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.

We predict the existence of rogue waves in Bose-Einstein condensates either loaded into a parabolic trap or embedded in an optical lattice. In the latter case, rogue waves can be observed in condensates with positive scattering length. They are immensely enhanced by the lattice. Local atomic density may increase up to tens times. We provide the initial conditions necessary for the experimental observation of the phenomenon. Numerical simulations illustrate the process of creation of rogue waves.

A self-consistent theory of a cylindrically shaped Bose-Einstein condensate (BEC) periodically modulated by a laser beam is presented. We show, both analytically and numerically, that modulational instability/stability is the mechanism by which wave functions of soliton type can be generated in a cylindrically shaped BEC subject to a one-dimensional optical lattice. The theory explains why bright solitons can exist in a BEC with positive scattering length and why condensates with negative scattering length can be stable and give rise to dark solitary pulses.

Switching on topology: Ways to control and design topological features in various systems are being studied intensively because the resulting properties tend to be robust against things such as scattering and defects, endowing the system with topological protection. Maczewsky et al. now look to another regime in optics to show that optical nonlinearity can induce a topological change in the properties of a photonic lattice. At low excitation power, probe light uniformly leaks into the rest of the lattice, an optically trivial phase. Above a threshold power, optical nonlinearity induces a topological change in the properties of the photonic lattice, and probe light is confined to propagate along the edge of the structure. These results illustrate a route to dynamically control the propagation of light. Science , this issue p. 701