I have a question about Feshbach resonance and the (generalized) Gross-Pitaevskii equation. If we consider a BEC at a finite temperature $T_{BEC}$ and a static mean-field thermal cloud at temperature $T\geq 0.4T_{BEC}$, then the equation of motion for the expectation value of the macroscopic wave function of the Bose gas, $\Phi(\vec{r},t)$, is

\begin{align} i\hbar\frac{\partial\Phi}{\partial t}=\left[-\frac{\hbar^2}{2m}\nabla^2+V_{trap}(\vec{r})+gn_c(\vec{r},t)+2g\tilde{n}(\vec{r},t)-iR(\vec{r},t)\right]\Phi\ , \end{align}

where $V_{trap}$ is the trapping potential, $n_c$ is the condensate density, $\tilde{n}$ is the noncondensate density, $R\sim g^2$ describes damping, $g=\dfrac{4\pi\hbar^2 a}{m}$, $a$ is the $s$-wave scattering length, and $m$ is the mass of the atoms in the BEC.

I know that Feshbach resonance changes the scattering length, such that \begin{align} a=a_{bg}\left(1-\frac{\Delta}{B-B_{peak}}\right)\ , \end{align} where $a_{bg}$ is the asymptotic background scattering length, $B$ is the magnetic field, $B_{peak}$ is the strength of the magnetic field on resonance, $\Delta=B_{zero}-B_{peak}$, and $B_{zero}$ is the strength of the magnetic field that gives $a=0$.

If I want to account for the effects of Feshbach resonance in the Gross-Pitaevksii equation, am I allowed to directly input the second formula into the first equation? Or do I need to include additional terms within my equation of motion? My concern is that when the magnetic field is chosen so that $a=0$, I will no longer have contributions arising from the condensate and non-condensate densities or damping. What do I do in the case that $a\to\pm\infty$?