Stability of solitons in

We report on detailed investigation of the stability of localized modes in the nonlinear Schrödinger equations with a nonlinear parity-time (alias ${\cal{PT}} $) symmetric potential. We are particularly focusing on the case where the spatially dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and tanh-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, which suggests that the relation between width of the modes and spatial size of the complex potential defines the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples