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We classify symmetry-protected and symmetry-breaking dynamical solutions fornonlinear saturable bosonic systems that display a non-hermitiancharge-conjugation symmetry, as realized in a series of recent groundbreakingexperiments with lasers and exciton polaritons. In particular, we show thatthese systems support stable symmetry-protected modes that mirror the conceptof zero-modes in topological quantum systems, as well as symmetry-protectedpower-oscillations with no counterpart in the linear case. In analogy totopological phases in linear systems, the number and nature ofsymmetry-protected solutions can change. The spectral degeneracies signallingphase transitions in linear counterparts extend to bifurcations in thenonlinear context. As bifurcations relate to qualitative changes in the linearstability against changes of the initial conditions, the symmetry-protectedsolutions and phase transitions can also be characterized by topologicalexcitations, which set them apart from symmetry-breaking solutions. Thestipulated symmetry appears naturally when one introduces nonlinear gain orloss into spectrally symmetric bosonic systems, as we illustrate forone-dimensional topological laser arrays with saturable gain andtwo-dimensional flat-band polariton condensates with density-dependent loss.

Topological dynamics and excitations in lasers and condensates with saturable gain or loss