Solitons in $\mathcal{P}\mathcal{T}$-symmetric periodic systems with the quadratic nonlinearity

Abstract

We introduce a one-dimensional system combining the $\mathcal{P}\mathcal{T}$-symmetric complex periodic potential and the ${\chi }^{(2)}$ (second-harmonic-generating) nonlinearity. The imaginary part of the potential, which represents spatially separated and mutually balanced gain and loss, affects only the fundamental-frequency (FF) wave, while the real potential acts on the second-harmonic (SH) component too. Soliton modes are constructed, and their stability is investigated (by means of the linearization and direct simulations) in semi-infinite and finite gaps in the corresponding spectrum, starting from the bifurcation which generates the solitons from edges of the gaps' edges. Families of solitons embedded into the conttinuous spectrum of the SH component are found too, and it is demonstrated that a part of the families of these \textit{embedded solitons} (ESs) is stable. The analysis is focused on effects produced by the variation of the strength of the imaginary part of the potential, which is a specific characteristic of the $\mathcal{P}\mathcal{T}$ system. The consideration is performed chiefly for the most relevant case of matched real potentials acting on the FF\ and SH components. The case of the real potential acting solely on the FF component is briefly considered too.

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Solitons in PT-symmetric nonlinear lattices
• A note on the PT-invariant periodic potential V(x)=4 cos^2 x + 4 i V_0 sin 2x
• Discrete solitons in PT-symmetric lattices
• Localized modes in ${\chi }^{(2)}$ media with $\PT$-symmetric localized potential
• Field patterns in periodically modulated optical parametric amplifiers and oscillators

Cited by in corpus (16)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Stability boundaries and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity
• Families of stationary modes in complex potentials
• Parity-Time Anti-Symmetric Parametric Amplifier
• Discrete solitons and vortices on two-dimensional lattices of $\mathcal{P}\mathcal{T}$-symmetric couplers
• A solvable model for solitons pinned to a PT-symmetric dipole
• Stationary modes and integrals of motion in nonlinear lattices with PT-symmetric linear part
• $\mathcal{P}\mathcal{T}$-symmetric coupler with ${\chi }^{(2)}$ nonlinearity
• Floquet Edge Multicolor Solitons
• CP symmetry in optical systems
• Quasi-energies, parametric resonances, and stability limits in ac-driven $\mathcal{P}\mathcal{T}$-symmetric systems
• Fundamental and vortex dissipative quadratic solitons supported by spatially localized gain
• Dissipative quadratic solitons supported by a localized gain
• A PT-symmetric dual-core system with the sine-Gordon nonlinearity and derivative coupling
• A systematic construction of parity-time ($\mathcal{P}\mathcal{T}$)-symmetric and non-$\mathcal{P}\mathcal{T}$-symmetric complex potentials from the solutions of various real nonlinear evolution equations
• Quantized transport of two-dimensional multifrequency solitons