quantum mechanics - Collective modes of charge density wave

The question is about collective modes of charge density waves, i.e., amplitude and phase fluctuations $\delta,\phi$ of the order parameter $\Delta(x,t)=(\Delta_0+\delta)e^{i\phi}$.

I read on p.1 of this paper that amplitude mode and phason mode can be expressed using the phonon normal coordinate $Q_{q\pm2k_F}$ respectively as ($2k_F$ is the density wave vector) $$\frac{1}{\sqrt{2}}(Q_{q+2k_F}\pm Q_{q-2k_F}).$$ But how to obtain this result? It's not quite obvious to me.

I also found in another review paper (p. 1135, eq(3.1)) the following. To the lowest order in the fluctuations $\delta,\phi$, the amplitude mode corresponds to $\Delta_{2k_F}+\Delta_{-2k_F}=2\Delta_0+2\delta$ and the phason mode corresponds to $\Delta_{2k_F}-\Delta_{-2k_F}=2\Delta_0\Delta\phi$. But I don't find a clear definition of $\Delta_{2k_F}$ in the paper. Still it's not clear how this is obtained.