A Do acoustic phonons disappear in charge density wave states?

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Summary
Do acoustic phonons disappear in charge density wave states?
In most standard exposition of (the mean-field theory of) charge density wave (CDW), phase and amplitude fluctuations are introduced as the collective excitations. Kohn anomaly in the acoustic phonon dispersion is also mentioned as temperature goes from the above till the CDW transition temperature, at which phonons condense at momentum Q=2kF.

But where are the phonons deep in, say, an incommensurate CDW phase?
Despite the condensation, are there still gapless acoustic phonons in the CDW phase, in addition to the phasons?
Or acoustic/optical phonons disappear and merge into the phasons and amplitudons?

Intuitively, I tend to think phason φ in cos(Q⋅x+φ) as a generalized acoutic phonon since phonon seems to be just setting Q=0 as a special case for normal solids.
 

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