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

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.Non UBCUnreviewedAuthor affiliation: The University of SydneyPostdoctora