Bilinear structures of the fourth-order lattice Gel'fand-Dikii equations

Abstract

In this paper we derive bilinear forms and present their solutions in Casoratians for several fourth-order lattice Gel'fand-Dikii (lattice GD-4) equations. These equations were recently formulated from the direct linearization approach and exhibit the multidimensionally consistent property in multi-component form. Based on the obtained soliton solutions, we are able to extend these equations by introducing a parameter $\delta$. These $\delta$-extended lattice GD-4 type equations are still consistent around the cube, and their bilinear forms together with Casoratian solutions are provided.