Geometry of the Stokes sets for families of functions of one variable (Q1365708)

The socalled Stokes lines play an important role for asymptotic approximations of the Airy function \(\text{Ai} (k,s)={1\over 2\pi i}\int_{0}^{\infty} e^{-kiF(x,s)} dx\) with the phase function \(F(x,s)={x^3\over 3}+sx\), \(s\in {\mathbb{C}}\setminus \{0\}\). This set is defined by \(\text{Re }F(x_1,s)= \text{Re }F(x_2,s)\), where \(x_1=x_1(s)\), \(x_2=x_2(s)\) are the two critical values of \(F( \cdot ,s)\). The main result is a characterisation of the Stokes set for more general phase functions \(F(x,s)={1\over \mu+1}x^{\mu+1}+{1\over \mu-1}s_{\mu-1}x^{\mu-1}+\cdots+ s_1 x\). Here, \(x_1(s)\) and \(x_2(s)\) are two nonfixed critical points of \(F( \cdot ,s)\), while the other critical points are fixed. In particular, the Stokes set for a special function considered by \textit{T. Pearcey} [Philos. Mag. 37, 311-317 (1946)] is described.