What drives AdS spacetime unstable?

We calculate the spectrum of linear perturbations of standing wave solutions discussed in [Phys. Rev. D 87, 123006 (2013)] as the first step of investigating the stability of the globally regular, asymptotically anti-de Sitter (AdS), time-periodic solutions discovered in [Phys. Rev. Lett. 111, 051102 (2013)]. We show that while this spectrum is only asymptotically nondispersive (as contrasted with the pure AdS case), putting a small standing wave solution on top of an AdS solution indeed prevents the turbulent instability. Thus we support the idea advocated in previous works that the nondispersive character of the spectrum of linear perturbations of AdS space is crucial for the conjectured turbulent instability.