On trilinear and quadrilinear equations associated with the lattice Gel’fand–Dikii hierarchy

Introduced in Zhang et al. (2012), 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.