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 $A_{N-1}^{(1)}$ invariant Painlevé equations. The $A_{N-1}^{(1)}$ 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 $\widehat{sl}(N)$ endowed with a principal gradation.

References in corpus (3)

• Miura and Generalized Bäcklund Transformation for KdV Hierarchy
• Symmetries and hamiltonians of Ince's XXXVIII and XLIX equations
• Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations

Cited by in corpus (4)

• Gauge Miura and Backlund Transformations for Generalized $A_n$-KdV Hierarchies
• On Rational Solutions of Dressing Chains of Even Periodicity
• On Hamiltonian Formalism for Dressing Chain Equations of Even Periodicity
• Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions