Single-wave solutions of the neutrino fast flavor system. Part II. Weak instabilities and their resonant behavior

Flavor instabilities in dense neutrino media trigger exponential growth of flavor waves, yet their nonlinear saturation remains poorly understood. We examine a simple proxy for this effect in the form of a single-wave solution of an axially symmetric fast flavor system. When the angular crossing is shallow and the growth rate of the instability correspondingly small, the flavor wave primarily affects resonant neutrinos that move in phase with it. The evolution of these resonant neutrinos becomes periodic, undergoing cycles of full flavor reversal. They feed power into the unstable wave, and subsequently return to their initial state, draining power back out. This new flavor pendulum captures the dynamics of weak, nearly monochromatic fast flavor instabilities. Since weakly unstable distributions always exhibit a narrow range of unstable wavenumbers, our model likely describes the earliest development of a flavor instability when it first appears. When the instability is not weak, the linear phase of a single-wave excitation does not connect to a regular nonlinear solution, unless the angle distribution consists of only two beams.