Multiple Darboux–Bäcklund transformations via truncated Painlevé expansion and Lie point symmetry approach**Project supported by the National Natural Science Foundation of China (Grant Nos. 11675055, 11175092, and 11205092), the Program from Shanghai Knowledge Service Platform for Trustworthy Internet of Things (Grant No. ZF1213), and K C Wong Magna Fund in Ningbo University.

Abstract

For a given truncated Painlevé expansion of an arbitrary nonlinear Painlevé integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, which is proved to be localized to Lie point symmetries for suitable prolonged systems. Taking the Korteweg–de Vries equation as an example, the n-th binary Darboux–Bäcklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.