Recent progress on nonlinear properties of parity-time ($\mathcal{PT}$)-symmetric systems is comprehensively reviewed in this article. $\mathcal{PT}$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\mathcal{PT}$ symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a $\mathcal{PT}$-symmetric system. The natural inclusion of nonlinearity into these $\mathcal{PT}$ systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above $\mathcal{PT}$-symmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear $\mathcal{PT}$-symmetric systems arising from various physical disciplines are presented, nonlinear properties of these systems are thoroughly elucidated, and relevant experimental results are described. In addition, emerging applications of $\mathcal{PT}$ symmetry are pointed out.

The existence of localized modes supported by the PT-symmetric nonlinear lattices is reported. The system considered reveals unusual properties: unlike other typical dissipative systems, it possesses families (branches) of solutions, which can be parametrized by the propagation constant; relatively narrow localized modes appear to be stable, even when the conservative nonlinear lattice potential is absent; and finally, the system supports stable multipole solutions.

By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schr\"odinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra does not imply similarity of the structure or stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. In addition, the stability is not directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use a graph representation of PT-symmetric networks allowing for a simple illustration of linearly equivalent networks and indicating their possible experimental design.

We study the families of nonlinear modes described by the nonlinear Schr\"odinger equation with the PT-symmetric harmonic potential x2$-${}2i$\alpha${}x. The nonlinear modes found display a number of interesting features. In particular, we have observed that modes bifurcating from different eigenstates of the underlying linear problem can actually belong to the same family of nonlinear modes. We also show that by proper adjustment of the coefficient $\alpha${} it is possible to enhance the stability of small-amplitude and strongly nonlinear modes compared to the well-studied case of the real harmonic potential.

Eliminating atoms of a Bose-Einstein condensate in a lattice from one cell is coherent perfect absorption of the quantum liquid.

We theoretically demonstrate the possibility of observing the macroscopic Zeno effect for nonlinear waveguides with localized dissipation. We show the existence of stable stationary flows, which are balanced by losses in the dissipative domain. The macroscopic Zeno effect manifests itself in the nonmonotonic dependence of the stationary flow on the strength of the dissipation. In particular, we highlight the importance of the dissipation parameters in observing the phenomenon. Our results are applicable to a large variety of systems, including the condensates of atoms or quasiparticles and optical waveguides.