... issue1
The strong analogy with the nonlinear Schroedinger equation would suggest that the question of the self consistency of the quantum weak turbulence theory without small $k$ damping might be revisited. 
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... approximation2
Here, we use the term Hartree-Fock approximation in the commonly used sense, which is not identical to the strict meaning of a quantum system with a Hartree-Fock wavefunction. In the latter, scattering and correlations are excluded. 
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... general3
On purpose we do not say "the most general solutions", as there may be some hidden symmetries of the QKE which generate more conserved quantities and thus more general solutions, corresponding to fluxes of those quantities. The statement that the general KE with $M$ conserved quantities has "a most general solution" depending on $2M$ parameters, is also not proven yet for the general collision integral. Remember also that we are considering the isotropic case. This is, of course, an idealization, and generally "drift" solutions may be of equal importance. 
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... small4
Even for broad band pumping the small amount (much smaller than in local band pumping) of $P$ and $Q$ fluxes are also excited, because the form of the pumping $f_k$ gets effectively multiplied by the Pauli blocking $(1 - n_k)$ term. But these fluxes are much smaller and relatively local in $k$ space. 
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...5
Much more carriers are pumped in the case of broad band pumping because of the pump distribution. Because there are fewer carriers in the narrow band pumping case,the "effective" chemical potential also decreases, which increases efficiency. 
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