of bright and dark solitons with localized $\mathcal{P}\mathcal{T}$-symmetric potentials
• Quasi-energies, parametric resonances, and stability limits in ac-driven $\mathcal{P}\mathcal{T}$-symmetric systems
• Edge and bulk dissipative solitons in modulated PT-symmetric waveguide arrays
• Fundamental and vortex dissipative quadratic solitons supported by spatially localized gain
• Dissipative quadratic solitons supported by a localized gain
• Exactly solvable $\mathcal{P}\mathcal{T}$-symmetric models in two dimensions
• Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity
• $\mathcal{P}\mathcal{T}$-symmetric currents of a Bose-Einstein condensate in a triple well
• Small-amplitude nonlinear modes under the combined effect of the parabolic potential, nonlocality and $\mathcal{P}\mathcal{T}$ symmetry
• Higher-dimensional soliton generation, stability and excitations of the PT-symmetric nonlinear Schrödinger equations
• Mini-review: Spatial solitons supported by localized gain