On a new matrix Schwarzian derivative

Abstract

We give a new definition of matrix Schwarzian derivative, which is simpler than the Lagrange Schwarzian derivative and also provides an alternative to other definitions which appear in the literature. Some basic properties are discussed, in particular, analogs of Möbius invariance and the result of a change of independent variable, these being the two properties of the scalar Schwarzian derivative often considered to account for its universality. We then use our new definition of matrix Schwarzian derivative to construct new Schwarzian matrix ordinary and partial differential equation hierarchies: a Schwarzian matrix second Painlevé hierarchy and a Schwarzian matrix Korteweg–de Vries hierarchy, respectively. In addition, we define a new matrix second Painlevé hierarchy.