Exactly solvable toy model for a SPASER

We propose an exactly solvable quasi-classical model of a spaser. The gain medium is described in terms of the nonlinear permittivity with negative losses. The model demonstrates the main features of a spaser: a self-oscillating state (spasing) arising without an external driving field if the pumping exceeds some threshold value, synchronization of a spaser by an external field within the Arnold tongue, and the possibility of compensating for Joule losses when the pumping is below threshold. Similar to the common laser, a transition to the spasing regime takes a form of the Hopf bifurcation.