Localized modes in χ ( 2 ) media with non- PT -symmetric complex localized potentials

We study the existence and the stability of solitons in quadratic nonlinear media with spatially localized non-$\mathcal{PT}$-symmetric tunable modulation of the linear refractive index. The properties of nonlinear modes bifurcating from a linear limit of a small fundamental harmonic field are investigated. The modes bifurcating from the linear mode of the second harmonic may exist even above the real phase breaking threshold. The stability intervals for different values of the propagation constant and gain-loss gradient are obtained. Examples of dynamics and excitations of solitons obtained by numerical simulations are also given.