Do acoustic phonons disappear in charge density wave states?

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sfman
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TL;DR
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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Please provide more background to your question. Not everyone has worked in your subfield. Of the many confusing points, I don't know what you means when you say phonons condense. Could you post a phonon spectrum to illustrate what you are thinking because I also don't know what you mean by gapless acoustic phonons.

One thing I can say is that acoustic phonons are so named because the low Q phonon spectrum is linear and the slope is a measure of the speed of sound of the material. If these didn't exist you would have an unphysical sample that wouldn't transmit sound.