,Multiple Darboux-B(a)cklund transformations via truncated Painlevé expansion and Lie point symmetry

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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-B(a)icklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.
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