The Extension of Noncommutative Modified KP Hierarchy and Its Quasideterminant Solutions

Abstract

The extended noncommutative modified KP (exncmKP) hierarchy is firstly constructed, which gives rise to two types of the ncmKP equation with self-consistent sources (ncmKPESCSs). Then, the noncommutative (NC) Miura transformation between the extended noncommutative KP (exncKP) hierarchy and the exncmKP hierarchy is presented, and the quasideterminant solutions of the exncmKP hierarchy are also given. As its byproduct, the quasideterminant solutions of two types of ncmKPESCSs are obtained. The matrix solutions of two types of ncmKPESCSs are finally investigated, and the impact of the source terms on the NC soliton is analyzed.

1. Introduction

It is known that noncommutative gauge theory has some important applications in D-branes. It was shown that noncommutative extension corresponds to the presence of background magnetic fields, and that NC solitons are nothing but the lower-dimensional D-branes in some situations [1]. Motivated by this, the extension of ordinary integrable systems to the NC counterparts has attracted much attention from many researchers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Some integrable-like properties of NC systems have been revealed, such as multisoliton [7,15,16], Hamilton structure [11], Backlund transformation [17], infinite conservation law [8], recursion operator [10], reduction [9], and so on. It is generally true that the solutions of NC integrable systems can be expressed in terms of a quasideterminant [2].

In 2007, Gilson and Nimmo used Darboux and binary Darboux transformation to investigate the ncKP equation and ncmKP equation [3,4]. Two kinds of quasideterminant solutions, quasi-Wronskian and quasigrammian solutions, were presented. These quasideterminant solutions were then verified directly by using the formulae for the derivative of quasideterminants. The matrix solutions of the ncKP equation and ncmKP equation were also obtained [5]. Notice that the authors did not specify the nature of noncommutativity and that the potential function and its derivatives are not assumed to be commutative. The results obtained in [3,4] are valid for the star product, the matrix, and the quarternion versions of integrable systems. Later, a similar approach was applied to discuss some other NC integrable systems such as the NC Davey–Stewartson equation [6], super KdV equation [12], NC Toda lattice [13], Nonisospectral KP equation [20], etc.

In 2010, the extension of the ncKP hierarchy was investigated in the frame of the noncommutative Sato theory. It was extended by introducing the square eigenfunction in noncommutative space–time [14]. The commutativity between the ncKP flow and the extended ncKP flow was shown to give rise to two types of ncKPESCSs. Recently, the quasideterminant solutions for the exncKP hierarchy were constructed by using a pseudo-differential operator [21]. The direct verification method was used to show that the quasi-Wronskian solutions we obtained are the solutions of ncKPESCSs. In [19], the source generation procedure is applied to the nc KP equation and the nc mKP equation. The ncKP equation with self-consistent sources (ncKPESCSs) and the ncmKP equation with self-consistent sources (ncmKPESCSs) are generated, and their quasi-Wronskian and quasigrammian solutions are presented, respectively. In addition, the Miura transformation between the ncKPESCS and the ncmKPESCS is established. As is known, ncmKPESCS is just an equation of the extended ncmKP hierarchy in the case of $n=2,K=3$. Although ncmKPESCS and its quasideterminant solutions were constructed in [19], the integrable extension of the whole ncmKP hierarchy and its related quasideterminant solutions have never been studied. Furthermore, an important question remains unexplored: what is the impact of source terms on the NC soliton of a noncommutative integrable system? In the present paper, we explore the extension of ncmKP hierarchy and the construction of quasideterminant solutions of this new hierarchy. We have found that when $N=0$, the extended ncmKP hierarchy can be reduced to ncmKP hierarchy. In this sense, the extended ncmKP hierarchy can be considered as an integrable generalization of ncmKP hierarchy. Furthermore, the related quasideterminant and matrix solutions of the extended ncmKP hierarchy are also reduced to those for ncmKP hierarchy in the case of $N=0$. We are sure that our results will fill the gap of the abovementioned literature and give an important supplement to the noncommutative Sato theory.

The outline of this paper is given as follows. In Section 2, the definition and some basic properties of quasideterminants are recalled. In Section 3, the exncKP hierarchy and the ncmKP hierarchy are briefly reviewed. In Section 4, the exncmKP hierarchy is constructed and two types of ncmKPESCSs are also obtained. In Section 5, the noncommutative Miura transformation between the exncKP hierarchy and the exncmKP hierarchy is presented. In Section 6, the quasideterminant solution of the exncmKP hierarchy is given, and as its byproducts, the quasideterminant solutions for two types of ncmKPESCSs are also obtained in Section 7. In Section 8, the matrix solutions for two types of ncmKPESCSs are finally investigated. Section 9 is devoted to a brief summary and discussion.

2. Quasideterminant

In this section, we will briefly recall the definition and some basic properties of quasideterminants. These can be found from the review paper [22].

2.1. The Definition of Quasidetermiant:

Suppose that A is an $n\times n$ matix over the noncommutative ring R; each element $a_{i,j}$ of A can lead to a quasideterminant ${\left|A\right|}_{i,j}$ defined by

$${\left|A\right|}_{i,j}=a_{i,j}-r_i^j{(A^{i,j})}^{-1}{c}_j^i,A^{-1}={(|A|}_{j,i}{)}^{-1}i,j=1,\cdots ,n.$$

(1)

In the above, $r_i^j$ represents the i-th row of A with the j-th element removed, $c_j^i$ represents the j-th column with the i-th element removed, and $A^{i,j}$ is an invertible submatrix obtained by removing the i-th row and the j-th column from A.

Quasideterminants can also be denoted as shown below by boxing the entry about which the expansion is made

$${\left|A\right|}_{i,j}=\left|\begin{array}{cc}{A}^{i,j}& c_j^i\\ r_i^j& \begin{array}{|c|}\hline a_{ij}\\ \hline\end{array}\end{array}\right|.$$

(2)

2.2. The Invariance Under Row and Column Operations

Similar to the determinant, the quasideterminant of a square matrix also has an invariance property [3,21].

$$\begin{array}{cc}\hfill {\left|\begin{array}{c}\left(\begin{array}{cc}E& 0\\ F& g\end{array}\right)·\left(\begin{array}{cc}A& B\\ C& d\end{array}\right)\end{array}\right|}_{n\times n}& ={\left|\begin{array}{cc}FA& EB\\ FA+gc& \begin{array}{|c|}\hline FB+gd\\ \hline\end{array}\end{array}\right|}_{n\times n}\hfill \\ \hfill & =g(d-C{A}^{-1}B)=g\left|\begin{array}{cc}A& B\\ C& \begin{array}{|c|}\hline d\\ \hline\end{array}\end{array}\right|,\hfill \end{array}$$

(3)

where A is an intertible matrix defined by (1).

Remark 1.

From the invariance property (3), we can easily show that

$$\left|\begin{array}{cc}A& B\\ A^k& \begin{array}{|c|}\hline B^k\\ \hline\end{array}\end{array}\right|=0$$

(4)

where $A^k,B^k$ stand for the k-th row of the square matrix A and the vector B, respectively.

2.3. The Derivative of Quasideterminant [3]

Let the prime denote the derivative with respect to the variable, then the derivative of quasideterminant is written as

$${\left|\begin{array}{cc}A& B\\ C& \begin{array}{|c|}\hline d\\ \hline\end{array}\end{array}\right|}^{\prime }=d^{\prime }-C^{\prime }{A}^{-1}B+C{A}^{-1}{A}^{\prime }{A}^{-1}B-C{A}^{-1}{B}^{\prime }.$$

(5)

It can also be expressed as

$${\left|\begin{array}{cc}A& B\\ C& \begin{array}{|c|}\hline d\\ \hline\end{array}\end{array}\right|}^{\prime }=\left|\begin{array}{cc}A& B\\ C^{\prime }& \begin{array}{|c|}\hline d^{\prime }\\ \hline\end{array}\end{array}\right|+\sum _{k=1}^N\left|\begin{array}{cc}A& e_k\\ C& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cc}A& B\\ {(A^k)}^{\prime }& \begin{array}{|c|}\hline {(B^k)}^{\prime }\\ \hline\end{array}\end{array}\right|,$$

(6)

or

$${\left|\begin{array}{cc}A& B\\ C& \begin{array}{|c|}\hline d\\ \hline\end{array}\end{array}\right|}^{\prime }=\left|\begin{array}{cc}A& B^{\prime }\\ C& \begin{array}{|c|}\hline d^{\prime }\\ \hline\end{array}\end{array}\right|+\sum _{k=1}^N\left|\begin{array}{cc}A& {(A_k)}^{\prime }\\ C& \begin{array}{|c|}\hline {(C_k)}^{\prime }\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cc}A& B\\ (e_k^T)& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,$$

(7)

where $A^k,B^k$ are defined as (4); $e_k$ is a column vector of N-th order with the k-th row being 1 and other rows being 0; and $A_k,C_k$ stand for the $k$-th column of the square matrix A and the vector C, respectively.

2.4. Noncommutative Jacobi Identity

There is a quasideterminant version of Jacobi identity for determinant. The simplest version of this identity is given by

$$\left|\begin{array}{ccc}A& B& C\\ D& f& g\\ E& h& \begin{array}{|c|}\hline i\\ \hline\end{array}\end{array}\right|=\left|\begin{array}{cc}A& C\\ E& \begin{array}{|c|}\hline i\\ \hline\end{array}\end{array}\right|-\left|\begin{array}{cc}A& B\\ E& \begin{array}{|c|}\hline h\\ \hline\end{array}\end{array}\right|·{\left|\begin{array}{cc}A& B\\ D& \begin{array}{|c|}\hline f\\ \hline\end{array}\end{array}\right|}^{-1}·\left|\begin{array}{cc}A& C\\ D& \begin{array}{|c|}\hline g\\ \hline\end{array}\end{array}\right|,$$

(8)

2.5. Homological Relations

As the direct result of the noncommutative Jocobi identity (8), the row and column homological relations can be obtained as follows

$$\left|\begin{array}{ccc}A& B& C\\ D& f& g\\ E& \begin{array}{|c|}\hline h\\ \hline\end{array}& i\end{array}\right|=\left|\begin{array}{ccc}A& B& C\\ D& f& g\\ E& h& \begin{array}{|c|}\hline i\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{ccc}A& B& C\\ D& f& g\\ 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}& 1\end{array}\right|,$$

(9)

and

$$\left|\begin{array}{ccc}A& B& C\\ D& f& \begin{array}{|c|}\hline g\\ \hline\end{array}\\ E& h& i\end{array}\right|=\left|\begin{array}{ccc}A& B& 0\\ D& f& \begin{array}{|c|}\hline 0\\ \hline\end{array}\\ E& h& 1\end{array}\right|·\left|\begin{array}{ccc}A& B& C\\ D& f& g\\ E& h& \begin{array}{|c|}\hline i\\ \hline\end{array}\end{array}\right|.$$

(10)

3. The ncmKP and exncKP Hierarchies

In this section, we briefly review the exncKP and ncmKP hierarchies in the spirit of the Gelfand–Dickey theory.

3.1. The Extended ncKP Hierarchy

The exncKP hierarchy is defined by [14,21]

$$L_{t_n}=[B_n,L],$$

(11)

$$L_{{\tau }_k}=[B_k+\sum _{i=1}^N{q}_i{\partial }^{-1}{r}_i,L],$$

(12)

where the Lax operator $L=\partial +u{\partial }^{-1}+u_2{\partial }^{-2}+\cdots$; $B_n={(L^n)}_{\ge 0}$ denotes the projection of powers of the operator on the differential part; ${\partial }^i$ denotes ${\displaystyle \frac{{\partial }^i}{\partial x^i}}$; $u,u_i,q_i,r_i(i=1,2,\cdots )$ depend on two sets of real variables $t=(t_1,t_2,t_3,\cdots )$ and $\tau =({\tau }_1,{\tau }_2,\cdots )$; and the commutator is defined as $[B_n,L]=B_{n}L-L{B}_n$. To derive the exncKP hierarchy and the ncKPESCS, we assume that $u,u_i$ and their related derivatives do not necessarily commute. $q_i$ and $r_i$ satisfy

$$q_{i,t_n}=B_n(q_i),r_{i,t_n}=-B_n^{*}(r_i).$$

(13)

where $B_n^{*}$ represents the adjoint operator of $B_n$. Given two functions, a and b, of both t and $\tau$, the adjoint mapping $a^{*}$ is defined by

$$a^{*}(b)=ba,$$

(14)

which immediately leads to

$$a^{*}{\partial }^m(b)=b^{(m)}a,\forall m\in \mathbb{Z}.$$

(15)

where $b^{(m)}$ denotes the m-th order derivative of b with respect to x.

Under the condition (13), the commutativity between (11) and (12) gives rise to

$$B_{n,{\tau }_k}-B_{k,t_n}+[B_n,B_k]+{[B_n,\sum _{i=1}^N{q}_j{\partial }^{-1}{r}_j]}_{\ge 0}=0.$$

(16)

Under (13), the Lax representation of (16) reads

$${\varphi }_{t_n}=B_n(\varphi ),$$

(17)

$${\varphi }_{{\tau }_k}=(B_k+\sum _{i=1}^N{q}_i{\partial }^{-1}{r}_i)(\varphi ).$$

(18)

When $n=2,k=3$ and $n=3,k=2$, (16) and (13) yield the first and second types of ncKPESCSs, respectively. It can be easily found that when $N=0$, that is, $q_i,r_i=0$, the exncKP hierarchy is reduced to the ncKP hierarchy.

3.2. Quasideterminant Solution of the exncKP Hierarchy [21]

Lemma 1.

Suppose that L is the solution of the ncKP hierarchy; $f_i,g_i(i=1,\cdots ,N)$ are $2N$ independent eigenfunctions of (17) and (18); and ${\theta }_i=f_i+b_i({\tau }_k)g_i$, $b_i({\tau }_k)$ are functions of ${\tau }_k$ satisfying $b_i^{\prime }({\tau }_k)={\beta }_i({\tau }_k){\eta }_i({\tau }_k)$. Then, the quasideterminant solution of the exncKP hierarchies (11) and (12) is given by

$$L[N]=T_{N}L{T}_N^{-1},T_N=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& q_i\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& {\partial }^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline {\partial }^{(N)}\\ \hline\end{array}\end{array}\right|,$$

(19)

$$q_i[N]={\beta }_i({\tau }_k)T_N(g_i)={\beta }_i({\tau }_k)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& g_i\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_N^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline g_i^{(N)}\\ \hline\end{array}\end{array}\right|$$

(20)

$$r_i[N]={\eta }_i({\tau }_k)\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 0\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-2)}& \cdots & {\theta }_i^{(N-2)}& \cdots & {\theta }_N^{(N-2)}& 0\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 1\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,i=1,\cdots ,N,$$

(21)

where ${\beta }_i({\tau }_k),{\eta }_i({\tau }_k)$ are functions of ${\tau }_k$ and both can commute with $f_i,g_i$ and their related derivatives.

3.3. The ncmKP Hierarchy

A pseudo-differential operator $\tilde{L}$ is defined by

$$\tilde{L}=\partial +w+w_1{\partial }^{-1}+w_2{\partial }^{-2}+\cdots ,$$

(22)

where $w,w_i(i=1,2,\cdots )$ and their related derivatives do not necessarily commute, and they are all the functions of $(t_1,t_2,t_3,\cdots )$ with $t_1=x$.

The ncmKP hierarchy is defined as follows [4]:

$${\tilde{L}}_{t_n}=[{\tilde{B}}_n,\tilde{L}],{\tilde{B}}_n={({\tilde{L}}^n)}_{\ge 1},$$

(23)

The compatibility of the $t_2$-flow and $t_3$-flow leads to the ncmKP equation below:

$$-4w_t+w_{xxx}-6w{w}_{x}w+3W_y+3[w_x,W]-3[w_{xx},w]-3[W,w^2]=0,$$

(24)

$$W_x-w_y+[w,W]=0,$$

(25)

where $t_2=y,t_3=t$, and the change in variables $w_1=-{\displaystyle \frac{1}{2}}(w_x+w^2-W)$ is performed.

The Lax pair for (24) and (25) is given by

$${\psi }_y={\psi }_{xx}+2w{\psi }_x,$$

(26)

$${\psi }_t={\psi }_{xxx}+3w{\psi }_{xx}+{\displaystyle \frac{3}{2}}(w_x+w^2+W){\psi }_x.$$

(27)

4. The exncmKP Hierarchy and ncmKP Equation with Self-Consistent Sources (ncmKPESCSs)

Similar to the construction of the exncKP hierarchies (11) and (12), the exncmKP hierarchy can also be defined as follows:

$${\tilde{L}}_{t_n}=[{\tilde{B}}_n,\tilde{L}],{\tilde{B}}_n={({\tilde{L}}^n)}_{\ge 1},$$

(28)

$${\tilde{L}}_{{\tau }_k}=[{\tilde{B}}_k+\sum _{i=1}^N{\tilde{q}}_i{\partial }^{-1}{\tilde{r}}_i\partial ,\tilde{L}],$$

(29)

where ${\tilde{q}}_i$ and ${\tilde{r}}_i$ satisfy

$${\tilde{q}}_{i,t_n}={\tilde{B}}_n({\tilde{q}}_i),{\tilde{r}}_{i,t_n}=-{(\partial {\tilde{B}}_n{\partial }^{-1})}^{*}({\tilde{r}}_i),$$

(30)

$\tilde{L}$ is defined as (22), and $w,w_i,\widetilde{q_i},\widetilde{r_i}$ are the functions of $t=(t_1,t_2,t_3,\cdots )$ and $\tau =({\tau }_1,{\tau }_2,\cdots )$. With the same proof as [14], under the conditions (30), we can show the commutativity between (28) and (29), which gives rise to

$${\tilde{B}}_{n,{\tau }_k}-{\tilde{B}}_{k,t_n}+[{\tilde{B}}_n,{\tilde{B}}_k]+{[{\tilde{B}}_n,\sum _{i=1}^N{\tilde{q}}_i{\partial }^{-1}{\tilde{r}}_i\partial ]}_{\ge 1}=0.$$

(31)

Under (30), the Lax representation of (31) reads

$${\psi }_{t_n}={\tilde{B}}_n(\psi ),$$

(32)

$${\psi }_{{\tau }_k}=({\tilde{B}}_k+\sum _{i=1}^N{\tilde{q}}_i{\partial }^{-1}{\tilde{r}}_i\partial )(\psi ).$$

(33)

When $n=2$ and $k=3$, (31) yields

$$\begin{array}{cc}\hfill & 2w_{{\tau }_3}-2w_{xxx}-3w_{1,xx}-6w{w}_{1,x}-3w_{1,t_2}-6w_x{w}_1-6w{w}_{xx}-6w_x^2-6w^2{w}_x\hfill \\ \hfill & -6[w^2,w_1]-2[\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i,w]+2\sum _{i=1}^N{({\tilde{q}}_i{\tilde{r}}_i)}_x=0.\hfill \end{array}$$

(34)

Setting ${\tau }_3=t,t_2=y$ and making the change in variables $w_1=-{\displaystyle \frac{1}{2}}(w_x+w^2-W)$, we obtain the first type of ncmKPESCS:

$$\begin{array}{cc}\hfill & -4w_t+w_{xxx}-6w{w}_{x}w+3W_y+3[w_x,W]-3[w_{xx},w]\hfill \\ \hfill & -3[W,w^2]+4[\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i,w]-4\sum _{i=1}^N{({\tilde{q}}_i{\tilde{r}}_i)}_x=0,\hfill \end{array}$$

(35)

$$W_x+w_y+[w,W]=0,$$

(36)

where ${\tilde{q}}_i$ and ${\tilde{r}}_i$ satisfy

$${\tilde{q}}_{i,y}={\tilde{q}}_{i,xx}+2w{\tilde{q}}_{i,x},{\tilde{r}}_{i,y}={\tilde{r}}_{i,xx}+2{\tilde{r}}_{i,x}w.$$

(37)

Under (37), the Lax representation for (35) and (36) reads

$$\begin{array}{cc}\hfill & {\psi }_y={\psi }_{xx}+2w{\psi }_x,\hfill \\ \hfill & {\psi }_t={\psi }_{xxx}+3w{\psi }_{xx}+{\displaystyle \frac{3}{2}}{w}_x{\psi }_x+{\displaystyle \frac{3}{2}}{w}^2{\psi }_x+{\displaystyle \frac{3}{2}}W{\psi }_x+\sum _{i=1}^N{\tilde{q}}_i\Omega (\psi ,{\tilde{r}}_i),\hfill \end{array}$$

(38)

where ${[\Omega (\psi ,{\tilde{r}}_i)]}_x={\tilde{r}}_i{\psi }_x$.

When $n=3$ and $k=2$, (31) yields

$$\begin{array}{cc}\hfill & 2w_{{\tau }_3}-2w_{xxx}-3w_{1,xx}-6w{w}_{1,x}-3w_{1,{\tau }_2}-6w_x{w}_1-6w{w}_{xx}-6w_x^2-6w^2{w}_x\hfill \\ \hfill & -6[w^2,w_1]+3[\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i,w_1]-3[\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i,w]+3\sum _{i=1}^N{({\tilde{q}}_i{\tilde{r}}_{i,x})}_x=0.\hfill \end{array}$$

(39)

Setting $t_3=t,{\tau }_2=y$, and making the change in variables

$$w_1=-{\displaystyle \frac{1}{2}}(w_x+w^2-W-\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i),$$

(40)

we obtain the second type of ncmKPESCS:

$$\begin{array}{cc}\hfill & -4w_t+w_{xxx}-6w{w}_{x}w+3W_y+3[w_x,W]-3[w_{xx},w]-3[W,w^2]\hfill \\ \hfill & +3\sum _{i=1}^N({\tilde{q}}_{i,xx}{\tilde{r}}_i-{\tilde{q}}_i{\tilde{r}}_{i,xx}+{({\tilde{q}}_i{\tilde{r}}_i)}_y+2w{({\tilde{q}}_i{\tilde{r}}_i)}_x+{\tilde{q}}_i{\tilde{r}}_i{w}_x+w_x{\tilde{q}}_i{\tilde{r}}_i)\hfill \\ \hfill & -3\sum _{i=1}^N[{\tilde{q}}_i{\tilde{r}}_i,w^2+W-w]=0,\hfill \end{array}$$

(41)

$$W_x-w_y+[w,W]=0,$$

(42)

where $q_i$ and $r_i$ satisfy

$${\tilde{q}}_{i,t}={\tilde{q}}_{i,xxx}+3w{\tilde{q}}_{i,xx}+({\displaystyle \frac{3}{2}}{w}_x+{\displaystyle \frac{3}{2}}{w}^2+{\displaystyle \frac{3}{2}}W+{\displaystyle \frac{3}{2}}\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i){\tilde{q}}_{i,x},$$

(43)

$${\tilde{r}}_{i,t}={\tilde{r}}_{i,xxx}-3{\tilde{r}}_{i,xx}w+{\tilde{r}}_{i,x}({\displaystyle \frac{3}{2}}{w}_x-{\displaystyle \frac{3}{2}}{w}^2-{\displaystyle \frac{3}{2}}W-{\displaystyle \frac{3}{2}}\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i).$$

(44)

Under (43) and (44), the Lax representation for (41) and (42) reads

$$\begin{array}{cc}\hfill & {\psi }_t={\psi }_{xxx}+3w{\psi }_{xx}+({\displaystyle \frac{3}{2}}{w}_x+{\displaystyle \frac{3}{2}}{w}^2+{\displaystyle \frac{3}{2}}W+{\displaystyle \frac{3}{2}}\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i){\psi }_x,\hfill \\ \hfill & {\psi }_y={\psi }_{xx}+2w{\psi }_x+\sum _{i=1}^N{\tilde{q}}_i\Omega (\psi ,{\tilde{r}}_i).\hfill \end{array}$$

(45)

Remark 2.

When $N=0$, that is, ${\tilde{q}}_i,{\tilde{r}}_i=0(i=1,\cdots ,N)$, both the first type and the second type of ncmKPESCSs reduce to the ncmKP Equations (24) and (25). Moreover, it can also be found that (35)–(37) and (41)–(44) reduce to the first type and the second type of mKPESCSs in the commutative case, respectively [20].

5. Noncommutative Miura Transformation Between exncKP Hierarchy and exncmKP Hierarchy

In this section, we will investigate the noncommutative Miura transformation between the exncKP hierarchies (11) and (12) and the exncmKP hierarchies (28) and (29), which is given by the following theorem.

Theorem 1.

Let $L,q_i,r_i(i=1,\cdots ,N)$ be the solution of the exncKP hierarchies (11) and (12), and f be a particular eigenfunction of (17) and (18); the noncommutative Miura transformation is thus defined by

$$\tilde{L}=f^{-1}Lf,$$

(46)

$${\tilde{q}}_i=f^{-1}{q}_i,{\tilde{r}}_{i,x}=r_{i}f,$$

(47)

and then $\tilde{L},{\tilde{q}}_i,{\tilde{r}}_i(i=1,\cdots ,N)$ satisfy (28)–(30).

Proof. 

The proof is similar to the counterpart in the commutative case, so we omit it here.

Notice that $L=\partial +u{\partial }^{-1}+u_1{\partial }^{-2}+\cdots ,\tilde{L}=\partial +w+w_1{\partial }^{-1}+\cdots ,$ (46) gives rise to

$$w=f^{-1}{f}_x,$$

(48)

$$w_1=f^{-1}uf.$$

(49)

Hence, we obtain the noncommutative Miura transformation between the first type of ncKPESCS and ncmKPESCS as follows:

$$\begin{array}{cc}\hfill & u=-{\displaystyle \frac{1}{2}}f(w_x+w^2-W)f^{-1},\hfill \\ \hfill & q_i=f{\tilde{q}}_i,r_i={\tilde{r}}_{i,x}{f}^{-1},\hfill \end{array}$$

(50)

where f is a special solution of (17) and (18) with $n=2,k=3$.

In addition, we also obtain the noncommutative Miura transformation between the second type of ncKPESCS and ncmKPESCS:

$$\begin{array}{cc}\hfill & u=-{\displaystyle \frac{1}{2}}f(w_x+w^2-W-\sum _{i=1}^N{\tilde{q}}_i{\tilde{r}}_i)f^{-1},\hfill \\ \hfill & q_i=f{\tilde{q}}_i,r_i={\tilde{r}}_{i,x}{f}^{-1},\hfill \end{array}$$

(51)

where f is a special solution of (17) and (18) with n = 3, k = 2. □

Remark 3.

Notice that the noncommutative Miura transformations (50) and (51) allow us to obtain the new solutions to first, second type of ncKPESCS from the known solutions to the first, second type of ncmKPESCS, respectively.

6. Quasideterminant Solution of the exncmKP Hierarchy

In this section, we will apply the noncommutative Miura transformation (46) and (47) to derive the quasideterminant solution of the exncmKP hierarchies (28)–(30).

Taking the special solution of (17) and (18)

$$f=T_N(1)=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,$$

(52)

we have

$$f^{-1}{T}_N=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& \begin{array}{|c|}\hline 0\\ \hline\end{array}\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& 1\end{array}\right|·\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& {\partial }^{N-1}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline {\partial }^N\\ \hline\end{array}\end{array}\right|.$$

(53)

Using the homological relation (10), we obtain

$${\tilde{T}}_N\triangleq f^{-1}{T}_N=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& \begin{array}{|c|}\hline 1\\ \hline\end{array}\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& {\partial }^{N-1}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& {\partial }^N\end{array}\right|.$$

(54)

Moreover, by using the derivative Formula (6) and (7) of the quasideterminant, we attain

$$\begin{array}{cc}\hfill & {\left(\begin{array}{c}\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|\end{array}\right)}_x\hfill \\ \hfill & =\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-2)}& \cdots & {\theta }_i^{(N-2)}& \cdots & {\theta }_N^{(N-2)}& 0\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 1\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|.\hfill \end{array}$$

(55)

It can be easily found that the right-hand side of (55) is just $r_{i}f$.

Combining Lemma 1 with Theorem 1, we obtain the quasiterminant solution of the exncmKP hierarchies (28) and (29), which is presented by Theorem 2.

Theorem 2.

Suppose that L is the solution of the ncKP hierarchy; $f_i,g_i(i=1,\cdots ,N)$ are independent eigenfunctions of (17) and (18) with $N=0$; ${\theta }_i=f_i+b_i({\tau }_k)g_i$; and the quasideterminant solution of the exncmKP hierarchies (28) and (29) is given by

$$\tilde{L}[N]=f^{-1}L[N]f={\tilde{T}}_{N}L{\tilde{T}}_N^{-1},$$

(56)

$${\tilde{q}}_i[N]={\beta }_i({\tau }_k)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& \begin{array}{|c|}\hline g_i\\ \hline\end{array}\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& g_i^{(N)}\end{array}\right|,$$

(57)

$${\tilde{r}}_i[N]={\eta }_i({\tau }_k)\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|$$

(58)

where $L[N]=\partial +u[N]{\partial }^{-1}+u_1[N]{\partial }^{-2}+\cdots ,\tilde{L}[N]=\partial +w[N]+w_1[N]{\partial }^{-1}+\cdots ,b_i({\tau }_k)$, ${\beta }_i({\tau }_k)$ and ${\eta }_i({\tau }_k)$ are the functions of ${\tau }_k$ satisfying $b_i^{\prime }({\tau }_k)={\beta }_i({\tau }_k)·{\eta }_i({\tau }_k)$, and ${\tilde{T}}_N$ is defined by (54).

As the byproduct of (56)–(58), we obtain the quasidetermiant solutions for two types of ncmKPESCSs (35)–(37) and (41)–(44), which will be explored in Section 7.

7. Quasideterminant Solutions for Two Types of ncmKPESCSs

In this section, we start from the vacuum solution $w=0,W=0$ for (35)–(37) and (41)–(44) with $N=0$, and apply (56)–(58) to construct the quasideterminant solutions for the first and second types of ncmKPESCSs (35)–(37) and (41)–(44), respectively. For convenience, some notations are introduced as follows:

$$Q(i,j)=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 0\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-j-1)}& \cdots & {\theta }_N^{(N-j-1)}& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N+i)}& \cdots & {\theta }_N^{(N+i)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,\widehat{Q}(i,j)=\left|\begin{array}{cccc}{\theta }_1^{(1)}& \cdots & {\theta }_N^{(1)}& 0\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-j)}& \cdots & {\theta }_N^{(N-j)}& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& 0\\ {\theta }_1^{(N+i+1)}& \cdots & {\theta }_N^{(N+i+1)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,$$

$$f(j)=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N+j)}& \cdots & {\theta }_N^{(N+j)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,$$

$Q\triangleq Q(0,0),\widehat{Q}\triangleq \widehat{Q}(0,0)$. Obviously, $f=f(0)$. In addition, it was also shown in [4] for the following two identities:

$$Q_x=Q(1,0)-Q(0,1)+Q^2,$$

(59)

$$f(i+1)=-\widehat{Q}(i,0)f.$$

(60)

We notice that (56) implies

$$w[N]=f^{-1}{f}_x,$$

(61)

$$w_1[N]=f^{-1}u[N]f.$$

(62)

7.1. The Quasideterminant Solutions for the First Type of ncmKPESCS

For the first type of ncmKPESCS, we have ${\theta }_i=f_i+b_i(t)g_i$ and

$$w_1[N]=-{\displaystyle \frac{1}{2}}\{{(w[N])}_x+w^2[N]-W[N]\}.$$

(63)

Substituting (62) into (63) and noting that $u[N]={\displaystyle \frac{{(v[N])}_x}{2}},v[N]=-2Q$, we have

$$W[N]={(w[N])}_x+w^2[N]-2f^{-1}{Q}_{x}f.$$

(64)

Substituting (61) into (64) leads to

$$W[N]=f^{-1}{f}_{xx}-2f^{-1}{Q}_{x}f.$$

(65)

From the derivative formulae of quasideterminant (22) and (23), a direct computation leads to

$$f_x=f(1)+Qf,$$

(66)

$$f_y=f(2)+Q(0,1)f+Qf(1),$$

(67)

$${\widehat{Q}}_x=\widehat{Q}(1,0)-\widehat{Q}(0,1)+{\widehat{Q}}^2.$$

(68)

Owing to (60), (66) and (67) become

$$f_x=(Q-\widehat{Q})f,$$

(69)

$$f_y=-\widehat{Q}(1,0)f+Q(0,1)f-Q\widehat{Q}f.$$

(70)

Differentiating both sides of (69) with respect to x,

$$f_{xx}={(Q-\widehat{Q})}_{x}f+(Q-\widehat{Q})f_x={(Q-\widehat{Q})}_{x}f+{(Q-\widehat{Q})}^{2}f.$$

(71)

Substituting (71) into (65), we attain

$$W[N]=f^{-1}{f}_{xx}-2f^{-1}{Q}_{x}f=f^{-1}(-\widehat{Q}(1,0)-Q\widehat{Q}+Q(1,0)+Q^2-Q_x)f.$$

(72)

Noting the formulae (59) and (70), we have

$$W[N]=f^{-1}{f}_{xx}-2f^{-1}{Q}_{x}f=f^{-1}(-\widehat{Q}(1,0)-Q\widehat{Q}+Q(1,0))f=f^{-1}{f}_y.$$

(73)

Therefore, we obtain the quasideterminant solution for the first type of ncmKPESCS, (35)–(37) given by

$$w[N]=f^{-1}{f}_x,$$

(74)

$$W[N]=f^{-1}{f}_y,$$

(75)

$${\tilde{q}}_i[N]={\beta }_i(t)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_n& \begin{array}{|c|}\hline g_i\\ \hline\end{array}\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& g_i^{(N)}\end{array}\right|,$$

(76)

$${\tilde{r}}_i[N]={\eta }_i(t)\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|,$$

(77)

where $f_i,g_i(i=1,\cdots ,N)$ are $2N$ independent eigenfunctions of (17) and (18) with $n=2$, $k=3,b_i^{\prime }(t)={\beta }_i(t){\eta }_i(t)$.

7.2. The Quasideterminant Solutions for the Second Type of ncmKPESCS

For the second type of ncmKPESCS, we have

$$\begin{array}{cc}\hfill & w_1[N]=-{\displaystyle \frac{1}{2}}\{{(w[N])}_x+w^2[N]-W[N]-\sum _{i=1}^N{\tilde{q}}_i[N]{\tilde{r}}_i[N]\}\hfill \\ \hfill & \Rightarrow fW[N]=f_{xx}-2Q_{x}f-f\sum _{i=1}^N{\tilde{q}}_i[N]{\tilde{r}}_i[N].\hfill \end{array}$$

(78)

Noting that $b_i^{\prime }(y)={\beta }_i(y)·{\eta }_i(y)$, we obtain

$$\begin{array}{cc}\hfill & f\sum _{i=1}^N{\tilde{q}}_i[N]{\tilde{r}}_i[N]\hfill \\ \hfill & =f\sum _{i=1}^N{\beta }_i(y)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& \begin{array}{|c|}\hline g_i\\ \hline\end{array}\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& g_i^{(N)}\end{array}\right|·{\eta }_i(y)\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_N& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|\hfill \\ \hfill & =f·f^{-1}\sum _{i=1}^N{b}_i^{\prime }(y)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& g_i\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline g_i^{(N)}\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|\hfill \\ \hfill & =\sum _{i=1}^N{b}_i^{\prime }(y)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& g_i\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline g_i^{(N)}\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|.\hfill \end{array}$$

(79)

Similar to the proof of Formula (5.6) in [21], we can prove that

$$\begin{array}{cc}\hfill & \sum _{i=1}^N{b}_i^{\prime }(y)\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& g_i\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& g_i^{(N-1)}\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline g_i^{(N)}\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccccc}{\theta }_1& \cdots & {\theta }_i& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_i^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ 0& \cdots & 1& \cdots & 0& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|=\hfill \\ \hfill & \left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ b_1^{\prime }(y)g_1^{(N)}& \cdots & b_N^{\prime }(y)g_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|+\sum _{k=1}^N\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 0\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(k-1)}& \cdots & {\theta }_N^{(k-1)}& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ b_1^{\prime }(y)g_1^{(k-1)}& \cdots & b_N^{\prime }(y)g_N^{(k-1)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|.\hfill \end{array}$$

(80)

Furthermore, we also find that

$$\begin{array}{cc}& {\theta }_i^{(N)}=f_i^{(N)}+b_i(y)g_i^{(N)}\Rightarrow {\theta }_{i,y}^{(N)}=f_{i,y}^{(N)}+b_i(y)g_{i,y}^{(N)}+b_i^{\prime }(y)g_i^{(N)}\hfill \\ & \Rightarrow {\theta }_{i,y}^{(N)}=f_i^{(N+2)}+b_i(y)g_i^{(N+2)}+b_i^{\prime }(y)g_i^{(N)}={\theta }_i^{(N+2)}+b_i^{\prime }(y)g_i^{(N)}.\hfill \end{array}$$

By using the derivative formulae of quasideterminant (22) and (23), we have

$$\begin{array}{cc}\hfill & f_y=\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_{1,y}^{(N)}& \cdots & {\theta }_{N,y}^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|+\sum _{k=1}^N\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 0\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(k-1)}& \cdots & {\theta }_N^{(k-1)}& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_{1,y}^{(k-1)}& \cdots & {\theta }_{N,y}^{(k-1)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|\hfill \\ \hfill & =f_{xx}-2Q_{x}f+\left(\begin{array}{cc}& \left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ b_1^{\prime }(y)g_1^{(N)}& \cdots & b_N^{\prime }(y)g_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|+\\ & \sum _{k=1}^N\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 0\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(k-1)}& \cdots & {\theta }_N^{(k-1)}& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ {\theta }_1^{(N)}& \cdots & {\theta }_N^{(N)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|·\left|\begin{array}{cccc}{\theta }_1& \cdots & {\theta }_N& 1\\ ⋮& & ⋮& ⋮\\ {\theta }_1^{(N-1)}& \cdots & {\theta }_N^{(N-1)}& 0\\ b_1^{\prime }(y)g_1^{(k-1)}& \cdots & b_N^{\prime }(y)g_N^{(k-1)}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|\end{array}\right).\hfill \end{array}$$

(81)

Noting that (79) and (80), and comparing (78) and (81), we attain

$$f·W[N]=f_y\Rightarrow W[N]=f^{-1}{f}_y.$$

(82)

Therefore, we obtain the quasideterminant solution for the second type of ncmKPESCS (41)–(44) defined as (74)–(77) with t repalaced by y, in which $f_i,g_i(i=1,\cdots ,N)$ are $2N$ independent eigenfunctions of (17) and (18) with $n=3,k=2,{\theta }_i=f_i+b_i(y)g_i$.

8. Matrix Solutions for ncmKPSCSs

In this section, inspired by [5], we will start from the vacuum solution $w=0,W=0$ for (35)–(37) and (41)–(44) with $N=0$ to construct the matrix solutions for two types of ncmKPESCSs. Here, we only investigate the one-soliton matrix solutions for them. It is noticed that the eigenfunctions $f_i$ and $g_i$ both satisfy (38) and (45) with $w=0$, $W=0$, $N=0$, that is,

$$f_{i,y}=f_{i,xx},f_{i,t}=f_{i,xxx},$$

(83)

$$g_{i,y}=g_{i,xx},g_{i,t}=g_{i,xxx}.$$

(84)

The simplest solutions of (83) and (84) are

$$f_i=I{e}^{{\zeta }_i},{\zeta }_i={\lambda }_i(x+{\lambda }_{i}y+{\lambda }_i^{2}t),$$

$$g_i=P_i{e}^{{\eta }_i},{\eta }_i={\mu }_i(x+{\mu }_{i}y+{\mu }_i^{2}t),$$

where $P_i$ is a projection matrix and I is a unit matrix.

Next, we will investigate the one-soliton matrix solutions for two types of ncmKPESCSs. In the case of $N=1$,

$$f=\left|\begin{array}{cc}{\theta }_1& 1\\ {\theta }_{1,x}& \begin{array}{|c|}\hline 0\\ \hline\end{array}\end{array}\right|=-{\theta }_{1,x}{\theta }_1^{-1}.$$

(85)

For the first type of ncmKPESCS,

$${\theta }_1=f_1+b_1(t)g_1=e^{{\zeta }_1}(I+b_1(t)e^{{\eta }_1-{\zeta }_1}{P}_1),$$

which leads to

$${\theta }_{1,x}=f_{1,x}+b_1(t)g_{1,x}=e^{{\zeta }_1}({\lambda }_{1}I+b_1(t){\mu }_1{e}^{{\eta }_1-{\zeta }_1}{P}_1),$$

(86)

$${\theta }_1^{-1}=f_1+b_1(t)g_1=e^{{\zeta }_1}{(I+b_1(t)e^{{\eta }_1-{\zeta }_1}{P}_1)}^{-1}.$$

(87)

Using the important formula [4],

$${(I-aP)}^{-1}=I+aP{(1-a)}^{-1},$$

(88)

we have, from (87),

$${\theta }_1^{-1}=e^{{\zeta }_1}(I-{\displaystyle \frac{b_1(t)e^{{\eta }_1-{\zeta }_1}}{1+b_1(t)e^{{\eta }_1-{\zeta }_1}}}{P}_1).$$

(89)

(85) and (86), together with (89), lead to

$$f=-{\lambda }_{1}I+({\lambda }_1-{\mu }_1){\displaystyle \frac{b_1(t)e^{{\eta }_1-{\zeta }_1}}{1+b_1(t)e^{{\eta }_1-{\zeta }_1}}}{P}_1.$$

(90)

From (74) to (77), we obtain a one-soliton matrix solution for the first type of ncmKPESCS (35)–(37) with $N=1$:

$$w[1]=-{\displaystyle \frac{({\lambda }_1-{\mu }_1)}{4\sqrt{{\lambda }_1{\mu }_1}}}{P}_{1}sech(\Delta +{\displaystyle \frac{1}{2}}ln{\displaystyle \frac{{\mu }_1}{{\lambda }_1}}),$$

(91)

$$W[1]=({\lambda }_1+{\mu }_1)w[1],$$

(92)

$${\tilde{q}}_1[1]={\beta }_1(t){\displaystyle \frac{{\lambda }_1-{\mu }_1}{\sqrt{{\lambda }_1{\mu }_1}}}{P}_{1}sech(\Delta +{\displaystyle \frac{1}{2}}ln{\displaystyle \frac{{\mu }_1}{{\lambda }_1}})e^{{\displaystyle \frac{{\eta }_1+{\zeta }_1-ln{b}_1(t)}{2}}},$$

(93)

$${\tilde{r}}_1[1]=-{\eta }_1(t)e^{-{\zeta }_1}(I-{\displaystyle \frac{1}{2}}{e}^{{\displaystyle \frac{\Delta }{2}}}sech\Delta ·P_1).$$

(94)

where

$$\Delta ={\displaystyle \frac{{\eta }_1-{\zeta }_1+ln{b}_1(t)}{2}},b_1^{\prime }(t)={\beta }_1(t)·{\eta }_1(t).$$

(95)

Similarly, we obtain the one-soliton matrix solution for the second type of ncmKPESCS (41)–(44) with $N=1$:

$$w[1]=-{\displaystyle \frac{({\lambda }_1-{\mu }_1)}{4\sqrt{{\lambda }_1{\mu }_1}}}{P}_{1}sech(\Delta +{\displaystyle \frac{1}{2}}ln{\displaystyle \frac{{\mu }_1}{{\lambda }_1}}),$$

(96)

$$W[1]=({\lambda }_1+{\mu }_1)w[1]+{\tilde{q}}_1[1]·{\tilde{r}}_1[1],$$

(97)

$${\tilde{q}}_1[1]={\beta }_1(y){\displaystyle \frac{{\lambda }_1-{\mu }_1}{\sqrt{{\lambda }_1{\mu }_1}}}{P}_{1}sech(\Delta +{\displaystyle \frac{1}{2}}ln{\displaystyle \frac{{\mu }_1}{{\lambda }_1}})e^{{\displaystyle \frac{{\eta }_1+{\zeta }_1-ln{b}_1(y)}{2}}},$$

(98)

$${\tilde{r}}_1[1]=-{\eta }_1(y)e^{-{\zeta }_1}(I-{\displaystyle \frac{1}{2}}{e}^{{\displaystyle \frac{\Delta }{2}}}sech\Delta ·P_1).$$

(99)

where $\Delta$ is defined by (95) with t replaced by y, $b_1^{\prime }(y)={\beta }_1(y)·{\eta }_1(y)$.

Remark 4.

When $N=0$, that is, ${\tilde{q}}_1[1]=0$ or ${\tilde{r}}_1[1]=0$, both (91)–(95) and (96)–(99) are reduced to a one-soliton matrix solution of ncmKP equation, which has been investigated in [5]. Comparing (91) and (92) with a one-soliton matrix solution of the ncmKP equation, we find that the velocity of propagation in the t direction of the one-soliton matrix solution for ncmKPESCS with $N=1$ can be modified by the choice of the function $b_1(t)$. Therefore, we conclude that for the first type of ncmKPESCS, the source terms result in a variation in the velocity of noncommutative soliton in the $t-$ direction. While comparing (96) and (97) with the one-soliton matrix solution of the ncmKP equation, not only can the velocity of propagation in the y direction be modified by the choice of the function $b_1(y)$, but also the shape, which shows that for the second type of ncmKPESCS, source terms result in both the variation in velocity in the $y-$ direction and the shape of noncommutative soliton.

9. Summary and Discussion

In this paper, the square eigenfunctions in nc space–time are introduced to extend the ncmKP hierarchy. The commutativity between the ncmKP flow and the extended ncmKP flow was shown to give rise to two types of ncmKPESCSs. The quasideterminant solution of the extended ncmKP hierarchy is constructed by using the noncommutative Miura transformation between the exncKP hierarchy and exncmKP hierarchy. From the matrix solutions for the different types of ncSESCSs, the impacts of the introduction of source terms on noncommutative solitons are shown to be different. We find that the method in our paper provides an efficient and unified way to construct extensions of nc integrable systems. It is believed that this approach can be used to produce extensions of other nc integrable systems such as the non-Abelian Toda lattice hierarchy. The extension of the supersymmetry integrable systems will be investigated in our future study.