On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy

TL;DR

This paper explores advanced lattice equations related to the Gel'fand-Dikii hierarchy, including derivations, dual equations, conservation laws, and the generation of Somos-like integer sequences, expanding understanding of integrable systems.

Contribution

It introduces new derivations and properties of trilinear and quadrilinear lattice equations within the Gel'fand-Dikii hierarchy, including their reductions and integer sequence applications.

Findings

Derivation of quadrilinear dual lattice equations

Identification of conservation laws and periodic reductions

Establishment of Somos-like integer sequences

Abstract

Introduced in 2012, by Zhang, Zhao, and Nijhoff, the trilinear Boussinesq equation is the natural form of the equation for the $\tau$-function of the lattice Boussinesq system. In this paper we study various aspects of this equation: its highly nontrivial derivation from the bilinear lattice AKP equation under dimensional reduction, a quadrilinear dual lattice equation, conservation laws, and periodic reductions leading to higher-dimensional integrable maps and their Laurent property. Furthermore, we consider a higher Gel'fand-Dikii lattice system, its periodic reductions and Laurent property. As a special application, from both a trilinear Boussinesq recurrence as well as a higher Gel'fand-Dikii system of three bilinear recurrences, we establish Somos-like integer sequences.