# Two Kinds of Darboux-Bäcklund Transformations for theq-Deformed KdV Hierarchy with Self-Consistent Sources

Authors: Hongxia Wu; Liangjuan Gao; Jingxin Liu; Yunbo Zeng
Journal: Advances in Mathematical Physics
 (2016)

---

## Abstract

Two kinds of Darboux-Bäcklund transformations (DBTs) are constructed for theq-deformed Nth KdV hierarchy with self-consistent sources (q-NKdVHSCS) by using the q-deformed pseudodifferential operators. Note that one of the DBTs provides a nonauto Bäcklund transformation for two q-deformed Nth KdV equations with self-consistent sources (q-NKdVESCS) with different degree. In addition, the soliton solution to the first nontrivial equation of q-KdVHSCS is also obtained.


---

## Body

## 1. Introduction

Theq-deformed integrable systems are regarded as the q-deformation of the related classical ones. The q-deformation is performed by using the q-derivative ∂q to take the place of usual derivative ∂x and it reduces to a classical integrable system as q→1. In recent years, some q-deformed integrable systems, especially the q-deformed Nth KdV hierarchy (q-NKdVH) and the q-deformed KP hierarchy (q-KPH), have attracted much interest both in mathematics and in physics [1–17]. It was shown that q-NKdVH inherited some integrable structures from the classical Nth KdV hierarchy, such as infinite conservation law [2], bi-Hamiltonian structure [3, 4], tau function [5, 6], Darboux-Bäcklund transformation [7], and q-Miura transformation [8]. In 1999, some elementary DBTs of the q-NKdVH (also called q-deformed Gelfand-Dickey hierarchy) were constructed by using the q-deformed pseudodifferential operators. The formula for the n-times repeated DBTs was also presented, which produces the new soliton solutions to the q-NKdVH [7]. For q-KPH, its bi-Hamiltonian structure, tau function, additional symmetries, q-effect in q-soliton, Virasoro constraints of tau function and integrable extension, and q-Bäcklund transformation were also explored in [9–17]. In 2008, based on the symmetry constraint for q-KPH, the new extension of this hierarchy was considered [16]. Two kinds of reductions of this new extended q-KP hierarchy were also studied, which give many 1+1 dimensional q-deformed soliton equations with self-consistent sources [16]. For example, the n-reduction L<0n=0 gives q-NKdVHSCS. However, to our knowledge, the DBTs and the soliton solution for q-NKdVHSCS still remain unexplored. It is known that the DBT is an important property to characterize the integrability of the hierarchy. Thus, it is necessary for us to explore the DBT for q-NKdVHSCS. We think our research results will deepen our understanding on soliton solutions of this hierarchy.The outline of this paper is as follows. In Section2, some notations in the q-calculus and the definition of the q-NKdVHSCS are briefly reviewed. In Section 3, we aim at the construction of auto DBTs for q-NKdVHSCS. In Section 4, the nonauto DBTs for q-NKdVHSCS are constructed. In Section 5, one soliton solution to the first nontrivial equation of q-NKdVHSCS is obtained by using nonauto DBTs. Section 6 is devoted to a brief summary.


## 2. Theq-Deformed Nth KdV Hierarchy with Self-Consistent Sources (q-NKdVHSCS)

In this section, we briefly review some notations in theq-calculus and the definition of the q-NKdVHSCS.Theq-derivative operator ∂q and q-shift operator θ are defined by (1)∂qfx=fqx-fxxq-1,(2)θfx=fqx.In this paper, we introduce two notations: [Pf] and P∘f=Pf, in which P is a q-pseudo-differential operator (q-PDO) given by (3)P=∑i=-∞npi∂qi.[Pf] denotes P acting on the function f, while Pf indicates the multiplication of P and f; that is, ∂qf=θ(f)∂q+[∂qf].It can be easily shown from (1) that when q→1, ∂q reduces to the ordinary differential operator ∂x and that θ and ∂q do not commute but satisfy(4)∂qθkf=qkθk∂qf,k∈Z.Let ∂q-1 be the formal inverse of ∂q such as ∂q∂q-1f=∂q-1∂qf=f. In general, the q-deformed Leibnitz rule holds(5)∂qnf=∑k≥0nkqθn-k∂qkf∂qn-k,n∈Z,where q-number and q-binomial are defined by(6)nq=qn-1q-1,nkq=nqn-1q⋯n-k+1q1q2q⋯kq,n0q=1.For a q-PDO P=∑i=-∞n‍pi∂qi, we separate P into the differential part P+=∑i=0n‍pi∂qi and the integral part P-=∑i≤-1‍pi∂qi. The conjugate operation P∗ is given by (7)P∗=∑i=-∞n∂q∗ipi,where ∂q∗=-∂qθ-1=-1/q∂1/q,(∂q-1)∗=(∂q∗)-1=-θ∂q-1.Theq-exponential function Eq(x) is defined as(8)Eqx=exp⁡∑k=1∞1-qkk1-qkxk,satisfying [∂qkEq(xz)]=zkEq(xz),k∈Z.The extendedq-KPH was given by [16] (9a)Ltn=Bn,L,Bn=Ln≥0,(9b)Lτk=Bk+∑j=1mϕj∂q-1ψj,L,Bk=Lk≥0,k≠n,(9c)ϕj,tn=Bnϕj,ψj,tn=-Bn∗ψj,j=1,…,m,where L=∂q+u0+u1∂q-1+u2∂q-2+⋯ and the coefficients ui(i=0,1,…) are the functions of t=(x,t1,…).The commutativity of (9a), (9b), and (9c) leads to the zero-curvature representation of q-KPH ((9a), (9b), and (9c)). As the n-reduction of the extended q-KPH, the q-NKdVHSCS is defined as follows [16]: (10a)Bn,τk=Bk+∑j=1mϕj∂q-1ψj,Bn,Bk=Bn≥0k/n(10b)Bnϕj=λjnϕj,(10c)Bn∗ψj=μjnψj,j=1,…,m.Under (10b) and (10c), the Lax representation for (10a) is (11a)Bnφ=λφ,(11b)φτk=Bk+∑j=1mϕj∂q-1ψjφ.We find that when m=0, q-NKdVHSCS ((10a), (10b), and (10c) and (11a) and (11b)) can be reduced to the q-NKdVH and its related Lax representation, respectively. In addition, when n=2, k=1 ((10a), (10b), and (10c)) becomes the first nontrivial soliton equation of q-KdVHSCS given by (12a)v1,τ1+e1+∑j=1mgj1=0,(12b)v0,τ3+e0+∑j=1mgj0=0,(12c)θv1-v1+u0-θ2u0=0,(12d)∂q2+v1∂q+v0φj-λj2φj=0,(12e)∂q2+v1∂q+v0∗ψj-μj2ψj=0,j=1,…,m.


## 3. The Auto Darboux-Bäcklund Transformation (DBT) forq-NKdVHSCS

In this section, we will focus on the construction of auto DBT forq-NKdVHSCS.Theorem 1.
AssumeBn, ϕj, ψj(j=1,…,m) be the solution of q-NKdVHSCS ((10a), (10b), and (10c)), and h1 satisfies (11a) and (11b) with λ=λ1n; the DBT is defined by (13a)B¯n=T1BnT1-1=∂qn+v¯n-1∂qn-1+⋯+v¯1∂q+v¯0,(13b)ϕ¯=T1ϕ=Wqh1,ϕh1,(13c)ϕ¯j=T1ϕj=Wqh1,ϕjh1,(13d)ψ¯j=T1-1∗ψj=-θΩh1,ψjθh1,j=1,…,m,Then B¯n, ϕ¯, ϕ¯j, ψ¯j(j=1,…,m) satisfy (10b) and (10c) and (11a) and (11b) and hence are the solution of q-NKdVHSCS ((10a), (10b), and (10c)), where T1=θ(h1)∂qh1-1=∂q-α1, α1=[∂qh1]/h1, and Wq[h1,ϕj] and Ωq[h1,ψj] are defined as follows: (14)Wqϕ1,ϕ=h1ϕ∂qh1∂qϕ,Ωh1,ψj=∂q-1h1ψj.Remark 2.
Here it should be pointed out that the formula holds(15)B¯k=T1BkT1-1+T1,τkT1-1,Bk=Bn≥0k/n,B¯k=B¯n≥0k/n,where the gauge operator T1 is defined above. The proof has been given in [7].Proof.
(1) We firstly show thatB¯n, ϕ¯j, ψ¯j(j=1,…,m) satisfy (10b) and (10c).
Noting thatBn, ϕj, ψj(j=1,…,m) are the solution of (10a), (10b), and (10c), we have (16)Bnϕj=λjnϕj,Bn∗ψj=μjnψj.Hence, (17)B¯nϕ¯j-λjnϕ¯j=T1BnT1-1T1ϕj-λjnϕ¯j=T1Bnϕj-λjnϕ¯j=λjnT1ϕj-ϕ¯j=0,B¯n∗ψ¯j-μjnψ¯j=T1-1∗Bn∗T1∗T1-1∗ψj-μjnψ¯j=T1-1∗Bn∗ψj-μjnψ¯j=0.
(2) We finally show thatB¯n, ϕ¯, ϕ¯j, ψ¯j satisfy (11a) and (11b).
Since the proof of (11a) is the same as the case (1), we only need to verify that B¯n, ϕ¯, ϕ¯j, ψ¯j satisfy (11b); that is, (18)ϕ¯τk-B¯k+∑j=1mϕ¯j∂q-1ψ¯jϕ¯=0,B¯k=B¯n≥0k/n.Noting that ϕ¯τk=[T1ϕ]τk=[T1,τkϕ]+[T1ϕτk] and ϕτk=[(Bk+∑j=1m‍ϕj∂q-1ψj)ϕ], we get (19)ϕ¯τk-B¯k+∑j=1mϕ¯j∂q-1ψ¯jϕ¯=T1,τkϕ+T1ϕτk-B¯k+∑j=1mϕ¯j∂q-1ψ¯jT1ϕ=T1,τk+T1Bk-B¯kT1+∑j=1mT1ϕj∂q-1ψj-∑j=1mϕ¯j∂q-1ψ¯jT1ϕ.According to Remark 2, we have (20)T1,τk+T1Bk-B¯kT1=0.Next, we prove (21)∑j=1mT1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯jT1=0.Since T1=∂q-α1 and ∂qϕj=θ(ϕj)∂q+[∂qϕj], then ∀j, we obtain by the tedious computation (22a)T1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯jT1=∂q-α1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯j∂q+ϕ¯j∂q-1ψ¯jα1=θϕjψj+ϕ¯jΩh1,ψjh1.In addition, we also have (22b)θϕjψj+ϕ¯jΩh1,ψjh1=θϕj∂qΩϕ1,ψj+T1h1Ωh1,ψjh1=∂qϕjΩh1,ψj-∂qϕjΩh1,ψj+∂qϕj-α1ϕjΩh1,ψjh1=∂q-α1ϕjΩh1,ψjh1=T1ϕj∂q-1ψjh1h1.Substituting (22b) into (22a) leads to (23)∑j=1mT1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯jT1=1h1T1∑j=1mϕj∂q-1ψjh1.Since h1 is the solution of (11a) and (11b) with λ=λ1n, we have (24)h1,τk=Bk+∑j=1mϕj∂q-1ψjh1⟹∑j=1mϕj∂q-1ψjh1=h1,τk-Bkh1.Moreover, by the property of determinant, we have(25)T1h1=Wqh1,h1h1=0.Differentiating both sides of (25) with respect to τk yields(26)T1,τkϕ1+T1ϕ1,τk=0⟹T1ϕ1,τk=-T1,τkϕ1.From (23), (24), and (26), we have(27)∑j=1mT1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯jT1=-1ϕ1T1,τk+T1Bkϕ1=-1ϕ1B¯kT1ϕ1=0.This completes the proof.Obviously, Theorem1 provides an auto DBT for q-NKdVHSCS ((10a), (10b), and (10c)). However, this DBT does not enable us to obtain the new solution of q-NKdVHSCS ((10a), (10b), and (10c)). So we have to seek for nonauto DBTs between the two q-NKdVHSCS ((10a), (10b), and (10c)) with different degrees of sources.


## 4. The Nonauto DBTs ofq-NKdVHSCS

In this section, we will construct the nonauto DBTs ofq-NKdVHSCS ((10a), (10b), and (10c)), which enables us to obtain the new solution of q-NKdVHSCS from the known solution of q-NKdVH.Theorem 3.
GivenBn, ϕj, ψj(j=1,…,m) the solution for q-NKdVHSCS ((10a), (10b), and (10c)), let f1,g1≡ϕm+1 be two independent eigenfunctions of (11a) and (11b) with λ=λm+1n. Let b1(τk) be a function of τk such that b1(τk)τk=(-1)m+1β1(τk)η1(τk). Denote h1=f1+b1(τk)g1.
The DBT is defined by(28a)B¯n=T1BnT1-1=∂qn+v¯n-1∂qn-1+⋯+v¯1∂q+v¯0,(28b)ϕ¯=T1ϕ=Wqh1,ϕh1,(28c)ϕ¯j=T1ϕj=Wqh1,ϕjh1,(28d)ψ¯j=T1-1∗ψj=-θΩh1,ψjθh1,j=1,…,m,(28e)ϕ¯m+1=-β1τkT1g1,(28f)ψ¯m+1=-1m+1η1τk1θh1,where T1=∂q-α1, α1=[∂qh1]/h1, and then B¯n, ϕ¯, ϕ¯j, ψ¯j(j=1,…,m), ϕ¯m+1, ψ¯m+1 satisfy (10b) and (11a) and (11b) with m replaced by m+1; hence B¯n, ϕ¯j, ψ¯j(j=1,…,m), ϕ¯m+1, ψ¯m+1 are the solution of q-NKdVHSCS ((10a), (10b), and (10c)) with m replaced by m+1.Proof.
(1) We firstly show thatB¯n, ϕ¯, ϕ¯j, ψ¯j(j=1,…,m), ϕ¯m+1, ψ¯m+1 are the solution of (10b) and (10c).
With the same proof as Theorem1, B¯n, ϕ¯, ϕ¯j, ψ¯j(j=1,…,m) can be shown to be the solution of (10b) and (10c). Here we only need to show that ϕ¯m+1, ψ¯m+1 are also the solution of (10b) and (10c). Consider(29)B¯nϕ¯m+1=-β1τkT1BnT1-1T1g1=-β1τkT1Bng1=-λm+1nβ1τkT1g1=λm+1nϕ¯m+1.Taking a proper solution ψm+1 of (10c) with μ=μm+1n such that Ω(h1,ψm+1)=-1, then we get (30)B¯n∗ψ¯m+1=-1m+1η1τkT1-1∗Bn∗T1∗1θh1=-1m+1η1τkT1-1∗Bn∗T1∗-Ωh1,ψm+1θh1.Noting that [(T1-1)∗ψm+1]=-(Ω(h1,ψm+1))/θ(h1), we derive from (30) (31)B¯n∗ϕ¯m+1=-1m+1η1τkT1-1∗Bn∗T1∗T1-1∗ψm+1=-1m+1η1τkT1-1∗Bn∗ψm+1=μm+1nψ¯m+1.
(2) We finally show thatB¯n, ϕ¯j, ψ¯j(j=1,…,m), ϕ¯m+1, ψ¯m+1 are the solution of (11a) and (11b) with m replaced by m+1. Evidently we only need to prove B¯n, ϕ¯j, ψ¯j(j=1,…,m), ϕ¯m+1, ψ¯m+1 satisfy (11b); that is, ϕ¯τk-[(B¯k+∑j=1m+1‍ϕ¯j∂q-1ψ¯j)ϕ¯]=0.
Noting thatϕ¯=[T1ϕ]⇒ϕ¯τk=[T1ϕ]τk=[T1,τkϕ]+[T1ϕτk], we have(32a)ϕ¯τk-B¯k+∑j=1m+1ϕ¯j∂q-1ψ¯jϕ¯=T1,τk+T1Bk-B¯kT1+∑j=1mT1ϕj∂q-1ψj-∑j=1m+1ϕ¯j∂q-1ψ¯jT1ϕ.From (15), a direct computation leads to (32b)T1,τk+T1Bk-B¯kT1+b1τkτkT1g1h1=0.Noticing that (33a)∑j=1mT1ϕj∂q-1ψj-∑j=1m+1ϕ¯j∂q-1ψ¯jT1=∑j=1mT1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯jT1-ϕ¯m+1∂q-1ψ¯m+1T1,then ∀j=1,…,m, we obtain by the tedious computation(33b)T1ϕj∂q-1ψj-ϕ¯j∂q-1ψ¯jT1=θϕjψj+ϕ¯jθΩh1,ψjθh1.Substituting (33b) into (33a), we get(33c)∑j=1mT1ϕj∂q-1ψj-∑j=1m+1ϕ¯j∂q-1ψ¯jT1=∑j=1mθϕjψj+ϕ¯jθΩh1,ψjθh1-ϕ¯m+1∂q-1ψ¯m+1∂q+ϕ¯m+1∂q-1ψ¯m+1α1=∑j=1mθϕjψj+ϕ¯jθΩh1,ψjθh1--1m+1η1τkϕ¯m+1∂q-1∂q1h1-∂q1h1+ϕ¯m+1∂q-1ψ¯m+1α1=∑j=1mθϕjψj+ϕ¯jθΩh1,ψjθh1+-1m+1β1τkη1τkT1g1h1.In addition, since f1,g1 are the solutions of (11a) and (11b), we have (34)f1,τk=Bk+∑j=1mϕj∂q-1ψjf1,g1,τk=Bk+∑j=1mϕj∂q-1ψjg1;hence (35)h1,τk=f1,τk+b1τkg1,τk+b1τkτkg1=Bk+∑j=1mϕj∂q-1ψjh1+b1τkτkg1⟹∑j=1mϕj∂q-1ψjh1=h1,τk-Bkh1-b1τkτkg1.Noting [T1h1]=Wq[h1,h1]/h1=0 and differentiating both sides of this equation with respect to τk lead to (36)T1,τkh1+T1h1,τk=0⟹T1h1,τk=-T1,τkh1.Rewriting (33c) leads to (37a)∑j=1mT1ϕj∂q-1ψj-∑j=1m+1ϕ¯j∂q-1ψ¯jT1=∑j=1mT1ϕj∂q-1ψjh1h1+-1m+1β1τkη1τkT1g1h1.Combining (32a) and (32b) and (35) and (36), we get (37b)∑j=1mT1ϕj∂q-1ψjh1h1=T1∑j=1mϕj∂q-1ψjh1h1=T1h1,τk-T1Bkh1-b1τkτkT1g1h1=-T1,τkh1+T1Bkh1h1-b1τkτkT1g1h1=-BkT1h1h1+b1τkτkT1g1h1-b1τkτkT1g1h1=-BkT1h1h1=0.Substituting (32b), (37a), and (37b) into (32a), we have(38)ϕ¯τk-B¯k+∑j=1m+1ϕ¯j∂q-1ψ¯jϕ¯=-b1τkτkT1g1h1+-1m+1β1τkη1τkT1g1h1.Noting b1(τk)τk=(-1)m+1β1(τk)η1(τk), we immediately get from (38) (39)ϕ¯τk-B¯k+∑j=1m+1ϕ¯j∂q-1ψ¯jϕ¯=0.This completes the proof.Theorem 4 (theN-times repeated nonauto DBT).
GivenBn,ϕ1,…,ϕm,ψ1…,ψm are the solution for q-NKdVHSCS ((10a), (10b), and (10c)), f1,…,fN,g1,…,gN are independent eigenfunctions of (11a) and (11b) with λ=λm+1n,…,λm+Nn. bi(τk),i=1,…,N, are functions of τk such that bi(τk)τk=(-1)m+Nβi(τk)ηi(τk).
Denotehi=fi+bi(τk)gi. The N-times repeated DBT is defined by (40a)BnN=TNBnTN-1=∂qn+vn-1N∂qn-1+⋯+v1N∂q+v0N,(40b)ϕN=TNϕ=Wqh1,h2,…,hN,ϕWqh1,h2,…,hN,(40c)ϕjN=TNϕj=Wqh1,h2,…,hN,ϕjWqh1,h2,…,hN,(40d)ψjN=TN-1∗ψj=-θGqh1,h2,…,hN,ψjθWqh1,h2,…,hN,j=1,…,m,(40e)ϕm+iN=-βiτsTNgi,(40f)ψm+iN=-1m+iηiτkθWqh1,…,hi-1,hi+1,…,hNθWqh1,h2,…,hN,i=1,…,N,where(41)TN=1Wqh1,h2,…,hNh1h2⋯hN1∂qh1∂qh2⋯∂qhN∂q⋮⋮⋮⋮∂qNh1∂qNh2⋯∂qNhN∂qN,Gqh1,h2,…,hN=h1h2⋯hN⋮⋮⋮⋮∂qN-2h1∂qN-2h2⋯∂qN-2hN∂q-1h1ψj∂q-1h2ψj⋯∂q-1hNψj,TN=DNDN-1⋯D1,Dk=∂q-αkk-1,αik=∂qhikhik,hik=Tkhi,k=0,1,…,N-1;then Bn(N),ϕj(N),ψj(N),(j=1,…,m) satisfy (10b) and (10c) and (11a) and (11b) with m replaced by m+N; hence Bn(N),ϕj(N),ψj(N),(j=1,…,m),ϕm+i(N),ψm+i(N),(i=1,…,N) are the solution of q-NKdVHSCS ((10a), (10b), and (10c)) with m replaced by m+N.Proof.
With the same method as Theorem3, we can show that ϕj(N),ψj(N),(j=1,…,m), ϕm+i(N),ψm+i(N),(i=1,…,N) satisfy (10b), (10c), and (11a). Here we only need to show Bn(N),ϕ(N),ϕj(N),ψj(N),(j=1,…,m),ϕm+i(N),ψm+i(N),(i=1,…,N) satisfy (11b). Next we will show it by the mathematical induction method. Theorem 3 indicates Bn(N),ϕj(N),ψj(N),(j=1,…,m),ϕm+i(N),ψm+i(N),(i=1,…,N) satisfy (11b) in the case of N=1.
Provided thatBl(N),ϕj(l),ψj(l),ϕm+i(l),ψm+i(l) satisfy (11b) for l≤N-1, (42a)ϕτkl=Bkl+∑j=1m+lϕjl∂q-1ψjlϕl,Bkl=Bnl≥0k/n,(42b)bjτkτk=-1l+iβiτkηiτk,l=1,…,N-1.Noticing that ϕ(N)=[DNϕ(N-1)], then when l=N, we have (43)ϕτkN-BkN+∑j=1m+NϕjN∂q-1ψjNϕN=DNϕN-1τk-BkNDN+∑j=1m+NϕjN∂q-1ψjNDNϕN-1=DN,τk+DNBkN-1-BkNDN+∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-∑j=1m+NϕjN∂q-1ψjNDNϕN-1;simplifying ∑j=1m+N-1‍DNϕj(N-1)∂q-1ψj(N-1)-∑j=1m+N‍ϕj(N)∂q-1ψj(N)DN leads to(44a)∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-∑j=1m+NϕjN∂q-1ψjNDN=∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-ϕjN∂q-1ψjNDN-ϕm+NN∂q-1ψm+NNDN.From (40f), we obtain (44b)ψm+NN=-1m+NηNτkθWqh1,h2,…,hi-1θWqh1,h2,…,hN=-1m+NηNτkθTN-1hN=-1m+NηNτkθhNN-1.Substituting (44b) into (44a) yields (45)∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-∑j=1m+NϕjN∂q-1ψjNDN=∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-ϕjN∂q-1ψjNDN--1m+NηNτkϕm+NNhNN-1+-1m+NηNτkϕm+NN∂q-1∂q1hNN-1+ϕm+NN∂q-1ψm+NNαNN-1=∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-ϕjN∂q-1ψjNDN--1m+NηNτkϕm+NN∂q-1∂qhNN-1θhNN-1hNN-1+ϕm+NN∂q-1ψm+NNαNN-1=∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-ϕjN∂q-1ψjNDN+-1m+NβNτkηNτkTNgNhNN-1.From (37a) for one DBT DN, we have (46)∑j=1m+N-1DNϕjN-1∂q-1ψjN-1-∑j=1m+NϕjN∂q-1ψjNDN=1hNN-1DN∑j=1m+N-1ϕjN-1∂q-1ψjN-1hNN-1.Note that hN(N-1) satisfies (47a)hN,τkN-1=BkN-1+∑j=1m+N-1ϕjN-1∂q-1ψjN-1hNN-1+bNτkτkgNN-1⟹∑j=1m+N-1ϕjN-1∂q-1ψjN-1hNN-1=hN,τkN-1-BkN-1hNN-1-bNτkτkgNN-1,and that (47b)TNhN=DNTN-1hN=DNhNN-1=0.Differentiating both sides of (47b) with respect to τk yields (48)DN,τkhNN-1+DNhN,τkN-1=0⟹DNhN,τkN-1=-DN,τkhNN-1;we obtain (49)1hNN-1DN∑j=1m+N-1ϕjN-1∂q-1ψjN-1hNN-1=-BkN-1DNhNN-1hNN-1=0.Combining (43), (45), (46), and (49), we get (50)ϕτkN-BkN+∑j=1m+NϕjN∂q-1ψjNϕN=-1m+NβNτkηNτkTNgNhNN-1-bNτkτkTNgNhNN-1=0.This completes the proof.


## 5. Soliton Solution ofq-KdVHSCS

It is known that KdV equation is the first nontrivial equation of the KdV hierarchy. However, the first nontrivial equation ofq-KdVHSCS is not the q-KdVESCS but (12a), (12b), (12c), (12d), and (12e). In this section, we aim to construct the soliton solution to (12a), (12b), (12c), (12d), and (12e). In order to get the soliton solution of (12a), (12b), (12c), (12d), and (12e), the following proposition is firstly presented.Proposition 5.
Letf1,g1 be two independent wave functions of (12e), h1≡f1+b1(τ1)g1, under the nonauto DBT, and the transformed coefficients are given by (51)v¯1-v1=xq-1v¯0-v0,where (52)v¯0=v0+∂qθf1,τ1+b1τ1g1,τ1h1+f1,τ1+b1τ1g1,τ1h1.Proof.
It was shown in [7] that formula (51) holds for (12a), (12b), (12c), (12d), and (12e) and that (53)v¯0-v0=∂qv1+α1+θα1.Noting that h1=f1+b1(τ1)g1,(B2)≥01/2=B1=∂q+u0, then we have (54)h1,τ1=B2≥01/2+∑j=1mϕj∂q-1ψjh1+b1τ1τ1g1=∂qh1+u0h1+∑j=1mϕjΩψj,h1+b1τ1τ1g1.From (54), we get (55)u0=h1,τ1-∂qh1h1-1h1∑j=1mϕjΩψj,h1-b1τ1τ1g1h1.Noticing that (12c) implies (56)v1=θu0+u0,we have (57)v¯0-v0=∂qθh1,τ1-∂qh1h1+h1,τ1-∂qh1h1-θ1h1∑j=1mϕjΩψj,h1+b1τ1τ1g1h1-1h1∑j=1mϕjΩψj,h1+b1τ1τ1g1h1+∂qh1h1+θ∂qh1h1=∂qθh1,τ1h1+h1,τ1h1-θ1h1∑j=1mϕjΩψj,h1+b1τ1τ1g1h1-1h1∑j=1mϕjΩψj,h1+b1τ1τ1g1h1=∂qθf1,τ1+b1τ1g1,τ1h1+f1,τ1+b1τ1g1,τ1h1-∂qθ1h1∑j=1mϕjΩψj,h1+1h1∑j=1mϕjΩψj,h1=∂qθf1,τ1+b1τ1g1,τ1h1+f1,τ1+b1τ1g1,τ1h1+qθ+1∂q1h1∑j=1mϕjΩψj,h1.Next we consider(58)∂q1h1∑j=1mϕjΩψj,h1=∑j=1mθϕjh1∂qΩψj,h1+∂qϕjh1Ωψj,h1=∑j=1mθϕjψjh1θh1+∂qϕjh1-ϕj∂qh1θh1h1Ωψj,h1=1θh1∑j=1mθϕjψjh1+∂qϕj-α1ϕjΩψj,h1=h1θh1∑j=1mθϕjψj+ϕ¯jΩψj,h1h1.Noting (37b), we can immediately derive (59)∂q1h1∑j=1mϕjΩψj,h1=0.Hence we obtain from (57) (60)v¯0=v0+∂qθf1,τ1+b1τ1g1,τ1h1+f1,τ1+b1τ1g1,τ1h1.This completes the proof.Next we will start from the trivial solution to (12a), (12b), (12c), (12d), and (12e) without sources, that is, v0=v1=0, and use Theorem 3 and Proposition 5 to construct one soliton solution to (12a), (12b), (12c), (12d), and (12e) with m=1. When v0=v1=0, then B2=∂q2; hence, the wave functions f1,g1 of Lax operator B2=∂q2 satisfy(61)∂q2φ=λ12φ,ϕτ1=∂qφ.We take the solution f1,g1 of system (61) as follows: (62)f1=Eqp1xexp⁡p1τ1,g1=Eq-p1xexp⁡-p1τ1,where Eq(x) denotes the q-exponential function satisfying (63)∂qEqp1x=p1Eqp1x,with an equivalent form (64)Eqx=∑k=0∞1kq!xk.Noting h1≡f1+b1(τ1)g1, where f1,g1 are defined by (62), we get from (51) and (52) (65a)v¯0=p1h1θ2f1-b1τ1g1-f1-b1τ1g1θ2h1xq-1h1θ2h1,(65b)v¯1=p1h1θ2f1-b1τ1g1-f1-b1τ1g1θ2h1h1θ2h1.In addition, by Theorem 3, we obtain(65c)ϕ¯1=-β1τ1h1∂qg1-g1∂qh1h12,(65d)ψ¯1=-η1τ1θ1h1,where b1(τ1),β1(τ1), and η1(τ1) satisfy (65e)b1τ1τ1=-βqτ1η1τ1.Then (65a), (65b), (65c), (65d), and (65e) present one soliton solution of (12a), (12b), (12c), (12d), and (12e) with m=1. In particular, when b1(τ1)=c, where c is an arbitrary constant, (65a), (65b), (65c), (65d), and (65e) can be reduced to one soliton solution to the first nontrivial equation of the q-KdV hierarchy [7]. Certainly, we also use Theorem 4 and Proposition 5 to construct the multisoliton solution to (12a), (12b), (12c), (12d), and (12e). But owing to the complexity of the computation, we omit it here.


## 6. Summary

Asn-reduction of the extended q-deformed KP hierarchy, q-NKdVHSCS is explored in this paper. Two kinds of DBTs are constructed and the soliton solution to the first nontrivial equation of q-KdVHSCS is also obtained. We find that one of the DBTs provides a nonauto Bäcklund transformation for the two q-NKdVESCS with different degree, which enables us to obtain the new solution of q-NKdVHSCS from the known solution of q-NKdVH. Noting that we only investigate DBT and solution of q-NKdVESCS, other integrable structures will be studied in our forthcoming paper such as infinite conservation law, tau function, and Hamiltonian structure.



---

*Source: 8153752-2016-09-07.xml*