Singularities of a simple elliptic operator
This paper continues the geometrical study of the operator \(B: f\to - f''+f^ 2/2\) acting on functions of \(0\leq x\leq 1\) which vanish at \(x=0,1\). This was initiated by \textit{H. P. McKean} and \textit{J. C. Scovel} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. (to appear)]. The present note shows that, on certain submanifolds of co-dimension \(<\infty\), B exhibits singularities of each of the types introduced by \textit{B. Morin} [C. R. Acad. Sci., Paris 260, 5662-5665; 6503-6506 (1965; Zbl 0178.268)]. These types are expressed in local coordinates by \[ (f_ 1,f_ 2,f_ 3,...)\to (f_ 1f_ 2+f^ 2_ 1f_ 3+...+f_ 1^{n-1}f_ n+f_ 1^{\quad n+1},f_ 2,f_ 3,...), \] i.e., a fold for \(n=1\), a cusp for \(n=2\), etc.
non-linear ordinary differential operators
scientific article; zbMATH DE number 3960612