$\mathcal{PT}$-symmetric systems are open systems where dissipative losses are exactly compensated by symmetrically arranged energy gain; that is, they preserve space reflection $\mathcal{P}$ and time-reversal along with complex conjugation $\mathcal{T}$ symmetries simultaneously: $\mathcal{PT} u(x,t) = \overline{u(-x,-t)}$. Such systems appeared in quantum mechanics as an attempt to relax Hermiticity property of operators describing particles [1]. A $\mathcal{PT}$-symmetric operator can have a real spectrum in certain regime, despite not being Hermitian, and can describe unitary time evolution, both essential properties in quantum mechanics. Many applications of $\mathcal{PT}$-symmetric operators appear in nonlinear optics, where $\mathcal{PT}$-symmetric systems have been observed experimentally [2]. The majority of discovered $\mathcal{PT}$-symmetric equations are variations of the NLS equation.

[1]: C.M. Bender, Making Sense of Non-Hermitian Hamiltonians, Rep. Prog. Phys. 70, 947–1018, 2007.

[2]: C.E. Ruter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev and D. Kip, Observation of parity–time symmetry in optics, Nat. Phys. 6, 192-195, 2010.