Hankel determinant solutions to several discrete integrable systems and the Laurent Property

Abstract

Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called $A_1$ $Q$-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Bäcklund transformation of the Somos-4 recurrence. The related topics about Somos polynomials are also studied.

16 pages

References in corpus (6)

• Noncommutative integrability, paths and quasi-determinants
• Laurent Polynomials and Superintegrable Maps
• Singularity confinement for maps with the Laurent property
• Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
• Diophantine non-integrability of a third order recurrence with the Laurent property
• Invariant number triangles, eigentriangles and Somos-4 sequences

Cited by in corpus (2)

• Some integrable maps and their Hirota bilinear forms
• Discrete integrable systems and condensation algorithms for Pfaffians