Note on the Stabilities of the Light-like Galileon Solutions

Light-like galileon solutions have been used to investigate the chronology problem in galileon-like theories, and in some cases may also be considered as solitons, evading a non-existence constraint from a zero-mode argument. Their stabilities have been analyzed via "local" approximation, which appears to suggest that all these light-like solutions are stable. We re-analyze the stability problem by solving the linear perturbation equation \emph{exactly}, and point out that the finite energy condition is essential for the light-like solitons to be stable. We also clarify potential ghost instabilities and why the zero-mode argument can not be naively generalized to include the light-like solitons.