  Flavor oscillations can occur on very short spatial and temporal scales in
the dense neutrino media in a core-collapse supernova (CCSN) or binary neutron
star merger (BNSM) event. Although the dispersion relations (DRs) of the fast
neutrino oscillations can be obtained by linearizing the equations of motion
(EoM) before the emergence of any significant flavor conversion, one largely
depends on numerical calculations to understand this interesting phenomenon in
the nonlinear regime. In this work we demonstrate that there exist nontrivial
solutions to the flavor EoM that govern the fast oscillations in
one-dimensional axisymmetric neutrino gases. These solutions represent a
coherent flavor isospin wave similar to the magnetic spin wave in a lattice of
magnetic dipoles. We also compute the DRs of such waves in some example cases
which are closely related to the DRs of the fast neutrino oscillations obtained
in the linear regime. This result sheds new light on the long-term behavior of
fast neutrino oscillations which can have various implications for the CCSN and
BNSM events.
