On the construction of elliptic solutions of integrable birational maps
arXiv:1409.1741 · doi:10.1080/10586458.2016.1166354
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: (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 3 and 4 particles.
26 pp. In the new version an Appendix by Yuri N. Fedorov is added, which treats the discrete time periodic Volterra chain with 3 particles by an alternative method, appendix by Yuri N. Fedorov