Solitons in a nonlinear Schrödinger equation with PT -symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes

We report branches of explicit expressions for nonlinear modes in parity-time $(\mathcal{PT})$-symmetric potentials of several types. For the single-well and double-well potentials the found solutions are two-parametric and appear to be stable even when the $\mathcal{PT}$ symmetry of respective underlying linear models is broken. Based on the examples of these solutions we describe an algorithm of excitation of a stable nonlinear mode in a model whose linear limit is unstable. The method is based on the adiabatic change of the control parameter driving the mode along a branch bifurcating from a stable linear mode. The suggested algorithm is confirmed by extensive numerical simulations.