A new extended q-deformed KP hierarchy

TL;DR

This paper introduces a new extended q-deformed KP hierarchy with Lax representation, encompassing various q-deformed soliton equations with sources, and generalizes classical integrable systems as q approaches 1.

Contribution

It constructs a novel extended q-deformed KP hierarchy with self-consistent sources and derives its reductions, expanding the framework of q-deformed integrable systems.

Findings

Contains two types of q-deformed KP equations with sources.

Reduces to classical systems as q approaches 1.

Provides a method to generate q-deformed soliton equations with sources.

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.