A new extended discrete KP hierarchy and generalized dressing method

TL;DR

This paper introduces an extended discrete KP hierarchy incorporating new flows and eigenfunctions, along with a generalized dressing method to obtain soliton solutions, advancing the understanding of integrable systems with self-consistent sources.

Contribution

It develops a new extended discrete KP hierarchy with additional flows and eigenfunctions, and proposes a generalized dressing method for solving it, including explicit soliton solutions.

Findings

Constructed the exDKPH with new flows and eigenfunctions.

Derived two types of discrete KP equations with self-consistent sources.

Presented N-soliton solutions for these equations.

Abstract

Inspired by the squared eigenfunction symmetry constraint, we introduce a new $\ta_k$-flow by ``extending'' a specific $t_n$-flow of discrete KP hierarchy (DKPH). We construct extended discrete KPH (exDKPH), which consists of $t_n$-flow, $\ta_k$-flow and $t_n$ evolution of eigenfunction and adjoin eigenfunctions, and its Lax representation. The exDKPH contains two types of discrete KP equation with self-consistent sources (DKPESCS). Two reductions of exDKPH are obtained. The generalized dressing approach for solving the exDKPH is proposed and the N-soliton solutions of two types of the DKPESCS are presented.