High dimensional Schwarzian derivatives and Painlev\'e integrable models

Abstract

Because of all the known integrable models possess Schwarzian forms with M\"obious transformation invariance, it may be one of the best way to find new integrable models starting from some suitable M\"obious transformation invariant equations. In this paper, the truncated Painlev\'e analysis is used to find high dimensional Schwarzian derivatives. Especially, a three dimensional Schwarzian derivative is obtained explicitly. The higher dimensional higher order Schwarzian derivatives which are invariant under the M\"obious transformations may be expressed by means of the lower dimensional lower order ones. All the known Schwarzian derivatives can be used to construct high dimensional Painlev\'e integrable models.