Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

Abstract

We identify the self-similarity limit of the second flow of sl(N ) mKdV hierarchy with the periodic dressing chain thus establishing a connection to A(1)N −1 Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra sl(N ) endowed with a principal gradation.

Generalized Bäcklund transformations for affine Toda hierarchies

“…(2.15) After multiplying both sides of relation (2.15) by the Cartan matrix K and making use of relation (2.14) we obtain,[19] …”