q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

Using the determinant representation of gauge transformation operator, we have shown that the general form of $\tau$ function of the $q$-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP ($q$-cKP) hierarchy, i.e. $l$-constraints of $q$-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of $q$-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear $q$-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of $q$-deformation ($q$-effects) in single $q$-soliton from the simplest $\tau$ function of the $q$-KP hierarchy and in multi-$q$-soliton from one-component $q$-cKP hierarchy, and their dependence of $x$ and $q$, were also presented. Finally, we observe that $q$-soliton tends to the usual soliton of the KP equation when $x\to 0$ and $q\to 1$, simultaneously.


Introduction
Study of the quantum calculus (or q-calculus) [1,2] has a long history, which may go back to the beginning of the twentieth century. F.H. Jackson was the first mathematician who studied the q-integral and q-derivative in a systematic way starting about 1910 [3,4] 1 . Since 1980's, the quantum calculus was re-discovered in the research of quantum group inspired by the studies on quantum integrable model that used the quantum inverse scattering method [5] and on noncommutative geometry [6]. In particular, S. Majid derived the q-derivative from the braided differential calculus [7,8].
The q-deformed integrable system (also called q-analogue or q-deformation of classical integrable system) is defined by means of q-derivative ∂ q instead of usual derivative ∂ with respect to x in a classical system. It reduces to a classical integrable system as q → 1. Recently, the q-deformation of the following three stereotypes for integrable systems attracted more attention. The first type is q-deformed N -th KdV (q-NKdV or q-Gelfand-Dickey hierarchy) [9,16], which is reduced to the N -th KdV (NKdV or Gelfand-Dickey) hierarchy when q → 1. The N -th q-KdV hierarchy becomes q-KdV hierarchy for N = 2. The q-NKdV hierarchy inherited several integrable structures from classical N -th KdV hierarchy, such as infinite conservation laws [10], bi-Hamiltonian structure [11,12], τ function [13,14], Bäcklund transformation [15]. The second type is the q-KP hierarchy [17,22]. Its τ function, bi-Hamiltonian structure and additional symmetries have already been reported in [20,21,18,22]. The third type is the q-AKNS-D hierarchy, and its bilinear identity and τ function were obtained in [23].
In order to get the Darboux-Bäcklund transformations, the two elementary types of gauge transformation operators, differential-type denoted by T (or T D ) and integral-type denoted by S (or T I ), for q-deformed N -th KdV hierarchy were introduced in [15]. Tu et al. obtained not only the q-deformed Wronskian-type but also binary-type representations of τ function of q-KdV hierarchy. On the basis of their results, He et al. [24] obtained the determinant representation of gauge transformation operators T n+k (n ≥ k) for q-Gelfand-Dickey hierarchy, which is a mixed iteration of n-steps of T D and then k-steps of T I . Then, they obtained a more general form of τ function for q-KdV hierarchy, i.e., generalized q-deformed Wronskian (q-Wronskian) IW q n+k [24]. It is important to note that for k = 0 IW q n+k reduces to q-deformed Wronskian and for k = n to binary-type determinant [15]. On the other hand, Tu introduced the q-deformed constrained KP (q-cKP) hierarchy [22] by means of symmetry constraint of q-KP hierarchy, which is a q-analogue of constrained KP (cKP) hierarchy [25,31].
The purpose of this paper is to construct the τ function of q-KP and q-cKP hierarchy, and then explore the q-effect in q-solitons. The main tool is the determinant representation of gauge transformation operators [32,33,34,35]. The paper is organized as follows: In Section 2 we introduce some basic facts on the q-KP hierarchy, such as Lax operator, Z-S equations, the existence of τ function. On the basis of the [15], two kinds of elementary gauge transformation operators for q-KP hierarchy and changing rule of q-KP hierarchy under it are presented in Section 3. In Section 4, we establish the determinant representation of gauge transformation operator T n+k for the q-KP hierarchy and then obtain the general form of τ function τ (n+k) q = IW q n+k . In particular, by taking n = 1, k = 0 we will show q-effect of single q-soliton solution of q-KP hierarchy. A brief description of the sub-hierarchy of q-cKP hierarchy is presented in Section 5, from the viewpoint of the symmetry constraint. In Section 6, we show that the q-Wronskian is one kind of forms of τ function of q-cKP if the functions in the q-Wronskian satisfy some restrictions. In Section 7 we consider an example which illustrates the procedure reducing q-KP to q-cKP hierarchy. q-effects of the q-deformed multi-soliton are also discussed. The conclusions and discussions are given in Section 8. Our presentation is similar to the relevant papers of classical KP and cKP hierarchy [32,34,36,37,38].
At the end of this section, we shall collect some useful formulae for reader's convenience. The q-derivative ∂ q is defined by and the q-shift operator is given by Let ∂ −1 q denote the formal inverse of ∂ q . We should note that θ does not commute with ∂ q , (∂ q θ k (f )) = q k θ k (∂ q f ), k ∈ Z.
In general, the following q-deformed Leibnitz rule holds: where the q-number and the q-binomial are defined by (n) q = q n − 1 q − 1 , n k q = (n) q (n − 1) q · · · (n − k + 1) q (1) q (2) q · · · (k) q , n 0 q = 1, and "•" means composition of operators, defined by ∂ q • f = (∂ q · f ) + θ(f )∂ q . In the remainder of the paper for any function f "·" is defined by ∂ q · f = ∂ q (f ) (∂ q f ). For a q-pseudodifferential operator (q-PDO) of the form P = n i=−∞ p i ∂ i q , we decompose P into the differential part P + = i≥0 p i ∂ i q and the integral partP − = i≤−1 p i ∂ i q . The conjugate operation " * " for P is We can write out several explicit forms of (1.1) for q-derivative ∂ q , as More explicit expressions of ∂ n q • f are given in Appendix A. In particular, ∂ −1 q • f has different forms, which will be used in the following sections. The q-exponent e q (x) is defined as follows Its equivalent expression is of the form The form (1.7) will play a crucial role in proving the existence [20] of τ function of q-KP hierarchy.

q-KP hierarchy
Similarly to the general way of describing the classical KP hierarchy [36,37], we shall give a brief introduction of q-KP based on [20]. Let L be one q-PDO given by which is called Lax operator of q-KP hierarchy. There exist infinite q-partial differential equations relating to dynamical variables {u i (x, t 1 , t 2 , t 3 , . . .), i = 0, −1, −2, −3, . . .}, and they can be deduced from generalized Lax equation, which are called q-KP hierarchy. Here B n = (L n ) + = n i=0 b i ∂ i q means the positive part of q-PDO, and we will use L n − = L n − L n + to denote the negative part. By means of the formulae given in (1.2)-(1.6) and in Appendices A and B, the first few flows in (2.2) for dynamical variables {u 0 , u −1 , u −2 , u −3 } can be written out as follows. The first flow is The second flow is The third flow is Obviously, ∂ t 1 = ∂ and equations of flows here are reduced to usual KP flows (4.10) and (4.11) in [39] when q → 1 and u 0 = 0. If we only consider the first three flows, i.e. flows of (t 1 , t 2 , t 3 ), In other words, u −1 = u(t 1 , t 2 , t 3 ) in the above equation when q → 1, and hence u −1 is called a q-soliton if u(t 1 , t 2 , t 3 ) = lim q→1 u −1 is a soliton solution of KP equation.
On the other hand, L in (2.1) can be generated by dressing operator S = 1 + ∞ k=1 s k ∂ −k q in the following way Further, the dressing operator S satisfies the Sato equation The q-wave function w q (x, t) and q-adjoint wave function w * q (x, t) for q-KP hierarchy are defined by and where t = (t 1 , t 2 , t 3 , . . .). Here, for a q-PDO P = i p i (x)∂ i q , the notation is used in (2.6). Note that w q (x, t) and w * q (x, t) satisfy following linear q-differential equations, (Lw q ) = zw q , ∂w q ∂t n = (B n w q ), Furthermore, w q (x, t) and w * q (x, t) can be expressed by sole function τ q (x, t) as From comparison of (2.5) and (2.8), the dressing operator S has the form of
Beside existence of the Lax operator, q-wave function, τ q for q-KP hierarchy, another important property is the q-deformed Z-S equation and associated linear q-differential equation. In other words, q-KP hierarchy also has an alternative expression, i.e., The "eigenfunction" φ and "adjoint eigenfunction" ψ of q-KP hierarchy associated to (2.11) are defined by ∂φ ∂t n = (B n φ), (2.12) where φ = φ(λ; x, t) and ψ = ψ(µ; x, t). Here (2.13) is different from the second equation in (2.7). φ i ≡ φ(λ i ; x, t) and ψ i ≡ ψ(µ i ; x, t) will be generating functions of gauge transformations.

Gauge transformations of q-KP hierarchy
The authors in [15] reported two types of elementary gauge transformation operator only for q-Gelfand-Dickey hierarchy. We extended the elementary gauge transformations given in [15], for the q-KP hierarchy. At the same time, we shall add some vital operator identity concerning to the q-differential operator and its inverse. Here we shall prove two transforming rules of τ function, "eigenfunction" and "adjoint eigenfunction" of the q-KP hierarchy under these transformations. Majority of the proofs are similar to the classical case given by [32,33] and [35], so we will omit part of the proofs. Suppose T is a pseudo-differential operator, and still holds for the transformed Lax operator L (1) ; then T is called a gauge transformation operator of the q-KP hierarchy.
If the initial Lax operator of q-KP is a "free" operator L = ∂ q , then the gauge transformation operator is also a dressing operator for new q-KP hierarchy whose Lax operator L (1) = T • ∂ q • T −1 , because of (3.2) becomes which is the Sato equation (2.4). In order to prove existence of two types of the gauge transformation operator, the following operator identities are necessary.
Lemma 2. Let f be a suitable function, and A be a q-deformed pseudo-differential operator, then Remark 1. This lemma is a q-analogue of corresponding identities of pseudo-differential operators given by [33].
Theorem 1. There exist two kinds of gauge transformation operator for the q-KP hierarchy, namely Type II : Here φ 1 and ψ 1 are defined by (2.12) and (2.13) that are called the generating functions of gauge transformation.

Taking this expression back into B
(1) n , we get and that indicates that T D (φ 1 ) is indeed a gauge transformation operator via Lemma 1. Second, we want to prove that the equation (3.2) holds for Type II case (see (3.7)). By direct calculation the left hand side of (3.2) is in the form of In the above calculation, the operator identity (3.5), (B n ) − = 0, (ψ 1 ) tn = −(B * n · ψ 1 ) were used. Moreover, with the help of (1.2), we have The two equations obtained above show that T I (ψ 1 ) satisfies (3.2), so T I (ψ 1 ) is also a gauge transformation operator of the q-KP hierarchy according to Lemma 1.

Remark 2.
There are two convenient expressions for T D and T I , In order to get a new solution of q-KP hierarchy from the input solution, we should know the transformed expressions of u i . The following two theorems are related to this. Before we start to discuss explicit forms of them, we would like to define the generalized q-Wronskian for a set of functions {ψ k , ψ k−1 , . . . , ψ 1 ; φ 1 , φ 2 , . . . , φ n } as which reduce to the q-Wronskian when k = 0, Suppose φ 1 (λ 1 ; x, t) is a known "eigenfunction" of q-KP with the initial function τ q , which generates gauge transformation operator T D (φ 1 ). Then we have new "eigenfunction", "adjoint eigenfunction" and τ function of the transformed q-KP hierarchy are in which (2.12) and (3.1) were used.
(2) Similarly, with the help of (B (3) According to the definition of T D in (3.6) and with the help of (3.8), L (1) can be expressed as Then This completes the proof of the theorem.
For the gauge transformation operator of Type II, there exist similar results. Let ψ 1 (µ 1 ; x, t) be a known "adjoint eigenfunction" of q-KP with the initial function τ q , which generates the gauge transformation operator T I (ψ 1 ). Then we have new "eigenfunction", "adjoint eigenfunction" and τ function of the transformed q-KP hierarchy are Proof . The proof is analogous to the proof of the previous theorem. So it is omitted.

Successive applications of gauge transformations
We now discuss successive applications of the two types of gauge transformation operators in a general way, which is similar to the classical case [32,34,35]. For example, consider the chain of gauge transformation operators, Here the index "i" in a gauge transformation operator means the i-th gauge transformation, and φ is transformed by j-step gauge transformations from the initial Lax operator L. Successive applications of gauge transformation operator in (4.1) can be represented by Our goal is to obtain φ (n+k) , ψ (n+k) , τ (n+k) q of L (n+k) transformed from L by the T n+k in the above chain. All of these are based on the determinant representation of gauge transformation operator T n+k . As the proof of the determinant representation of T n+k is similar extremely to the case of classical KP hierarchy [34], we will omit it.
Lemma 3. The gauge transformation operator T n+k has the following determinant representation (n > k): .

Lemma 4.
Under the case of n = k, T n+k is given by In the above lemmas, T n+k are expanded with respect to the last column collecting all subdeterminants on the left of the symbols ∂ i q (i = −1, 0, 1, 2, . . . , n − k); T −1 n+k are expanded with respect to the first column by means of collection of all minors on the right of φ i ∂ −1 q . Basing on the determinant representation, first of all, we would like to consider the case of k = 0 in (4.1), i.e.
whose corresponding equivalent gauge transformation operator is Theorem 4. Under the gauge transformation T n (n ≥ 1), Proof . (1) Successive application of Theorem 2 implies Using the determinant representation of T n in it leads to φ (n) . Here T (1) (2) Similarly, according to Theorem 2 we have Then ψ (n) can be deduced by using the determinant representation of T −1 n in the Lemma 3 with k = 0. Here we omit the generating functions in T with the help of the determinant representation of Lemma 3 with k = 0. Here W q 1 (φ 1 ) = φ 1 .
It should be noted that there is a θ action in (4.4), which is the main difference between the q-KP and classical KP beside different elements in determinant. Furthermore, for more complicated chain of gauge transformation operators in (4.1), φ (n+k) , ψ (n+k) , τ (n+k) q of L (n+k) can be expressed by the generalized q-Wronskian.
Theorem 5. Under the gauge transformation T n+k (n > k > 0), Proof . (1) The repeated iteration of Theorems 2 and 3 according to the ordering of T I and T D deduces Then taking in it φ (n) = (T n · φ) from (4.3) , we get Therefore the determinant form of φ (n+k) is given by Lemma 3.
Remark 3. There exists another complicated chain of gauge transformation operators for q-KP hierarchy (that may be regarded as motivated by the classical KP hierarchy) that can lead to another form of τ (n+k) q . This is parallel to the classical case of [32].
If the initial q-KP is a "free" operator, then L = ∂ q that means the initial τ function is 1. We can write down the explicit form of q-KP hierarchy generated by T n+k . Under this situation, (2.12) and (2.13) become that possess set of solution {φ i , ψ i } as follows After the (n+k)-th step gauge transformation T n+k , the final form of τ q can be given in following corollary, which can be deduced directly from Theorems 4 and 5.
Corollary 1. The gauge transformation can generate the following two forms of τ function of the q-KP hierarchy, (4.8) Here {φ i , ψ i } are def ined by (4.6) and (4.7).
On the other hand, we know from (3.3) that T n defined by (4.2) is a dressing operator if its generating functions are given by (4.6). Therefore we can define one q-wave function Corollary 2. The relationship in (2.8) between the q-wave function and τ q is satisf ied by τ (n) q in (4.8) and q-wave function in (4.9), i.e., Proof . We follow the Dickey's method on page 100 of [37] to prove the corollary. By direct computations, Comparing the fraction above of the determinant term with (4.9), we can see that they are similar, although the form of the determinant in the numerator is different. The determinant in the numerator of (4.9) can be reduced to the same form of (4.10) if the second row, divided by z, is subtracted from the first one, the third from the second etc.

Symmetry constraint of q-KP: q-cKP hierarchy
We know that there exists a constrained version of KP hierarchy, i.e. the constrained KP hierarchy (cKP) [25,31], introduced by means of the symmetry constraint from KP hierarchy.   With inspiration from it, the symmetry of q-KP was established in [22]. In the same article the authors defined one kind of constrained q-KP (q-cKP) hierarchy by using the linear combination of generators of additional symmetry. In this section, we shall briefly introduce the symmetry and q-cKP hierarchy [22]. The linearization of (2.2) is given by We call δL = δu 0 + δu 1 ∂ −1 q + · · · the symmetry of the q-KP hierarchy, if it satisfies (5.1). Let L be a "dressed" operator from ∂ q , we find where δS = δs 1 ∂ −1 q + δs 2 ∂ −2 q + · · · , and K = δSS      2). So the q-KP hierarchy admits a reduction defined by (L n ) − = 0, which is called q-deformed n-th KdV hierarchy. For example, n = 2, it leads to q-KdV hierarchy, whose q-Lax operator is There is also another symmetry called additional symmetry, which is K = (M m L l ) − [22], and it also satisfies (5.3). Here the operator M is defined by and Γ q is defined as The more general generators of additional symmetry are in form of which are constructed by combination of K = (M m L l ) − . The operator Y q (µ, λ) can be expressed as , t; λ)).  Du -1, u -1 Figure 9.  In order to define the q-analogue of the constrained KP hierarchy, we need to establish one special generator of symmetry further φ(t) and ψ(t) satisfy (2.12) and (2.13). In other words, we get a new symmetry of q-KP hierarchy, where φ(λ; x, t) and ψ(µ; x, t) is an "eigenfunction" and an "adjoint eigenfunction", respectively. We can regard from the process above that K = φ(λ; x, t) • ∂ −1 q • ψ(µ; x, t) is a special linear combination of the additional symmetry generator (M m L l ) − . It is obvious that generator K in (5.4) satisfies (5.3), because of the following two operator identities, Here A is a q-PDO, and a and b are two functions. Naturally, q-KP hierarchy also has a multicomponent symmetry, i.e.
It is well known that the integrable KP hierarchy is compatible with generalized l-constraints of this type (L l ) − = i q i • ∂ −1 x • r i . Similarly, the l-constraints of q-KP hierarchy Du -1, u -1 Figure 12. Du -1, u -1 Figure 13. also lead to q-cKP hierarchy. The flow equations of this q-cKP hierarchy It can be obtained directly by using the operator identities in (5.5). An important fact is that there exist two m-th order q-differential operators such that AL l and L l B are differential operators. From (AL l ) − = 0 and (L l B) − = 0, we get that A and B annihilate the functions φ i and ψ i , i.e., A(φ 1 ) = · · · = A(φ m ) = 0, B * (ψ 1 ) = · · · = B * (ψ m ) = 0, that implies φ i ∈ Ker (A). It should be noted that Ker (A) has dimension m. We will use this fact to reduce the number of components of the q-cKP hierarchy in the next section.
6 q-Wronskian solutions of q-cKP hierarchy We know from Corollary 1 that q-Wronskian is a τ function of q-KP hierarchy. Here φ i (i = 1, 2, . . . , N ) satisfy linear q-partial differential equations, ∂φ i ∂t n = (∂ n q φ i ), n = 1, 2, 3, . . . . (6.2) In this section, we will reduce τ (N ) q in (6.1) to a τ function of q-cKP hierarchy. To this end, we will find the additional conditions satisfied by φ i except the linear q-differential equation (6.2).
Corollary 1 also shows that the q-KP hierarchy with Lax operator L (N ) = T N • ∂ q • T −1 N is generated from the "free" Lax operator L = ∂ q , which has the τ function τ (N ) q in (6.1). In order to get the explicit form of such Lax operator L (N ) , the following lemma is necessary. .
Proof . The proof is a direct consequence of Lemma 3 and Theorem 4 from the initial "free" Lax operator L = ∂ q . The generating functions {φ i , i = 1, 2, . . . , N } of T N satisfies equations (6.2), which is obtained from definition of "eigenfunction" (2.12) of the KP hierarchy under B n = ∂ n q .

Remark 4.
This theorem is a q-analogue of the classical theorem on cKP hierarchy given by [38].
Proof . The q-Wronskian identity proven in Appendix C implies equivalence between (6.4) and (6.5). Using T N and T −1 N in Lemma 5 and the operator identity in (5.5) we have where g i is given by (6.3) and is automatically a tau function of N -component q-cKP hierarchy with the form (6.6). Next, we can reduce the N -component to the M -component (M < N ) by a suitable constraint of φ i .
Suppose that the M -component (M < N ) q-cKP hierarchy is obtained by constraint of qKP hierarchy generated by T N , i.e., there exist suitable functions {q i , r i } such that As we pointed out in previous section, for a Lax operator whose negative part is in the form of exists an M -th order q-differential operator A such that AL l is a q-differential operator, then we have . Therefore, at most M of these functions T N (∂ l q φ i ) can be linearly independent because the Kernel of A has dimension M . So (6.5) is deduced.
Conversely, suppose (6.5) is true, we will show that there exists one M-component q-ckP (M < N ) constrained from (6.6). The equation (6.5) implies that at most M of functions T N (∂ l q φ i ) (i = 1, 2, . . . , N ) are linearly independent. Then we can find suitable M functions {q 1 , q 2 , . . . , q M }, which are linearly independent, to express functions T N (∂ l q φ i ) as c ij q j , i = 1, · · · , N with some constants c ij . Taking this back into (6.6), it becomes which is an M -component q-cKP hierarchy as we expected.
7 Example reducing q-KP to q-cKP hierarchy To illustrate the method in Theorem 6 reducing the q-KP to multi-component a q-cKP hierarchy, we discuss the q-KP generated by T N | N =2 . In order to obtain the concrete solution, we only consider the three variables (t 1 , t 2 , t 3 ) in t. Furthermore, the q 1 , r 1 and u −1 are constructed in this section. According to Theorem 6, the q-KP hierarchy generated by T N | N =2 possesses a tau function Here 1, 2, 3), and ξ = d + µt 1 + µ 2 t 2 + µ 3 t 3 , c i and d are arbitrary constants. These functions satisfy the linear equations ∂φ i ∂t n = ∂ n q φ i , n = 1, 2, 3, i = 1, 2, as a special case of (6.2). From (6.6), the q-KP hierarchy generated by T N | N =2 is in the form of Here q 1 and r 1 are undetermined, which can be expressed by φ 1 and φ 2 as follows. According to (6.4), the restriction for φ 1 and φ 2 to reduce (7.2) to (7.3) is given by Obviously, we can let µ = λ 2 and d = c 2 such that (7.4) holds for φ 1 and φ 2 . Then the τ function of a single component q-cKP defined by (7.3) is which is deduced from (7.1). That means we indeed reduce the τ function τ (2) q in (7.1) of the q-KP hierarchy generated by T N | N =2 to the τ function τ qcKP of the one-component q-cKP hierarchy. Furthermore, we would like to get the explicit expression of (q 1 , r 1 ) of q-cKP in (7.3). Using the determinant representation of T N | N =2 and T −1 N | N =2 , we have under the restriction µ = λ 2 and d = c 2 . One can find that f 1 and f 2 are linearly dependent, and (λ l × θ e −(ξ 1 +ξ 2 +ξ 3 ) (λ l 3 − λ l 2 )e q (λ 1 x)e ξ 1 + (λ l 3 − λ l 1 )e q (λ 2 x)e ξ 2 + (λ l 2 − λ l 1 )e q (λ 3 x)e ξ 3 τ qcKP .

Conclusions and discussions
In this paper, we have shown in Theorem 1 that there exist two types of elementary gauge transformation operators for the q-KP hierarchy. The changing rules of q-KP under the gauge transformation are given in Theorems 2 and 3. We mention that these two types of elementary gauge transformation operators are introduced first by Tu et al. [15] for q-NKdV hierarchy. Considering successive application of gauge transformation, we established the determinant representation of the gauge transformation operator of the q-KP hierarchy in Lemma 3 and the corresponding results on the transformed new q-KP are given in Theorem 5. For the q-KP hierarchy generated by T n+k from the "free" Lax operator L = ∂ q (i.e. the Lax operator is n+k ), Corollary 1 shows that the generalized q-Wronskian IW q k,n of functions {φ i , ψ j } (i = 1, 2, . . . , n; j = 1, 2, . . . , k) is a general τ function of it, and q-Wronskian W q n of functions φ i (i = 1, 2, . . . , n) is also a special one. Here {φ i } and {ψ j } satisfy special linear q-partial differential equations (4.5).
The symmetry and symmetry constraint of q-KP (q-cKP) hierarchy are discussed in Section 5. On the basis of the representation of T N in Lemma 5, the q-KP hierarchy whose Lax operator L l = T N • ∂ l q • T −1 N is generated from the "free" Lax operator L = ∂ q . The explicit form of its negative part L l − is given in (6.6), which is called l-constraint of the q-KP hierarchy. Further we found necessary and sufficient conditions that are given in Theorem 6, reducing a q-Wronskian solution in (6.1) of the q-KP hierarchy to solutions of the multi-component q-cKP hierarchy. One example is given in Section 7 to illustrate the method, i.e., the q-KP generated by T N | N =2 is reduced to one-component q-cKP hierarchy. By taking finite variables (t 1 , t 2 , t 3 ) in t, the   component q 1 and r 1 are written out. Our results can be reduced to the classical results in [38].
As we pointed out in Section 2, u −1 is the q-analogue of the solution of classical KP equation if we only consider three variables (t 1 , t 2 , t 3 ) in t. Therefore, the solution u −1 is called q-soliton of the q-KP equation, although we do not write out the q-KP equation on u −1 . One can find that the equations of dynamical variables {u 0 ,u −i } in q-KP hierarchy are coupled with each other and can not get one q-partial differential equation associated only with one dynamical variable, like classical KP equation has one dynamical variable u −1 . The reason is that the q-Leibnitz rule contains q-differential operation and θ operation, however, the Leibnitz rule of the standard calculus only contains one differential operation. We get a single q-soliton u −1 by means of the simplest τ function τ q = W q 1 (φ 1 ) = φ 1 in Section 4. Meanwhile, the multi-q-soliton u −1 is obtained from one-component q-cKP hierarchy in Section 7. Figures of q-effect △u −1 show that q-soliton u −1 indeed goes to classical soliton of KP equation when x → 0 and q → 1 and q-deformation does not destroy the rough profile of the q-soliton. In other worlds, the figure of q-soliton is similar to the classical soliton of KP equation. We also show the trends of the q-effect △u −1 depends on x and q; x plays a role of the amplifier of q-effects. In conclusion, the figures of q-effects △u −1 let us know the process of q-deformation in integrable systems for the first time. Of course, it is a long way to explore the physical meaning of q from the soliton theory.
In comparison with the research of classical soliton theory [40], in particular, the KP hierarchy [36,37], the cKP [25,31] hierarchy and the AKNS [40] hierarchy, there exist at least several topics needed to be discussed in order to research the integrability property of nonlinear q-partial differential equations. For instance, the Hamiltonian structure the q-cKP hierarchy and its q-W-algebra; the gauge transformation of the q-cKP hierarchy; the q-Hirota equation associated with the bilinear identity of the q-KP hierarchy; the symmetry analysis of q-differential   equation and q-partial differential equations; the interaction of q-solitons; the q-AKNS hierarchy and its properties. Since the KP hierarchy has B-type and C-type sub-hierarchies, what are q-analogues of them? In particular, we showed in the previous sections that convergence of e q (x) affects the q-soliton, so the analytic property of e q (x) is a basis for research the interaction of q-solitons. We will try to investigate these questions in the future.

(C.2)
The left hand side of (C.1) equals the left hand side of (C.2), which is followed by It should be noted that the proof above is independent of the form of φ k , so we can replace φ N +j with (∂ l q φ N +j ). This completes the proof of the q-Wronskian identity.