Bäcklund transformations for the Burgers equation via localization of residual symmetries

We obtain the non-local residual symmetry related to truncated Painlevé expansion of Burgers equation. In order to localize the residual symmetry, we introduce new variables to prolong the original Burgers equation into a new system. By using Lie's first theorem, we obtain the finite transformation for the localized residual symmetry. More importantly, we also localize the linear superposition of multiple residual symmetries to find the corresponding finite transformations. It is interesting to find that the n-th Bäcklund transformation for Burgers equation can be expressed by determinants in a compact way.