I worked through much of 3.3.1 (Laser Threshold) in Strogatz's Nonlinear Dynamics and Chaos, but I'm struggling to understand the adiabatic elimination he does and when it's allowable.
We have a system modeling a laser where $n$ is the number of photons in the laser and $N$ is the number of excited atoms. The equations are:
$$\dot n= GnN-kn$$ $$\dot N= -GnN-fN+p$$
where $G$, $k$, $f$, and $p$ are various control parameters.
To convert it from a one-dimensional system, we make the 'quasi-static' approximation $\dot N \approx 0$, which Strogatz says represents "$N$ relaxing more rapidly than $n$".
This approximation is the part I'm confused about:
a) If $\dot N\approx0$, do we assume that $N$ is constant? Or are these different assumptions? How can $\dot N\approx0$ be true when $\dot N$ has the constant, non-zero $p$ term?
b) In the 4th part of the question, we are asked to find the range of parameters for which this approximation is acceptable. I tried the 'dimensionless' groups approach from earlier in the book, but that led to a dead-end. Is there a good introduction to when adiabatic elimination is allowed that isn't in the context of complex Quantum Mechanics?