  An analytical theory is proposed for the kinetic electrostatic electron
nonlinear (KEEN) waves originally found in simulations by Afeyan et al
[arXiv:1210.8105]. We suggest that KEEN waves represent saturated states of the
negative mass instability (NMI) reported recently by Dodin et al [Phys. Rev.
Lett. 110, 215006 (2013)]. Due to the NMI, trapped electrons form
macroparticles that produce field oscillations at harmonics of the bounce
frequency. At large enough amplitudes, these harmonics can phase-lock to the
main wave and form stable nonlinear dissipationless structures that are
nonstationary but otherwise similar to Bernstein-Greene-Kruskal modes. The
theory explains why the formation of KEEN modes is sensitive to the excitation
scenario and yields estimates that agree with the numerical results of Afeyan
et al. A new type of KEEN wave may be possible at even larger amplitudes of the
driving field than those used in simulations so far.
