Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 002, 19 pages      arXiv:1308.4233
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

Christopher M. Ormerod
Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA

Received September 19, 2013, in final form December 28, 2013; Published online January 03, 2014

We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlevé equation with $E_6^{(1)}$ symmetry. We present a description of a set of symmetries of the reduced equations and their relations to the symmetries of the discrete Painlevé equation. Finally, we exploit the simple symmetric form of the reduced equations to find rational and hypergeometric solutions of this discrete Painlevé equation.

Key words: difference equations; integrability; reduction; isomonodromy.

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