The combined effect of dissipation with gain on dynamics of a physical system is a fundamental issue either in classical physics or in quantum theories. Recently, the renewed interest in systems with gain and losses was triggered by the discovery of the so-called parity-time (PT) symmetry. In its original understanding the PT-symmetry is a property of a system to be invariant under a combined action of a spatial reflection and time-reversal. Interest in PT-symmetric systems was stimulated by the observation that non-Hermitian PT-symmetric systems may possess purely real spectrum. This originated intensive discussion about possibilities of non-Hermitian extension of the quantum mechanics.In the meanwhile, it was suggested that PT-symmetry has natural applications in nonlinear physics, and, in particular, in optics and in the theory of Bose-Einstein condensates (BECs). Presence of nonlinearity essentially enriches the problem and originates completely novel behavior. We address several fundamental issues of applications of PT-symmetry in nonlinear systems. We consider nonlinear modes in PT-symmetric waveguides and study their existence, stability, bifurcations, and other interesting properties introduced by the nonlinearity. Our interest is related to both discrete PT-symmetric lattices (networks) and spatially continuous systems. We pay particular attention to the phenomena that are only possible as a result of the interplay between nonlinearity and PT symmetry. One of such unusual features is the possibility of existence of continuous families of PT-symmetric dissipative solitons, which can only be explained if one accounts for peculiar properties induced by the PT symmetry. We also investigate dynamics of the nonlinear modes and, in particular, phenomena related to the finite-time blowup, mechanisms for controllable switching and amplification in PT-symmetric waveguides, and dynamics at the exceptional points which separate the regimes of the unbroken and broken PT symmetries.

- Nonlinear modes in finite-dimensional PT-symmetric systems, D. A. Zezyulin and V. V. Konotop, Physical Review Letters 108, 213906 (2012).URL: http://prl.aps.org/abstract/
PRL/v108/i21/e213906 Supercritical blowup in coupled parity-time-symmetric nonlinear Schrödinger equations, J.-P. Dias, M. Figueira, V. V. Konotop and D. A. Zezyulin, Studies in Applied Mathematics 133, 422–440 (2014).URL: http://dx.doi.org/10.1111/ sapm.12063 CPT-symmetric spin-orbit–coupled condensate, Y. V. Kartashov, V. V. Konotop and D. A. Zezyulin, EPL 107, 50002 (2014).URL: http://dx.doi.org/10.1209/ 0295-5075/107/50002