Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 007, 30 pages      arXiv:2206.14403      https://doi.org/10.3842/SIGMA.2023.007

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

Guesh Yfter Tela a, Song-Lin Zhao b and Da-Jun Zhang a
a) Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
b) Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, P.R. China

Received July 02, 2022, in final form February 06, 2023; Published online February 21, 2023

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.

Key words: lattice Gel'fand-Dikii type equation; direct linearization approach; multi-dimensional consistency; Lax pair.

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