  Nonlinear wave propagation in parity-time ($\mathcal{PT}$) symmetric
localized potentials is investigated analytically near a phase-transition point
where a pair of real eigenvalues of the potential coalesce and bifurcate into
the complex plane. Necessary conditions for phase transition to occur are
derived based on a generalization of the Krein signature. Using multi-scale
perturbation analysis, a reduced nonlinear ODE model is derived for the
amplitude of localized solutions near phase transition. Above phase transition,
this ODE model predicts a family of stable solitons not bifurcating from linear
(infinitesimal) modes under a certain sign of nonlinearity. In addition, it
predicts periodically-oscillating nonlinear modes away from solitons. Under the
opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below
phase transition, solution dynamics is predicted as well. All analytical
results are compared to direct computations of the full system and good
agreement is observed.
