Gauge Symmetry Origin of B\"acklund Transformations for Painlev\'e Equations

TL;DR

This paper reveals that B"acklund transformations for Painlevé equations originate from gauge symmetries in the mKdV hierarchy, connecting integrable systems, algebraic structures, and Painlevé equations through a self-similarity limit.

Contribution

It establishes a novel link between gauge symmetries in the mKdV hierarchy and B"acklund transformations of Painlevé equations, providing a new algebraic perspective.

Findings

Identifies the self-similarity limit of mKdV hierarchy with the dressing chain.

Derives B"acklund symmetries from gauge transformations.

Connects Painlevé equations to affine Lie algebra structures.

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 $A^{(1)}_{N-1}$ invariant Painlev\'e equations. The $A^{(1)}_{N-1}$ B\"acklund symmetries of dressing equations and Painlev\'e equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra $\widehat{sl}(N)$ endowed with a principal gradation.