Analytic solutions and integrability for bilinear recurrences of order six

Abstract

Somos sequences are integer sequences generated by recurrence relations that are in bilinear form, meaning that they can be written as a quadratic relation between adjacent sets of iterates. Such sequences have appeared in number theory, statistical mechanics and algebraic combinatorics, as well as arising from reductions of bilinear partial difference equations in the theory of discrete integrable systems. This article is concerned with the general form of the Somos-6 recurrence, which is a three-parameter family of bilinear recurrences of order six. After explaining how it arises by reduction from the bilinear discrete BKP equation (Miwa's equation), an invariant Poisson bracket for Somos-6 is presented. Four independent Casimirs of this bracket, which are the invariants under the action of a group of gauge transformations, lead to an associated map on a four-dimensional reduced phase space. Two rational first integrals for this map are constructed, and (for certain parameter choices) these are found to be in involution for a non-degenerate Poisson bracket associated with a symplectic form on the reduced phase space, so that the map is Liouville integrable. For generic parameter values the explicit analytic solution of the Somos-6 recurrence is given in terms of the Kleinian sigma function for a curve of genus two.

Keywords:

• Somos sequences
• Laurent phenomenon
• integrable maps
• Poisson bracket
• theta function
• Kleinian sigma function

AMS Subject Classifications::

• 11B37
• 33E30
• 37J35

Acknowledgements

Some of this work was carried out with a Visiting Fellowship at the Isaac Newton Institute, Cambridge, during the Programme on Discrete Integrable Systems. I have benefited from various correspondence and conversations with several people, in particular J. Propp, N. Elkies, Y. Suris and A. Bobenko. Finally, I am grateful to the referees for useful comments which have improved the final version of the article.