We report on detailed investigation of the stability of localized modes in
the nonlinear Schrodinger equations with a nonlinear parity-time (alias 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, what suggests that
the relation between width of the modes and spatial size of the complex
potential define the stability in the general case. These results were
confirmed using the direct propagation of the solutions for the mentioned
examples.Comment: 6 pages, 4 figures; accepted to Europhysics Letters,
https://www.epletters.net
