The Gross-Pitaevskii equation $$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2\nabla^2}{2m}\psi+V\psi+U_0|\psi|^2\psi $$ ($V$ is the external potential, $U_0$ is the interaction constant) in conventionally used to describe dynamics of a Bose-Einstein condensate.
Also it is often used to find excitation spectrum. For example, in the uniform case $V=0$, assuming small perturbations [following C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases] $$ \psi=\left\{\sqrt{n}+[ue^{i(\mathbf{qr}-\omega t)}-v^*e^{-i(\mathbf{qr}-\omega t)}]\right\}e^{-i\mu t/\hbar} $$ and taking the equilibrium concentration $n$ and chemical potential $\mu=U_0n$, we get $$ \hbar\omega=\sqrt{\left(\frac{\hbar^2q^2}{2m}+U_0n\right)^2-(U_0n)^2.} $$ This is usual expression for the spectrum of Bogolyubov excitations which can also be found from microscopic mean-field theory. Excitation energies in more complicated nonuniform cases are also often found using the linearized Gross-Pitaevskii equation.
Now the question:
Why does the linearized Gross-Pitaevskii equation, which is aimed to describe coherent motion of the condensate, at the same time describe the spectrum of incoherent thermal excitations, which do not belong to the condensate?
Does any qualitative difference between coherent oscillations of the condensate and thermal excitations exist, while they are described by the same Gross-Pitaevskii equation?