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