Cauchy matrix approach to novel extended semidiscrete KP-type systems

Two novel extended semidiscrete KP-type systems, namely, partial differential–difference systems with one continuous and two discrete variables, are investigated. Introducing an arbitrary function into the Cauchy matrix function or the plane wave factor allows implementing extended integrable systems within the Cauchy matrix approach. We introduce the bilinear $D\Delta^2$KP system, the extended $D\Delta^2$pKP, $D\Delta^2$pmKP, and $D\Delta^2$SKP systems, all of which are based on the Cauchy matrix approach. This results in a diversity of solutions for these extended systems as contrasted to the usual multiple soliton solutions.