Residual symmetries and B\"acklund transformations

TL;DR

This paper demonstrates that residual symmetries derived from Painlevé expansions are nonlocal symmetries that can be localized to Lie point symmetries, and these are equivalent to Darboux-Bäcklund transformations across several integrable systems.

Contribution

It establishes a unified approach to derive Bäcklund transformations from residual symmetries for various integrable equations.

Findings

Residual symmetries are nonlocal symmetries from Painlevé expansions.

Residual symmetries can be localized to Lie point symmetries.

Darboux-Bäcklund transformations are equivalent to finite residual symmetry transformations.

Abstract

It is proved that for a given truncated Painlev\'e expansion of an arbitrary nonlinear Painlev\'e integrable system, the residue with respect to the singularity manifold is a nonlocal symmetry. The residual symmetries can be localized to Lie point symmetries after introducing suitable prolonged systems. The finite transformations of the residual symmetries are equivalent to the second type of Darboux-B\"acklund transformations. The once B\"acklund transformations related to the residual symmetries are same for many integrable systems including the Korteweg-de Vries, Kadomtsev-Petviashvili, Boussinesq, Sawada-Kortera and Kaup-Kupershmidt equations. For the Korteweg-de Vries equation, the $n^{th}$ Darboux transformations can also be obtained from the Lie point symmetry approach via the localization of the residual symmetries.