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Reduction from ABS equations to $A_4^{(1)}$-surface q-Painleve equations

Banff International Research Station for Mathematical Innovation and Discovery

In this talk, I show that a reduction from a 4-dimensional hypercube to a rhombic dodecahedron causes the reduction from ABS equations (partial difference equations) [ABS,Boll] to q-Painleve equations which are $A_4^{(1)}$-surface type in Sakai's classification [Sakai]. Moreover, I also present Lax pairs of the q-Painleve equations constructed by using this reduction. This work has been done in collaboration with Prof. Nalini Joshi and Dr Yang Shi and supported by an Australian Laureate Fellowship FL120100094 and grant DP130100967 from the Australian Research Council. [ABS]V.E. Adler, A.I. Bobenko, and Y.B. Suris. Classification of integrable equations on quad-graphs. The consistency approach. Comm. Math. Phys., 233(3):513-543, 2003. [Boll]R. Boll. Classification of 3D consistent quad-equations. J. Nonlinear Math. Phys., 18(3):337-365, 2011. [Sakai]H. Sakai. Rational surfaces associated with affine root systems and geometry of the Painleve equations. Comm. Math. Phys., 220(1):165-229, 2001.

Mathematics; Ordinary differential equations; Dynamical systems and ergodic theory; Dynamical systems

video/mp4

Author affiliation: The University of Sydney

2017-04-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61095

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/