I am doing long-running project that involves asymptotic saddle-point estimation of integrals (for flavour, it's this sort of stuff) and I would like to ask if there are established ways in the literature to deal with the hand-off between regions where different pairs of saddle-points interact, and with how to transition from uniform approximations to simpler saddle-point expressions once you get out of the regions where the former is necessary.

To give a concrete example, consider the Pearcey-like integral $$ F(\Omega;u,k)=\int_{-\infty}^\infty e^{ik\left[\frac14 t^4-\frac32 t^2+(\Omega+i u)t\right]} \mathrm dt, $$ where the integral should really be understood as going from $-\infty e^{i\pi/8}$ to $\infty e^{i\pi/8}$, i.e. falling in the valleys of the dominating $e^{\frac14 ikt^4}$ term. Here $k$ and $u$ should be understood as parameters of the problem, while $\Omega$ is the main variable of interest, which 'drives' the saddle points on 'tracks' laid out by $u$.

The core feature that this model embodies is the fact that those 'tracks' have two distinct conjunctions. At each of those conjunctions, a pair of saddle points have a close approach (and, if $u=0$, coalesce), the appropriate contour topology changes, and then the saddle points separate. Moreover, one of those saddle points (the middle one, to the extent that the branch cuts allow unambiguous identification) plays a role in both of those conjunctions.

From physical grounds, I am quite happy taking the independent-saddles approximation where each of them is seen and integrated as an independent gaussian, and I am OK with keeping the leading-order terms in that asymptotic expansion. On the other hand, at the conjunctions and their neighbourhoods, I do need to step in and apply a uniform approximation where the relevant saddle-point pair is seen as an Airy integral, to keep the dependence of $F(\Omega;u,k)$ on $\Omega$ continuous and non-divergent and as smooth as possible. However, if I do that with the saddle-point pair on the right, then that 'locks' the middle saddle point when it begins to interact with the saddle on the left, where I would need to 'release' it so I can do a uniform approximation of the other conjunction.

So, my core question is: is there an established / robust / well-tested way to 'release' the saddles from their Airy-function identity during the uniform-approximation handling of the individual conjunctions to go back into an independent-saddles view in preparation for a future encounter that will require a different re-organization of the saddles? How is this handled in the literature and when trying to calculate / approximate / understand integrals like this one?

(I appreciate that the physicist's asymptotic analysis I'm doing, where I only care about the first term and its physical interpretation, with priority over accuracy of the approximation, may well be quite at odds with an analyst's views of this configuration, but I think it's worth asking.)

Intuitively, I suspect that what's probably best for this problem is taking the asymptotic expansion for the Airy function and using the $1/\zeta$ estimate of the subleading term to set the points at which it is no longer crucial and at which it can be ignored, and then do some form of interpolation between the uniform and non-uniform takes over that region. However, that feels like a relatively kludgy approach and I'd like to know if there's any better ones out there.