Various truncations in Painlev\'e analysis of PDEs

Abstract

The ``truncation procedure'' initiated by Weiss et al. is best understood as a Darboux transformation. If it leads to the Lax pair of the PDE under study, the B\"acklund transformation follows by an elimination, thus proving the integrability. We present the state of the art of this powerful technique. The easy situations were all handled by the WTC one-family truncation and its homographically invariant version. An updated version of this method has been recently developed, which is now able to handle the Kaup-Kupershmidt and Tzitz\'eica equations. It incorporates a new feature, namely the distinction between two entire functions usually mingled, which are shown to be linked by formulae established by Gambier for his classification.