Interacting Stokes lines (Q2744093)

Here, the author develops [see, \textit{S. J. Chapman, J. R. King} and \textit{K. L. Adams}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454, No. 1978, 2733-2755 (1998; Zbl 0916.34017)] optimal truncation methods to analyse a number of problems in asymptotics beyond all orders in which the Stokes lines run parallel to the real axis. He analyses two simplest examples that exhibit such phenomena: NEWLINE\[NEWLINE \varepsilon\frac{d\varphi}{dz} + \varphi = \frac{2z}{z^{2} +\pi^{2}/4},\quad \text{and}\quad \varepsilon\frac{d\varphi}{dz} + \varphi = \tanh z,\quad \varphi \rightarrow 0 \text{ as }z \rightarrow -\infty.NEWLINE\]NEWLINE These cases are used sake for transparency and brevity, but the relevant parts of the analysis carry over largely unchanged to equations such as NEWLINE\[NEWLINE -\varepsilon^{2}\frac{d^{3}\varphi}{dz^{3}}+\frac{d\varphi}{dz} = F(\varphi)\quad\text{and}\quad -\varepsilon^{2}\frac{d^{4}\varphi}{dz^{4}}+ \frac{d^{2}\varphi}{dz^{2}} = F(\varphi).NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].