On the Construction of Elliptic Solutions of Integrable Birational Maps

ABSTRACT

We present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are the following: (i) application of classical addition theorems for elliptic functions and (ii) experimental technique to detect an algebraic curve containing a given sequence of points in a plane. These methods are applied to Kahan–Hirota–Kimura discretizations of the periodic Volterra chains with three and four particles.

KEYWORDS:

• elliptic function
• birational map
• integrable map

2000 AMS SUBJECT CLASSIFICATION:

• 14H70
• 37J35
• 70H06
• 33E05
• 34M05
• 65P10

Acknowledgment

The authors of the article would like to thank the referee for valuable remarks. The author of the Appendix is grateful to Yu. Suris for useful discussions.

Funding

This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” Yuri Fedorov was supported by Spanish MINECO-FEDER Grant MTM2012-37070.