Linear relations for Laurent polynomials and lattice equations

TL;DR

This paper studies nonlinear recurrences with the Laurent property, showing they are linearizable, and introduces new lattice equations with linearizable properties, expanding understanding of their structure and initial value problem solutions.

Contribution

It generalizes known recurrences with the Laurent property, demonstrates their linearizability, and introduces new lattice equations with similar properties.

Findings

Recurrences are linearizable with constant coefficient relations.

New lattice equations with the Laurent property are introduced.

Constructs initial value problems with the Laurent property for these equations.

Abstract

A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. We consider a family of nonlinear recurrences with the Laurent property, which were derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalize the Heideman-Hogan recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. The latter are obtained from linear relations with periodic coefficients, which were found recently by Kamiya et al. from travelling wave reductions of a linearizable lattice equation on a 6-point stencil. We introduce another linearizable lattice equation on the same stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6-point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.