Rainbow Nambu-Goldstone Modes under a Shear Flow

We study an $O(N)$ scalar model under shear flow and its Nambu-Goldstone modes associated with spontaneous symmetry breaking $O(N) \to O(N-1)$. find that the mode splits into infinite number of gapless modes, which we call rainbow modes. They have different group velocities fractional dispersion relation $\omega \sim k_1^{2/3}$, where $k_1$ is wavenumber along flow. Such behaviors do not counterparts in equilibrium state.