“Completeness” of the Painlevé Test—General Considerations—Open Problems

Abstract

Though it may sound strange to laypersons and professionals alike, mathematics exhibits a variety of cultures, some of them known by a common name even though they are quite distinct. For example, in some circles if you mention “dynamics,” it is taken for granted that you are referring to Hamiltonian systems, very likely even time-independent ones. To some other mathematicians, “dynamics” is taken to deal with evolutionary systems, those for which there is a “time variable” and which are well posed in the sense of Hadamard, i.e., whose solutions exist and are uniquely determined by an initial condition (and possibly boundary conditions, of course) and depend continuously on the given data—what arc often called “marching problems” ; the characteristic mathematical structure of the evolution in time is the semigroup.

Lecture notes prepared by the author and Christian Scheen.