Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear $\mathcal{P}\mathcal{T}$-symmetric defect

Abstract

Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded $\mathcal{P}\mathcal{T}$-symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and loss. Fundamental solitons in tightly knit lattices, as well as all dipole modes, exist above a finite threshold value of the total power. However, the threshold vanishes for fundamental solitons in loosely knit lattices. The stability of the discrete solitons is investigated analytically by means of the Vakhitov-Kolokolov (VK) criterion, and, in the full form, via the computation of eigenvalues for perturbation modes. Fundamental and dipole solitons tend to be stable at smaller and larger values of the total power (norm), respectively. The increase of the strength of the coupling between the two defect-forming sites leads to strong expansion of the stability areas. The scattering problem for linear lattice waves impinging upon the defect is considered too.

arXiv comment11 pages, 8 figures

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• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
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• Families of stationary modes in complex potentials
• Revisiting the optical $PT$-symmetric dimer
• Discrete solitons in self-defocusing systems with $\mathcal{P}\mathcal{T}$-symmetric defects
• Staggered $\mathcal{P}\mathcal{T}$-symmetric ladders with cubic nonlinearity
• Dynamical evolutions in non-Hermitian triple-well system with complex potential
• Spatiotemporal solitary modes in a twisted cylinder waveguide pipe with the self-focusing Kerr nonlinearity
• Emergence of Exceptional Points in Periodic Metastructures with Hidden PT-symmetric Defects
• Self trapping transition for a nonlinear impurity within a linear chain