  In an environment with high-density neutrinos formed in a core-collapse
supernova (CCSN), the neutrinos exhibit nonlinear and complex oscillation
behaviors due to their self-interactions. The onset of this nonlinear
oscillation can be investigated by linearizing the evolution equation for small
perturbations around the flavor eigenstates. While the condition under which
the flavor eigenstates are unstable has been investigated in many studies, how
the perturbations evolve in spacetime has yet to be elucidated. In this paper,
we analytically and correctly derive the asymptotic behaviors of the linear
perturbations in 4-dimensional spacetime in the linear regime for a 2-beam
neutrino model using the recently proposed Lefschetz thimble formulation. The
result suggests that the perturbations grow in the directions between the two
neutrino beams. We also briefly discuss the possible effects of neutrino flavor
conversion on the explosion mechanism of a CCSN. In particular, the result
implies that the flavor instability in the preshock region may propagate into
the postshock region, contrary to the previous study focusing on the group
velocity in 1-dimensional space. How to treat the case of a more realistic
continuous spectrum is also discussed.
