We study the existence and stability of solitons in the quadratic nonlinear media with spatially localized parity-time-(PT )-symmetric modulation of the linear refractive index. Families of stable one-and two-hump solitons are found. The properties of nonlinear modes bifurcating from a linear limit of small fundamental harmonic fields are investigated. It is shown that the fundamental branch, bifurcating from the linear mode of the fundamental harmonic is limited in power. The power maximum decreases with the strength of the imaginary part of the refractive index. The modes bifurcating from the linear mode of the second harmonic can exist even above the PT -symmetry-breaking threshold. We found that the fundamental branch bifurcating from the linear limit can undergo a secondary bifurcation colliding with a branch of two-hump soliton solutions. The stability intervals for different values of the propagation constant and gain or loss gradient are obtained. The examples of dynamics and excitations of solitons obtained by numerical simulations are also given.
We introduce a one-dimensional system combining the PT -symmetric complex periodic potential and the χ (2) (second-harmonic-generating) nonlinearity. The imaginary part of the potential, which represents spatially separated and mutually balanced gain and loss, affects only the fundamentalfrequency (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 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 PT 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.
The existence, spatial properties, and stability of localized modes in nonlinear quadratic materials with periodically modulated linear refractive index and quadratic nonlinearity (the latter having the zero mean value along the structure) are investigated. The branches of stationary solutions existing in the semi-infinite and three higher gaps are computed. It is shown that modes may possess different symmetries of the fundamental field (i.e., profiles described either by even or odd functions of the transverse coordinate), while requiring the second harmonic to be symmetric. We found that only small-amplitude gap solitons can be stable. The respective modes can bifurcate only from an edge of a total gap when it coincides with the band edge of the linear spectrum of the fundamental field. Moreover, in this case stable modes can be excited with their centers belonging to a slab with either higher or lower refractive index. This simultaneous existence of different stable localized modes can be viewed as a bistability of gap solitons. Examples of the dynamics and excitations of gap solitons are also given.
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