Bilinearization and solutions of the fourth-order lattice Gel'fand-Dikii type equations

TL;DR

This work develops Hirota bilinear forms and Casoratian solutions for all fourth-order lattice Gel'fand-Dikii (GD-4) type equations (A-2, B-2, C-3) arising from direct linearization, and introduces a parameter-based $\delta$-extension that preserves the multidimensional consistency (CAC). By transforming nonlinear GD-4 systems to bilinear form, the authors construct explicit Casoratian soliton solutions, including single and multi-soliton configurations as well as Jordan-block and quasi-rational cases, with extended inputs $\phi_s$ and auxiliary equations (CES) guiding the generalized solutions. The $\delta$-extension yields new bilinear and nonlinear forms $\mathcal{H}^{\delta}$ and deformed GD-4 equations $B_{21}^{\delta}$, $A_{21}^{\delta}$, $C_{31}^{\delta}$, etc., while preserving CAC and enabling a broader family of Casoratian solutions. Overall, the paper advances the understanding of higher-order discrete GD hierarchies, demonstrating that their bilinear structures admit rich solution spaces and remain integrable under δ-extensions, thereby enriching the toolbox for discrete integrable systems and potential reductions to lattice KP-type equations.

Abstract

In this paper we derive bilinear forms and solutions in Casoratians for some fourth-order lattice Gel'fand-Dikii (lattice GD-4) type equations. These equations were recently formulated from the direct linearization approach and exhibit multidimensionally consistent property in multi-component form. The obtained solitons and Casoratian forms enable us to extend these equations by introducing a parameter $δ$. These $δ$-extended lattice GD-4 type equations are still consistent around the cube, and their bilinear forms together with Casoration solutions are presented.

Theorems & Definitions (17)

• Proposition 1
• Proposition 2
• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• proof
• Remark 1
• Theorem 5