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From discrete integrable equations to Laurent recurrences

journal contribution
posted on 25.03.2022, 05:54 authored by Khaled HamadKhaled Hamad, Pieter Van Der KampPieter Van Der Kamp
We show how to obtain relations for the divisors of terms generated by a homogenized version of a rational recurrence. When the rational recurrence confines singularities the relations take the form of a rational recurrence, possibly with periodic coefficients. As the recurrence generates polynomials one expects it to possess the Laurent property. The method we develop uses ultra-discretization and recursive factorization. It is applied to certain QRT-maps which gives rise to Somos-k (k = 4, 5) sequences with periodic coefficients. Novel (N + 3)-rd order recurrences are obtained from the Nth order DTKQ-equation (N = 2, 3). In each case the resulting recurrence equation has the Laurent property. The method is equally applicable to non-integrable or non-confining equations. However, in the latter case the degree and the order of the relation might display unbounded growth. We demonstrate the difference, by considering different parameter choices in a generalized Lyness equation.

Funding

This research was supported by the Australian Research Council and by the La Trobe University Disciplinary Research Program in Mathematical and Computer Sciences.

History

Publication Date

01/06/2016

Journal

Journal of Difference Equations and Applications

Volume

22

Issue

6

Pagination

28p. (p. 789-816)

Publisher

Taylor & Francis

ISSN

1563-5120

Rights Statement

© 2016 Taylor & Francis