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We describe linear and nonlinear modes in a ring-shaped waveguide withlocalized gain and dissipation modeled by two Dirac $\delta$ functions locatedsymmetrically. The strengths of the gain and dissipation are equal, i.e., thesystem obeys the parity-time symmetry. This configuration describes atomicBose-Einstein condensates with local loading and local elimination of atoms,polaritonic condensates, or optical ring resonators with local pump andabsorption. We discuss the linear spectrum of such a system and show if thelocation of the $\delta$ functions is slightly asymmetric, then the system canbe driven through a series of exceptional points by the change of thegain-and-loss coefficient. In the nonlinear case, the system admits solutionswith spatially constant and periodic densities which are presented in the exactanalytical form. These solutions are supported by the current directed from thegain potential towards the absorbing potential. The system also admitscurrentless solutions. Stability and bifurcations of nonlinear solutions arealso described.

Nonlinear currents in a ring-shaped waveguide with balanced gain and dissipation