On an \(n\)th-order linear ordinary differential equation with a turning-singular point
The author considers the system \(\sum^n_{k=1} \varepsilon^{(n-k)h}P_k (x, \varepsilon)y^{(n-k)} =\varepsilon^{nh} y^{(n)}\), in a suitable identified space with favorable conditions imposed on the coefficients, with the asymptotic form \(P_k(x, \varepsilon)= p_k.x^{k m}+ O((\varepsilon x^{-1/ \alpha})^l)\) for all \(k\) with \(\alpha= 1/(m+r)>0\), and \(x\) in the region \(K\varepsilon^\alpha \leq| x|\leq x_0\), for some positive constant \(K\). Adopting a stretching change \(x=t \varepsilon^\alpha\), the above differential equation is transformed to the system: \(\varepsilon^{nh}y^{(n)}= \sum\varepsilon^{(n-k) h^.} p_k.P^k(t)y^{(n-k)}\), \(h'=h-(m+1)\), \(\alpha>0\). The author constructs solutions to the system according to the WKB method and uses the asymptotic expansion for the inner and outer solutions to prove that every solution to the original system converges uniformly. The topology of the local and global Stokes curves is discussed and brilliant graphs of these curves are represented. Many references are given related to the subject. The reviewer finds this whole article exposed in such an intriguing way that is very enjoyable to read.
scientific article; zbMATH DE number 1207094