On the Fourth-Order Lattice Gel'fand-Dikii Equations

TL;DR

This paper investigates fourth-order lattice Gel'fand-Dikii equations, presenting their extended forms, analyzing their integrability, and exploring reductions to simpler components within a multi-dimensional framework.

Contribution

It introduces new higher-order lattice Gel'fand-Dikii equations, discusses their integrability via direct linearization and multi-dimensional consistency, and explores their reductions.

Findings

Equations are related to a quartic discrete dispersion relation.

Some equations are multi-dimensionally consistent.

Reductions to four-component forms are possible.

Abstract

The fourth-order lattice Gel'fand-Dikii equations in quadrilateral form are investigated. Utilizing the direct linearization approach, we present some equations of the extended lattice Gel'fand-Dikii type. These equations are related to a quartic discrete dispersion relation and can be viewed as higher-order members of the extended lattice Boussinesq type equations. The resulting lattice equations given here are in five-component form, and some of them are multi-dimensionally consistent by introducing extra equations. Lax integrability is discussed both by direct linearization scheme and also through multi-dimensional consistent property. Some reductions of the five-component lattice equations to the four-component forms are considered.