A new extended q-deformed KP hierarchy

Abstract

A method is proposed in this paper to construct a new extended q-deformed KP (q-KP) hiearchy and its Lax representation. This new extended q-KP hierarchy contains two types of q-deformed KP equation with self-consistent sources, and its two kinds of reductions give the q-deformed Gelfand-Dickey hierarchy with self-consistent sources and the constrained q-deformed KP hierarchy, which include two types of q-deformed KdV equation with sources and two types of q-deformed Boussinesq equation with sources. All of these results reduce to the classical ones when q goes to 1. This provides a general way to construct (2+1)-and (1+1)-dimensional q-deformed soliton equations with sources and their Lax representations.

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

“…As we know that most of the (1 + 1)-dimensional soliton hierarchies with self-consistent sources were constructed by constrained flows [18,36]. This work and the authors' previous works [15,19,33,35] suggest that one might unify the constructions of (1 + 1)-dimensional soliton hierarchies with self-consistent sources in the framework of symmetries generating functions too. We may consider this possibility in the future.…”

“…However, dressing method cannot be applied directly to the extended integrable hierarchies. To find solutions for the extended hierarchy, the authors developed a generalized dressing method which can be used to give general Wronskian (discrete Wronskian, q-deformed Wronskian, respectively) solutions to the extended KP hierarchy, the extended discrete KP hierarchy and the extended q-deformed KP hierarchy [15,17,35]. This method is emerged from the idea of introducing some arbitrary functions in dressing approach, which is similar to the method of 'variation of constant' for solving ODEs.…”

“…Recently, inspired by the squared eigenfunction symmetries [23,24], the authors proposed a systematic approach to construct new (2 + 1)-dimensional integrable systems, by extending certain flow of the system with the help of squared eigenfunction symmetries. In this way, the authors have found many new integrable hierarchies, which are called the 'extended integrable hierarchies', such as the extended KP and mKP, BKP, CKP hierarchies, the extended discrete KP hierarchy, the extended q-deformed KP hierarchy as well as the extended two-dimensional Toda lattice hierarchy [15,17,19,20,[33][34][35].…”

Compatibility equations of the extended KP hierarchy

“…An extended KP hierarchy and an extended q-deformed KP hierarchy were presented recently by using the dressing operator and its corresponding Baker-Akhiezer function [17,18]. The dressing technique (see, e.g., [19]) was used to determine the extended KP flows and the extended mKP flows, and Sato theory (see, e.g., [20]) was generalized to present Wronskian type solutions of the extended KP and mKP flows [21].…”

The KP hierarchy with self–consistent sources: construction, Wronskian solutions and bilinear identities

“…In order to find a unified framework to study the two types of SESCSs, a systematical method was proposed by the authors on the basis of Sato's theory [13]. This is a systematic method to generate the SESCSs, and can be used to study the case of BKP, CKP [22], q-deformed KP [10,11], and some other cases. A generalized dressing method was also derived for these soliton hierarchy with sources, and their Wronskian solutions (including soliton solutions) were obtained [14].…”