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I am sorry but I slightly disagree with what you exactly added here. There are many situations with different asymptotic expansions in different regions and not all are about Stokes phenomenon. The matter of jumps given by matrix faxtors at Stokes lines is one of the specifics. It is not only about property of single functions but also about whole modules of solutions of certain differential equations for example. I will see how to repair but I think the added material somewhat misleads. For now it is OK, but it will have to be more specific later, when we write more about asymptotic analysis. It would be nice to have entries on Borel summability, asymptotic expansions and Stokes sheaf.
Unfortunately the description is still vague and can not deserve name "Details" as there is no mathematical exact detail given at all yet.
It is rather generic. I do not know how you define "behave like them". Only for Fuchsian equations the formal solutions are converging, as long as one has irregular singular point for linear meromorphic ODE, one deals with the Stokes phenomenon. At least this is my understanding.
This definition (expressing in terms of Airy function) would be too narrow. At least every linear meromorphic connection with an irregular singularity exhibits the Stokes phenomenon, but there are certainly some nonlinear cases as well. For example, the PainlevĂ© equations exhibit the "quasilinear Stokes phenomenon" which has been investigated by Kapaev and Its, and many others.
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