We study the existence and stability of solitons in the quadratic nonlinear media with spatially localized parity-time- ($\mathcal{PT}$)-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 $\mathcal{PT}$-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.

• Received 14 September 2012

DOI:https://doi.org/10.1103/PhysRevA.86.053815