# Links between(γn,σk)-KP Hierarchy, (γn,σk)-mKP Hierarchy, and (2+1)-(γn,σk)-Harry Dym Hierarchy

Authors: Yehui Huang; Yuqin Yao; Yunbo Zeng
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

The new (2+1)-(γn,σk)-Harry Dym hierarchy and (γn,σk)-mKP hierarchy with two new time series γn and σk, which consist of γn-flow, σk-flow, and mixed γn and σk evolution equations of eigenfunctions, are proposed. Gauge transformations and reciprocal transformations between (γn,σk)-KP hierarchy, (γn,σk)-mKP hierarchy, and (2+1)-(γn,σk)-Harry Dym hierarchy are studied. Their soliton solutions are presented.


---

## Body

## 1. Introduction

Generalizations of soliton hierarchies are important topics since last century. In 2008, KdV6 equation is studied as a nonholonomic deformation of KdV equation. Kupershmidt developed a deformation from the bi-Hamiltonian structure of the soliton equations and KdV6 equation could be seen as the deformation of the KdV equation [1]. We generalized this Kupershmidt deformed system and find that KdV6 equation could be seen as the Rosochatius deformation of the KdV equation with self-consistent sources and firstly found the bi-Hamiltonian structure for the KdV6 equation [2]. The Kadomtsev-Petviashvili (KP) hierarchy is an important 2+1 dimensional integrable system and the generalizations of KP hierarchy (KPH) attract a lot of interests from both physical and mathematical points of view [3–11]. One kind of generalization is the multicomponent KP hierarchy [3], which contains many physical relevant nonlinear integrable systems such as Davey-Stewartson equation, two-dimensional Toda lattice, and three-wave resonant integrable equations. Another kind of generalization of KP equation is the so called KP equation with self-consistent sources (KPESCS) [10, 11]. The third kind of generalization is the extended KP hierarchy (exKPH), which is constructed by introducing another time series {τk} [12–14]. The exKPH consists of tn-flow of KP hierarchy, τk-flow, and tn-evolution equations of eigenfunctions. To make difference, we may call the exKPH as (tn,τk)-KPH.Recently, we generalize the(tn,τk)-KPH to (γn,σk)-KPH by introducing two new time series γn and σk with two parameters αn and βk [15]:(1a)Lγn=Bn+αn∑i=1Nqi∂-1ri,L,Bn=L≥0n,(1b)Lσk=Bk+βk∑i=1Nqi∂-1ri,L,(1c)αnqi,σk-Bkqi-βkqi,γn-Bnqi=0,αnri,σk+Bk∗ri-βkri,γn+Bn∗ri=0,i=1,2,…,N,where the pseudodifferential operator L with potential functions ui is defined as (2)L=∂+u1∂-1+u2∂-2+⋯and ∗ denotes the adjoint operator. The compatibility of γn-flow (1a) and σk-flow (1b) under (1c) gives rise to the zero-curvature representation for (1a), (1b), and (1c),(3a)Bn,σk-Bk,γn+Bn,Bk+βkBn,∑i=1Nqi∂-1ri++αn∑i=1Nqi∂-1ri,Bk+=0,(3b)αnqi,σk-Bkqi-βkqi,γn-Bnqi=0,αnri,σk+Bk∗ri-βkri,γn+Bn∗ri=0,i=1,2,…,N,with the Lax representation(4)ψγn=Bn+αn∑i=1Nqi∂-1riψ,ψσk=Bk+βk∑i=1Nqi∂-1riψ.The (γn,σk)-KPH consists of γn-flow, σk-flow, and one mixed γn and σk evolution equation of eigenfunctions. The (γn,σk)-KPH can be reduced to the KPH and (tn,τk)-KPH and contains first type and second type as well as mixed type of KPESCS as special cases. We also develop the dressing method to solve the (γn,σk)-KPH [15].The (2+1)-Harry Dym equation has been firstly defined by Konopelchenko and Dubrovsky in 1984 [16]. The Harry Dym hierarchy is the third hierarchy of soliton hierarchies presented by the pseudodifferential operator technique after the KP hierarchy and mKP hierarchy [17]. In this paper, we first constructed the (2+1)-(γn,σk)-Harry Dym hierarchy ((2+1)-(γn,σk)-HDH), which consists of γn-flow, σk-flow, and one mixed γn and σk evolution equation of eigenfunctions. The (2+1)-(γn,σk)-HDH can be reduced to the (2+1)-Harry Dym hierarchy ((2+1)-HDH) and (tn,τk)-HDH [18] and contains first type and second type as well as mixed type of Harry Dym equation with self-consistent sources (HDESCS) as special cases. Then the (γn,σk)-mKPH is proposed. The generalized gauge transformations and generalized reciprocal links between these three kinds of (γn,σk) hierarchies are studied. Further, based on these transformations and the solutions of the (γn,σk)-KPH [15], the solutions of (γn,σk)-mKPH and (2+1)-(γn,σk)-HDH are obtained, respectively.The paper is organized as follows. In Section2, we propose a new (γn,σk)-HDH and present its reduction. A new (γn,σk)-mKPH is proposed in Section 3. Section 4 is devoted to studying the links between the three kinds of (γn,σk) hierarchies. Section 5 presents the soliton solutions and a conclusion is given in the last section.


## 2. A New (2+1)-(γn,σk)-Harry Dym Hierarchy

### 2.1. A New (2+1)-(γn,σk)-Harry Dym Hierarchy

It is well known that the pseudodifferential operatorL¯ for (2+1)-HDH with potential functions wi is defined as (5)L¯=w∂+w0+w1∂-1+w2∂-2+⋯.The (2+1)-HDH is given by [17](6)L¯tn=B¯n,L¯,n≥1,where B¯n=L¯+n=(L¯n)≥2,  n≥1.The compatibility of thetn-flow and tk-flow of (6) leads to the zero-curvature representation of (2+1)-HDH:(7)B¯n,tk-B¯k,tn+B¯n,B¯k=0.In particular, when taking B¯2=w2∂2 and  B¯3=w3∂3+3w2(wx+w0)∂2 and setting t2=y,  t3=t, and w0=(1/2)∂-1(wy/w2)-(1/2)wx, (7) yields the (2+1)-HD equation (8)4wt=w3wxxx-31ww2∂-1wyw2y,which is reduced to the Harry Dym equation. Consider(9)4wt=w3wxxxby dropping the y-dependence.Based on the squared eigenfunction symmetry constraint(10)L¯k=B¯k+∑i=1Nq¯i∂-1r¯i∂2,q¯i,tn=B¯nq¯i,r¯i,tn=-∂-2B¯n∗∂2r¯i,which is compatible with (2+1)-HDH [17], we propose the following generalized (2+1)-HDH with two generalized time series γn and σk: (11a)L¯γn=B¯n+αn∑i=1Nq¯i∂-1r¯i∂2,L¯,(11b)L¯σk=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2,L¯,(11c)αnq¯i,σk-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,αnr¯i,σk+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0,i=1,2,…,N.We will prove the compatibility of (11a) and (11b) under (11c) in the following theorem.Theorem 1.
Theγn-flow (11a) and σk-flow (11b) under (11c) are compatible.Proof.
Denote(12)B^n=B¯n+αn∑i=1Nq¯i∂-1r¯i∂2,B^k=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2.In order to prove L¯γn,σk=L¯σk,γn, that is, (13)B^n,σk-B^k,γn+B^n,B^k,L¯=0,we only need to prove(14)B^n,σk-B^k,γn+B^n,B^k=0.
For convenience, we omit∑‍. We can find that (15)B^n,σk=B¯n,σk+αnq¯∂-1r¯∂2σk=B¯k+βkq¯∂-1r¯∂2,L¯n++αnq¯∂-1r¯∂2σk=B¯k,L¯n++βkq¯∂-1r¯∂2,L¯n++αnq¯σk∂-1r¯∂2+αnq¯∂-1r¯σk∂2,and, similarly, (16)B^k,γn=B¯n,L¯k++αnq¯∂-1r¯∂2,L¯k++βkq¯γn∂-1r¯∂2+βkq¯∂-1r¯γn∂2.Moreover, on the other hand, under the formula [18](17)B¯n,∑i=1Nq¯i∂-1r¯i∂2-=∑i=1NB¯nq¯i∂-1r¯i∂2-∑i=1Nq¯i∂-3B¯n∗r¯i∂4,we have (18)B^n,B^k=B¯n,B¯k+B¯n,βkq¯∂-1r¯∂2+αnq¯∂-1r¯∂2,B¯k=L¯n-L¯n-,L¯k-L¯k-++βkB¯n,q¯∂-1r¯∂2++αnq¯∂-1r¯∂2,B¯k++βkB¯n,q¯∂-1r¯∂2-+αnq¯∂-1r¯∂2,B¯k-=B¯n,L¯k++L¯n,B¯k+-L¯n-,L¯k-++βkB¯n,q¯∂-1r¯∂2++αnq¯∂-1r¯∂2,B¯k++βkB¯nq¯∂-1r¯∂2-βkq¯∂-3B¯n∗r¯∂4-αnB¯kq¯∂-1r¯∂2+αnq¯∂-3B¯k∗r¯∂4.Then (15), (16), and (18) under (11c) yield (19)B^n,σk-B^k,γn+B^n,B^k=αnq¯σk-B¯kq-βkq¯γn-B¯nq¯∂-1r¯∂2+q¯∂-1αnr¯σk+∂-2B¯k∗∂2r¯-βkr¯γn+∂-2B¯n∗∂2r¯∂2=0.The compatibility ofγn-flow (11a) and σk-flow (11b) under (11c) gives rise to the zero-curvature representation for (11a), (11b), and (11c): (20)B¯n+αn∑i=1Nq¯i∂-1r¯i∂2σk-B¯k+βk∑i=1Nq¯i∂-1r¯i∂2γn+B¯n+αn∑i=1Nq¯i∂-1r¯i∂2,B¯k+βk∑i=1Nq¯i∂-1r¯i∂2=0which under (11c) can be simplified as follows. Then we have the following.Theorem 2.
The commutativity of (11a) and (11b) under (11c) gives rise to the zero-curvature equation for the generalized (2+1)-HDH with two generalized time series: (21a)B¯n,σk-B¯k,γn+B¯n,B¯k+βkB¯n,∑i=1Nq¯i∂-1r¯i∂2++αn∑i=1Nq¯i∂-1r¯i∂2,B¯k+=0,(21b)αnq¯i,σk-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,αnr¯i,σk+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0,i=1,2,…,N, with the Lax representation(22)ψ¯γn=B¯n+αn∑i=1Nq¯i∂-1r¯i∂2ψ¯,ψ¯σk=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2ψ¯.We briefly call (11a), (11b), and (11c) and (21a) and (21b) as (2+1)-(γn,σk)-HDH. It is easy to see that (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c) and (21a) and (21b)) for αn=βk=0 reduces to (2+1)-HDH ((6) and (7)) and (2+1)-(γn,σk)-HDH for αn=0,  βk=1 reduces to (tn,τk)-HDH [18]. So (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c) and (21a) and (21b)) presents a more generalized (2+1)-HDH which contains the (2+1)-HDH and (2+1)-(tn,τk)-HDH as the special cases.Example 3.
Let us taken=2 and k=3, and set γ2=y,  σ3=t. Then (21a) and (21b) become (23a)B¯2,t-B¯3,y+B¯2,B¯3+β3B¯2,∑i=1Nq¯i∂-1r¯i∂2++α2∑i=1Nq¯i∂-1r¯i∂2,B¯3+=0,(23b)α2q¯i,t-B¯3q¯i-β3q¯i,y-B¯2q¯i=0,α2r¯i,t+∂-2B¯3∗∂2r¯i-β3r¯i,y+∂-2B¯2∗∂2r¯i=0,i=1,2,…,N,which gives the following nonlinear equation: (24a)2wwt-3w2wx+w0y+w3wwxxx+6wxwxx+3ww0xx+12wxw0x+2β3w2∑i=1Nq¯ir¯ix-2β3wwx∑i=1Nq¯ir¯i-3α2w3∑i=1Nq¯i,xxr¯i+q¯i,xr¯ix-6α2w2wx+w0∑i=1Nq¯ir¯ix+3α2w2wx+w0x∑i=1Nq¯ir¯i=0,(24b)wy-w2wxx+2w0x+α2w∑i=1Nq¯ir¯ix-α2wx∑i=1Nq¯ir¯i=0,(24c)α2q¯i,t-w3q¯i,xxx-3w2wx+w0q¯i,xx-β3q¯i,y-w2q¯i,xx=0,α2r¯i,t-w3r¯i,xxx-3w2w0r¯i,xx-β3r¯i,y+w2r¯i,xx=0,i=1,2,…,N,with the Lax representation as follows: (25)ψ¯y=w2∂2+α2∑i=1Nq¯i∂-1r¯i∂2ψ¯,(26)ψ¯t=w3∂3+3w2wx+w0∂2+β3∑i=1Nq¯i∂-1r¯i∂2ψ¯.Particularly, when takingα2=β3=0; α2=0, β3=1; α2=1, β3=0; and α2=1, β3=1, respectively, (24a), (24b), and (24c) and (26) reduce to the (2+1)-HD equation, the first type of (2+1)-HD equation with self-consistent sources ((2+1)-HDESCS), the second type of (2+1)-HDESCS, and the mixed type of (2+1)-HDESCS and their Lax representations, respectively.


### 2.2. Reduction

Consider the constraint given by(27)L¯k=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2.Then (11b) yields(28)L¯kσk=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2,L¯k=0,(29)B¯k,σk=L¯σkk+=0,∑i=1Nq¯i∂-1r¯i∂2σk=L¯σkk-=0,which imply that L¯, B¯k, q¯i, and r¯i under (27) are independent of σk. Subsequently, q¯i,σk and r¯i,σk in (11c) should be replaced by λiq¯i and -λir¯i as in the case of constrained flow of KP [19, 20]; namely, (11c) under the constraint (27) should be replaced by (30)αnλiq¯i-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,(31)αn-λir¯i+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0.We will show that constraint (27) is invariant under γn-flow (11a) and (31). In fact, making use of (11a), (17), and (31), we have (32)
L
¯
k
-
B
¯
k
γ
n
=
L
¯
γ
n
k
-
=
B
^
n
,
L
¯
k
-
,
β
k
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
γ
n
=
β
k
∑
i
=
1
N
q
¯
i
,
γ
n
∂
-
1
r
¯
i
∂
2
+
q
¯
i
∂
-
1
r
¯
i
,
γ
n
∂
2
=
∑
i
=
1
N
β
k
B
¯
n
q
¯
i
∂
-
1
r
¯
i
∂
2
+
α
n
λ
i
q
¯
i
-
B
¯
k
q
¯
i
∂
-
1
r
¯
i
∂
2
-
β
k
q
¯
i
∂
-
1
B
¯
n
r
¯
i
∂
2
+
α
n
q
¯
i
∂
-
1
-
λ
i
r
¯
i
+
∂
-
2
B
¯
k
∗
∂
2
r
¯
i
∂
2
=
B
¯
n
,
β
k
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
-
B
¯
k
,
α
n
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
=
B
^
n
,
β
k
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
-
B
¯
k
,
α
n
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
=
B
^
n
,
L
¯
k
-
-
B
^
n
,
B
¯
k
-
-
B
¯
k
,
B
^
n
-
+
B
¯
k
,
B
¯
n
-
=
B
^
n
,
L
¯
k
-
.Then (33)L¯k-B¯k-βk∑i=1Nq¯i∂-1r¯i∂2γn=0.This means that the submanifold determined by k-constraint (27) is invariant under the γn-flow (11a) and (31).Therefore, the constrained flow of (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c) and (21a) and (21b)) under (27) reads (34a)B¯k,γn+B¯k,B¯n+βk∑i=1Nq¯i∂-1r¯i∂2,B¯n++αnB¯k,∑i=1Nq¯i∂-1r¯i∂2+=0,(34b)αnλiq¯i-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,αn-λir¯i+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0,i=1,2,…,N.(34c)withB¯n=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2+n/k.Example 4.
Whenn=2, k=3, γ2=y, (34a), (34b), and (34c) give (35a)3w2wx+w0y-w3wwxxx+6wxwxx+3ww0xx+12wxw0x-2β3w2∑i=1Nq¯ir¯ix+2β3wwx∑i=1Nq¯ir¯i+3α2w3∑i=1Nq¯i,xxr¯i+q¯i,xr¯ix+6α2w2wx+w0∑i=1Nq¯ir¯ix-3α2w2wx+w0x∑i=1Nq¯ir¯i=0,(35b)wy-w2wxx+2w0x+α2w∑i=1Nq¯ir¯ix-α2wx∑i=1Nq¯ir¯i=0,(35c)α2λiq¯i-w3q¯i,xxx-3w2wx+w0q¯i,xx-β3q¯i,y-w2q¯i,xx=0,α2-λir¯i-w3r¯i,xxx-3w2w0r¯i,xx-β3r¯i,y+w2r¯i,xx=0,i=1,2,…,N,which is the k-constraint (2+1)-(γn,σk)-Harry Dym equation.


## 2.1. A New (2+1)-(γn,σk)-Harry Dym Hierarchy

It is well known that the pseudodifferential operatorL¯ for (2+1)-HDH with potential functions wi is defined as (5)L¯=w∂+w0+w1∂-1+w2∂-2+⋯.The (2+1)-HDH is given by [17](6)L¯tn=B¯n,L¯,n≥1,where B¯n=L¯+n=(L¯n)≥2,  n≥1.The compatibility of thetn-flow and tk-flow of (6) leads to the zero-curvature representation of (2+1)-HDH:(7)B¯n,tk-B¯k,tn+B¯n,B¯k=0.In particular, when taking B¯2=w2∂2 and  B¯3=w3∂3+3w2(wx+w0)∂2 and setting t2=y,  t3=t, and w0=(1/2)∂-1(wy/w2)-(1/2)wx, (7) yields the (2+1)-HD equation (8)4wt=w3wxxx-31ww2∂-1wyw2y,which is reduced to the Harry Dym equation. Consider(9)4wt=w3wxxxby dropping the y-dependence.Based on the squared eigenfunction symmetry constraint(10)L¯k=B¯k+∑i=1Nq¯i∂-1r¯i∂2,q¯i,tn=B¯nq¯i,r¯i,tn=-∂-2B¯n∗∂2r¯i,which is compatible with (2+1)-HDH [17], we propose the following generalized (2+1)-HDH with two generalized time series γn and σk: (11a)L¯γn=B¯n+αn∑i=1Nq¯i∂-1r¯i∂2,L¯,(11b)L¯σk=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2,L¯,(11c)αnq¯i,σk-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,αnr¯i,σk+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0,i=1,2,…,N.We will prove the compatibility of (11a) and (11b) under (11c) in the following theorem.Theorem 1.
Theγn-flow (11a) and σk-flow (11b) under (11c) are compatible.Proof.
Denote(12)B^n=B¯n+αn∑i=1Nq¯i∂-1r¯i∂2,B^k=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2.In order to prove L¯γn,σk=L¯σk,γn, that is, (13)B^n,σk-B^k,γn+B^n,B^k,L¯=0,we only need to prove(14)B^n,σk-B^k,γn+B^n,B^k=0.
For convenience, we omit∑‍. We can find that (15)B^n,σk=B¯n,σk+αnq¯∂-1r¯∂2σk=B¯k+βkq¯∂-1r¯∂2,L¯n++αnq¯∂-1r¯∂2σk=B¯k,L¯n++βkq¯∂-1r¯∂2,L¯n++αnq¯σk∂-1r¯∂2+αnq¯∂-1r¯σk∂2,and, similarly, (16)B^k,γn=B¯n,L¯k++αnq¯∂-1r¯∂2,L¯k++βkq¯γn∂-1r¯∂2+βkq¯∂-1r¯γn∂2.Moreover, on the other hand, under the formula [18](17)B¯n,∑i=1Nq¯i∂-1r¯i∂2-=∑i=1NB¯nq¯i∂-1r¯i∂2-∑i=1Nq¯i∂-3B¯n∗r¯i∂4,we have (18)B^n,B^k=B¯n,B¯k+B¯n,βkq¯∂-1r¯∂2+αnq¯∂-1r¯∂2,B¯k=L¯n-L¯n-,L¯k-L¯k-++βkB¯n,q¯∂-1r¯∂2++αnq¯∂-1r¯∂2,B¯k++βkB¯n,q¯∂-1r¯∂2-+αnq¯∂-1r¯∂2,B¯k-=B¯n,L¯k++L¯n,B¯k+-L¯n-,L¯k-++βkB¯n,q¯∂-1r¯∂2++αnq¯∂-1r¯∂2,B¯k++βkB¯nq¯∂-1r¯∂2-βkq¯∂-3B¯n∗r¯∂4-αnB¯kq¯∂-1r¯∂2+αnq¯∂-3B¯k∗r¯∂4.Then (15), (16), and (18) under (11c) yield (19)B^n,σk-B^k,γn+B^n,B^k=αnq¯σk-B¯kq-βkq¯γn-B¯nq¯∂-1r¯∂2+q¯∂-1αnr¯σk+∂-2B¯k∗∂2r¯-βkr¯γn+∂-2B¯n∗∂2r¯∂2=0.The compatibility ofγn-flow (11a) and σk-flow (11b) under (11c) gives rise to the zero-curvature representation for (11a), (11b), and (11c): (20)B¯n+αn∑i=1Nq¯i∂-1r¯i∂2σk-B¯k+βk∑i=1Nq¯i∂-1r¯i∂2γn+B¯n+αn∑i=1Nq¯i∂-1r¯i∂2,B¯k+βk∑i=1Nq¯i∂-1r¯i∂2=0which under (11c) can be simplified as follows. Then we have the following.Theorem 2.
The commutativity of (11a) and (11b) under (11c) gives rise to the zero-curvature equation for the generalized (2+1)-HDH with two generalized time series: (21a)B¯n,σk-B¯k,γn+B¯n,B¯k+βkB¯n,∑i=1Nq¯i∂-1r¯i∂2++αn∑i=1Nq¯i∂-1r¯i∂2,B¯k+=0,(21b)αnq¯i,σk-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,αnr¯i,σk+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0,i=1,2,…,N, with the Lax representation(22)ψ¯γn=B¯n+αn∑i=1Nq¯i∂-1r¯i∂2ψ¯,ψ¯σk=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2ψ¯.We briefly call (11a), (11b), and (11c) and (21a) and (21b) as (2+1)-(γn,σk)-HDH. It is easy to see that (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c) and (21a) and (21b)) for αn=βk=0 reduces to (2+1)-HDH ((6) and (7)) and (2+1)-(γn,σk)-HDH for αn=0,  βk=1 reduces to (tn,τk)-HDH [18]. So (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c) and (21a) and (21b)) presents a more generalized (2+1)-HDH which contains the (2+1)-HDH and (2+1)-(tn,τk)-HDH as the special cases.Example 3.
Let us taken=2 and k=3, and set γ2=y,  σ3=t. Then (21a) and (21b) become (23a)B¯2,t-B¯3,y+B¯2,B¯3+β3B¯2,∑i=1Nq¯i∂-1r¯i∂2++α2∑i=1Nq¯i∂-1r¯i∂2,B¯3+=0,(23b)α2q¯i,t-B¯3q¯i-β3q¯i,y-B¯2q¯i=0,α2r¯i,t+∂-2B¯3∗∂2r¯i-β3r¯i,y+∂-2B¯2∗∂2r¯i=0,i=1,2,…,N,which gives the following nonlinear equation: (24a)2wwt-3w2wx+w0y+w3wwxxx+6wxwxx+3ww0xx+12wxw0x+2β3w2∑i=1Nq¯ir¯ix-2β3wwx∑i=1Nq¯ir¯i-3α2w3∑i=1Nq¯i,xxr¯i+q¯i,xr¯ix-6α2w2wx+w0∑i=1Nq¯ir¯ix+3α2w2wx+w0x∑i=1Nq¯ir¯i=0,(24b)wy-w2wxx+2w0x+α2w∑i=1Nq¯ir¯ix-α2wx∑i=1Nq¯ir¯i=0,(24c)α2q¯i,t-w3q¯i,xxx-3w2wx+w0q¯i,xx-β3q¯i,y-w2q¯i,xx=0,α2r¯i,t-w3r¯i,xxx-3w2w0r¯i,xx-β3r¯i,y+w2r¯i,xx=0,i=1,2,…,N,with the Lax representation as follows: (25)ψ¯y=w2∂2+α2∑i=1Nq¯i∂-1r¯i∂2ψ¯,(26)ψ¯t=w3∂3+3w2wx+w0∂2+β3∑i=1Nq¯i∂-1r¯i∂2ψ¯.Particularly, when takingα2=β3=0; α2=0, β3=1; α2=1, β3=0; and α2=1, β3=1, respectively, (24a), (24b), and (24c) and (26) reduce to the (2+1)-HD equation, the first type of (2+1)-HD equation with self-consistent sources ((2+1)-HDESCS), the second type of (2+1)-HDESCS, and the mixed type of (2+1)-HDESCS and their Lax representations, respectively.


## 2.2. Reduction

Consider the constraint given by(27)L¯k=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2.Then (11b) yields(28)L¯kσk=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2,L¯k=0,(29)B¯k,σk=L¯σkk+=0,∑i=1Nq¯i∂-1r¯i∂2σk=L¯σkk-=0,which imply that L¯, B¯k, q¯i, and r¯i under (27) are independent of σk. Subsequently, q¯i,σk and r¯i,σk in (11c) should be replaced by λiq¯i and -λir¯i as in the case of constrained flow of KP [19, 20]; namely, (11c) under the constraint (27) should be replaced by (30)αnλiq¯i-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,(31)αn-λir¯i+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0.We will show that constraint (27) is invariant under γn-flow (11a) and (31). In fact, making use of (11a), (17), and (31), we have (32)
L
¯
k
-
B
¯
k
γ
n
=
L
¯
γ
n
k
-
=
B
^
n
,
L
¯
k
-
,
β
k
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
γ
n
=
β
k
∑
i
=
1
N
q
¯
i
,
γ
n
∂
-
1
r
¯
i
∂
2
+
q
¯
i
∂
-
1
r
¯
i
,
γ
n
∂
2
=
∑
i
=
1
N
β
k
B
¯
n
q
¯
i
∂
-
1
r
¯
i
∂
2
+
α
n
λ
i
q
¯
i
-
B
¯
k
q
¯
i
∂
-
1
r
¯
i
∂
2
-
β
k
q
¯
i
∂
-
1
B
¯
n
r
¯
i
∂
2
+
α
n
q
¯
i
∂
-
1
-
λ
i
r
¯
i
+
∂
-
2
B
¯
k
∗
∂
2
r
¯
i
∂
2
=
B
¯
n
,
β
k
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
-
B
¯
k
,
α
n
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
=
B
^
n
,
β
k
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
-
B
¯
k
,
α
n
∑
i
=
1
N
q
¯
i
∂
-
1
r
¯
i
∂
2
-
=
B
^
n
,
L
¯
k
-
-
B
^
n
,
B
¯
k
-
-
B
¯
k
,
B
^
n
-
+
B
¯
k
,
B
¯
n
-
=
B
^
n
,
L
¯
k
-
.Then (33)L¯k-B¯k-βk∑i=1Nq¯i∂-1r¯i∂2γn=0.This means that the submanifold determined by k-constraint (27) is invariant under the γn-flow (11a) and (31).Therefore, the constrained flow of (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c) and (21a) and (21b)) under (27) reads (34a)B¯k,γn+B¯k,B¯n+βk∑i=1Nq¯i∂-1r¯i∂2,B¯n++αnB¯k,∑i=1Nq¯i∂-1r¯i∂2+=0,(34b)αnλiq¯i-B¯kq¯i-βkq¯i,γn-B¯nq¯i=0,αn-λir¯i+∂-2B¯k∗∂2r¯i-βkr¯i,γn+∂-2B¯n∗∂2r¯i=0,i=1,2,…,N.(34c)withB¯n=B¯k+βk∑i=1Nq¯i∂-1r¯i∂2+n/k.Example 4.
Whenn=2, k=3, γ2=y, (34a), (34b), and (34c) give (35a)3w2wx+w0y-w3wwxxx+6wxwxx+3ww0xx+12wxw0x-2β3w2∑i=1Nq¯ir¯ix+2β3wwx∑i=1Nq¯ir¯i+3α2w3∑i=1Nq¯i,xxr¯i+q¯i,xr¯ix+6α2w2wx+w0∑i=1Nq¯ir¯ix-3α2w2wx+w0x∑i=1Nq¯ir¯i=0,(35b)wy-w2wxx+2w0x+α2w∑i=1Nq¯ir¯ix-α2wx∑i=1Nq¯ir¯i=0,(35c)α2λiq¯i-w3q¯i,xxx-3w2wx+w0q¯i,xx-β3q¯i,y-w2q¯i,xx=0,α2-λir¯i-w3r¯i,xxx-3w2w0r¯i,xx-β3r¯i,y+w2r¯i,xx=0,i=1,2,…,N,which is the k-constraint (2+1)-(γn,σk)-Harry Dym equation.


## 3. The(γn,σk)-mKPH

In the same way, the(γn,σk)-mKPH can be formulated. The L~ operator of mKP hierarchy is defined by (36)L~=∂+v0+v1∂-1+v2∂-2+⋯.The Lax equation of mKP hierarchy is given by(37)L~tn=B~n,L~,B~n=L~≥1n,n≥1.The commutativity of ∂tn and ∂tk flows gives the zero-curvature equation (7).Since the squared eigenfunction symmetry constraint, given by(38)L~k=B~k+∑i=1Nq~i∂-1r~i∂,q~i,tn=B~nq~i,r~i,tn=-∂-1B~n∗∂r~i,is compatible with mKP hierarchy [21, 22], we have the following.Definition 5.
The(γn,σk)-mKPH is defined by (39a)L~γn=B~n+αn∑i=1Nq~i∂-1r~i∂,L~,(39b)L~σk=B~k+βk∑i=1Nq~i∂-1r~i∂,L~,(39c)αnq~i,σk-B~kq~i-βkq~i,γn-B~nq~i=0,αnr~i,σk+∂-1B~k∗∂r~i-βkr~i,γn+∂-1B~n∗∂r~i=0,i=1,2,…,N.(39a), (39b), and (39c) have the Lax representation(40)ψ~γn=B~n+αn∑i=1Nq~i∂-1r~i∂ψ~,ψ~σk=B~k+βk∑i=1Nq~i∂-1r~i∂ψ.In the same way as in Section 2, we can verify the compatibility of (39a) and (39b) under (39c).By using the formula(41)B~n,∑i=1Nq~i∂-1r~i∂-=∑i=1NB~nq~i∂-1r~i∂-∑i=1Nq~i∂-2B~n∗∂r~i∂,the zero-curvature equation for (γn,σk)-mKPH can be written as (42a)B~n,σk-B~k,γn+B~n,B~k+βkB~n,∑i=1Nq~i∂-1r~i∂++αn∑i=1Nq~i∂-1r~i∂,B~k+=0,(42b)αnq~i,σk-B~kq~i-βkq~i,γn-B~nq~i=0,αnr~i,σk+∂-1B~k∗∂r~i-βkr~i,γn+∂-1B~n∗∂r~i=0,i=1,2,…,N.It is easy to see that (γn,σk)-mKPH ((39a), (39b), and (39c) and (42a) and (42b)) for αn=βk=0 reduces to mKPH (37) and (γn,σk)-mKPH for αn=0, βk=1 reduces to exmKPH [23].Example 6.
Let us taken=2 and k=3, and set γ2=y, σ3=t, and v0=v. Then (42a) and (42b) become(43a)B~2,t-B~3,y+B~2,B~3+β3B~2,∑i=1Nq~i∂-1r~i∂++α2∑i=1Nq~i∂-1r~i∂,B~3+=0,(43b)α2q~i,t-B~3q~i-β3q~i,y-B~2q~i=0,α2r~i,t+∂-1B~3∗∂r~i-β3r~i,y+∂-1B~2∗∂r~i=0,i=1,2,…,N, which gives the following nonlinear equation: (44a)4vt-vxxx+6v2vx-3∂-1vyy-6vx∂-1vy+4β3∑i=1Nq~ir~ix+α2∑i=1N3q~ir~i,xx-q~i,xr~i-3q~ir~iy-6vq~ir~ix=0,(44b)α2q~i,t-q~i,xxx-3vq~i,xx-32∂-1vyq~i,x-32vxq~i,x-32v2q~i,x-32∑j=1Nq~jr~jq~ix-β3q~i,y-q~i,xx-2vq~ix=0,α2r~i,t-r~i,xxx-3vr~i,xx-32∂-1vyr~i,x-32vxr~i,x-32v2r~i,x-32∑j=1Nq~jr~jr~ix-β3r~i,y+r~i,xx-2vr~ix=0,i=1,2,…,N.Particularly, when takingα2=β3=0; α2=0, β3=1; α2=1, β3=0; and α2=1, β3=1, respectively, (44a) and (44b) reduce to the mKP equation, the first type of mKP equation with self-consistent sources, the second type of mKP equation with self-consistent sources, and the mixed type of mKP equation with self-consistent sources, respectively.


## 4. Links between(γn,σk)-KPH and (γn,σk)-mKPH and (γn,σk)-mKPH and (2+1)-(γn,σk)-HDH

In [17] the gauge transformations between KP and mKP hierarchies and the reciprocal transformation between mKP and HD hierarchies are proposed. The KP, mKP, and HD hierarchy are intimately related under these gauge transformations and reciprocal links. In [22], the constrained KP hierarchy and the constrained modified KP hierarchy are studied. And in [24], the gauge transformation between KP and mKP hierarchies with self-consistent sources is given. It is natural to think whether these transformations can be generalized to the (γn,σk)-KPH, (γn,σk)-mKPH, and (2+1)-(γn,σk)-HDH. The answer is positive and the main results are as follows.Theorem 7.
(a) SupposeL,qi,ri satisfy (γn,σk)-KPH (1a), (1b), and (1c) and q and f are independent eigenfunctions for Lax pair (4); then(45)L~=q-1Lq,q~i=q-1qi,r~i=-∂-1qri,q~=q-1fsatisfy the (γn,σk)-mKPH ((39a), (39b), and (39c)) and its Lax pair (40).
(b) SupposeL~,  q~i,  r~i satisfy the (γn,σk)-mKPH ((39a), (39b), and (39c)); q~ is eigenfunction for Lax pair (40). After the transformation x¯=q~(x,γn,σk),  γ¯n=γn,  σ¯k=σk; then(46)L¯=L~,q¯i=q~i,r¯i=∂-1q~r~ix-r~iq~satisfy the (2+1)-(γn,σk)-HDH ((11a), (11b), and (11c)).Proof.
For convenience, we omit∑‍.(a) Consider(47)L~γn=-q-2qγnLq+q-1Lγnq+q-1Lqγn=-q-2L≥0n+αnqi∂-1riqLq+q-1L≥0n+αnqi∂-1ri,Lq+q-1LL≥0n+αnqi∂-1riq=q-1Lq,q-1L≥0n+αnqi∂-1riq+q-1L≥0n+αnqi∂-1ri,Lq=q-1L≥0n+αnqi∂-1riq-q-1L≥0n+αnqi∂-1riq,L~=q-1L≥0nq-q-1L≥0nq,L~+αnq-1qi∂-1riq-q-1qi∂-1riq,L~=L~≥1n,L~+αnq~i∂-1r~i∂,L~=L~≥1n+αnq~i∂-1r~i∂,L~.In the same way, (39b) can be proved. The two equations in (39c) can be proved similarly. So we only prove the first one. Consider (48)αnq~i,σk-L~≥1kq~i=-αnq-2qσkqi+αnq-1qi,σk-αnL~≥1kq~i=-αnq-2L≥0k+βkqi∂-1riqqi+q-1βkqi,γn-L≥0nqi+αnL≥0kqi-αnL~≥1kq~i=-αnq-1L~≥0kq-1qqi-αnβkq-2qi∂-1riqqi+βkq-1qi,γn-L≥0nqi+αnL~≥0kqiq-1-αnL~≥1kq~i=-αnq-1L~≥0k1qi+αnL~≥0kqiq-1-αnβkq-2qi∂-1riqqi+βkq-1qi,γn-L≥0nqi-αnL~≥1kq~i=αnL~≥1kq~i-αnL~≥1kq~i-αnβkq-2qi∂-1riqqi+βkq-1qi,γn-L≥0nqi=-αnβkq-2qi∂-1riqqi+βkq-1qi,γn-L≥0nqi.Similarly, we have(49)βkq~i,γn-L~≥1nq~i=-αnβkq-2qi∂-1riqqi+αnq-1qi,σk-L≥0kqi.Thus, we have(50)αnq~i,σk-L~≥1kq~i-βkq~i,γn-L~≥1nq~i=q-1βkqi,γn-L≥0nqi-αnqi,σk-L≥0kqi=0.In the following, we will prove thatq~=q-1f satisfies Lax pair (40). Consider (51)q~γn=-q-2qγnf+q-1fγn=-q-1L≥0n+αnqi∂-1riqq-1f+q-1L≥0n+αnqi∂-1rif=-q-1qL~≥0nq-1qq-1f+q-1qL~≥0nq-1f-αnq-1qi∂-1riqq-1f+αnq-1qi∂-1rif=-L~≥0n1q~+L~≥0nq~+αnq~q~ir~i+αnq~i∂-1rif=L~≥1n+αnq~i∂-1r~i∂q~.Similarly, we can prove thatq~=q-1f satisfies the second equation in (40).(b) Since the first order coefficient ofL¯≥1¯n=L~≥1n is given by [L¯≥1¯nx¯], we have(52)L¯≥2¯n=L~≥1n-L~≥1nq~∂x¯.L~γn=[∂γn,L~]=[∂γ¯n+q~γn∂x¯,L¯]=L¯γn+[q~γnq~x-1∂x,L~] yields (53)L¯γn=L~γn-q~γnq~x-1∂x,L~=L~≥1n+αnq~i∂-1r~i∂,L~-L~≥1n+αnq~i∂-1r~i∂q~q~x-1∂x,L~=L~≥1n-L~≥1nq~∂x¯,L¯+αnq~i∂-1r~i∂-αnq~i∂-1r~iq~x∂x¯,L~=L¯≥2¯n,L¯+αnq~i∂-1r~i∂-αnq~ir~iq~-∂-1r~ixq~∂x¯,L¯=L¯≥2¯n,L¯+-αnq¯i∂x¯-1r¯ix∂x¯+αnq¯ir¯i∂x¯,L¯=L¯≥2¯n,L¯+αnq¯i∂x¯-1r¯i∂x¯2,L¯,which implies that (11a) holds. In the same way, we can prove (11b). Consider(54)αnq¯i,σk-L¯≥2¯kq¯i=αnq~i,σk-αnq~σkq¯i,x¯-αnL¯≥2¯kq¯i=βkq~i,γn-L~≥1nq~i+αnL~≥1kq~i-αnL~≥1k+βkq~i∂-1r~i∂xq~q¯ix¯-αnL¯≥2¯kq¯i=αnL~≥1kq~i-αnL~≥1kq~q¯ix¯-αnβkq~i∂-1r~i∂xq~q¯ix¯+βkq~i,γn-L~≥1nq~i-αnL¯≥2¯kq¯i=-αnβkq~i∂-1r~i∂xq~q¯ix¯+βkq~i,γn-L~≥1nq~i.Similarly(55)βkq¯i,γn-L¯≥2¯nq¯i=-αnβkq~i∂-1r~i∂xq~q¯ix¯+αnq~i,σk-L~≥1kq~i.So(56)αnq¯i,σk-L¯≥2¯kq¯i-βkq¯i,γn-L¯≥2¯nq¯i=βkq~i,γn-L~≥1nq~i-αnq~i,σk-L~≥1kq~i=0.In this way, we present the connection between the solutions of(γn,σk)-KPH, (γn,σk)-mKPH, and (γn,σk)-HDH under the gauge transformations and reciprocal transformations. In [17], the original three hierarchies are intimately summarized in a diagram. In our generalized system, we can find that similar results remain correct after we add some constraints on the hierarchies. These provide us with a convenient way to obtain the solutions of (γn,σk)-mKPH and (γn,σk)-HDH from the solutions of (γn,σk)-KPH.


## 5. Solutions for(γn,σk)-mKPH and (2+1)-(γn,σk)-HDH

We first briefly recall the generalized dressing method for the(γn,σk)-KPH proposed in [15]. Let fi, gi satisfy (57a)fi,γn=∂nfi,fi,σk=∂kfi,(57b)gi,γn=∂ngi,gi,σk=∂kgi,i=1,…,N,and let hi be the linear combination of fi and gi:(58)hi=fi+Fiαnγn+βkσkgii=1,…,N,with Fi(X) being a differentiable function of X, X=αnγn+βkσk. The dressing operator W is defined by(59)W=Wrh1,…,hN,∂Wrh1,…,hN,where (60)Wrh1,…,hN=h1h2⋯hNh1′h2′⋯hN′⋮⋮⋮⋮h1N-1h2N-1⋯hNN-1,Wrh1,…,hN,∂=h1h2⋯hN1h1′h2′⋯hN′∂⋮⋮⋮⋮⋮h1Nh2N⋯hNN∂Nis the Wronskian determinant. Define(61)qi=-F˙iWgi,ri=-1N-iWrh1,…,h^i,…,hNWrh1,…,hN,i=1,…,N,where the hat ^ means ruling out this term from the Wronskian determinant, F˙i=dFi/dX. Then we have the following.Theorem 8 (see [15]).
LetW be defined by (59) and (58), let L=W∂W-1, and let qi and ri be given by (61); then W, L, qi, ri satisfy (γn,σk)-KPH ((1a), (1b), and (1c) and (3a) and (3b)).Choose(62)q=W1=-1NWrh1′,h2′,…,hN′Wrh1,h2,…,hNas the particular eigenfunction for (4); based on Theorem 7, we have the Wronskian solution for (γn,σk)-mKPH: (63a)L~=Wrh1,…,hN,∂Wrh1′,…,hN′∂Wrh1,…,hN,∂Wrh1′,…,hN′-1,(63b)q~i=-F˙iWrh1,…,hN,giWrh1′,…,hN′,(63c)r~i=Wrh1′,…,h^i′,…,hN′Wrh1,…,hN.Choose(64)f=WeηN=eηNWrh1,…,hNh1h2⋯hN1h1′h2′⋯hN′μN⋮⋮⋮⋮⋮h1Nh2N⋯hNNμNNas another eigenfunction for (4). Based on Theorem 7, we have (65)q~=-1NeηNWrh1′,…,hN′h1h2⋯hN1h1′h2′⋯hN′μN⋮⋮⋮⋮⋮h1Nh2N⋯hNNμNN.Usingx¯=q~,  ∂x=q~x∂x¯, based on Theorem 7, we have (66)
L
¯
=
L
~
x
¯
=
h
¯
1
h
¯
2
⋯
h
¯
N
1
q
~
x
∂
x
¯
h
¯
1
q
~
x
∂
x
¯
h
¯
2
⋯
q
~
x
∂
x
¯
h
¯
N
q
~
x
∂
x
¯
⋮
⋮
⋮
⋮
⋮
q
~
x
N
∂
x
¯
N
h
¯
1
q
~
x
N
∂
x
¯
N
h
¯
2
⋯
q
~
x
N
∂
x
¯
N
h
¯
N
q
~
x
N
∂
x
¯
N
q
~
x
∂
x
¯
h
¯
1
q
~
x
∂
x
¯
h
¯
2
⋯
q
~
x
∂
x
¯
h
¯
N
q
~
x
2
∂
x
¯
2
h
¯
1
q
~
x
2
∂
x
¯
2
h
¯
2
⋯
q
~
x
2
∂
x
¯
2
h
¯
N
⋮
⋮
⋮
⋮
q
~
x
N
∂
x
¯
N
h
¯
1
q
~
x
N
∂
x
¯
N
h
¯
2
⋯
q
~
x
N
∂
x
¯
N
h
¯
N
q
~
x
∂
x
¯
(67)
h
¯
1
h
¯
2
⋯
h
¯
N
1
q
~
x
∂
x
¯
h
¯
1
q
~
x
∂
x
¯
h
¯
2
⋯
q
~
x
∂
x
¯
h
¯
N
q
~
x
∂
x
¯
⋮
⋮
⋮
⋮
⋮
q
~
x
N
∂
x
¯
N
h
¯
1
q
~
x
N
∂
x
¯
N
h
¯
2
⋯
q
~
x
N
∂
x
¯
N
h
¯
N
q
~
x
N
∂
x
¯
N
q
~
x
∂
x
¯
h
¯
1
q
~
x
∂
x
¯
h
¯
2
⋯
q
~
x
∂
x
¯
h
¯
N
q
~
x
2
∂
x
¯
2
h
¯
1
q
~
x
2
∂
x
¯
2
h
¯
2
⋯
q
~
x
2
∂
x
¯
2
h
¯
N
⋮
⋮
⋮
⋮
q
~
x
N
∂
x
¯
N
h
¯
1
q
~
x
N
∂
x
¯
N
h
¯
2
⋯
q
~
x
N
∂
x
¯
N
h
¯
N
-
1
=
q
~
x
N
N
+
1
/
2
h
¯
1
h
¯
2
⋯
h
¯
N
1
h
¯
1
′
h
¯
2
′
⋯
h
¯
N
′
∂
x
¯
⋮
⋮
⋮
⋮
⋮
h
¯
1
N
h
¯
2
N
⋯
h
¯
N
N
∂
x
¯
N
q
~
x
N
N
+
1
/
2
h
¯
1
′
h
¯
2
′
⋯
h
¯
N
′
h
¯
1
′
′
h
¯
2
′
′
⋯
h
¯
N
′
′
⋮
⋮
⋮
⋮
h
¯
1
N
h
¯
2
N
⋯
h
¯
N
N
q
~
x
∂
x
¯
h
¯
1
h
¯
2
⋯
h
¯
N
1
h
¯
1
′
h
¯
2
′
⋯
h
¯
N
′
∂
x
¯
⋮
⋮
⋮
⋮
⋮
h
¯
1
N
h
¯
2
N
⋯
h
¯
N
N
∂
x
¯
N
h
¯
1
′
h
¯
2
′
⋯
h
¯
N
′
h
¯
1
′
′
h
¯
2
′
′
⋯
h
¯
N
′
′
⋮
⋮
⋮
⋮
h
¯
1
N
h
¯
2
N
⋯
h
¯
N
N
-
1
=
W
r
h
¯
1
,
…
,
h
¯
N
,
∂
x
¯
W
r
h
¯
1
′
,
…
,
h
¯
N
′
q
~
x
∂
x
¯
W
r
h
¯
1
,
…
,
h
¯
N
,
∂
x
¯
W
r
h
¯
1
′
,
…
,
h
¯
N
′
-
1
,where the elements h¯i=hi(x¯) and h¯i′ denotes ∂h¯i/∂x¯. Similarly, we have (68a)q¯i=-F˙iWrh¯1,…,h¯N,g¯iWrh¯1′,…,h¯N′,(68b)r¯i=-∂x-1q~xr~i=-∂x¯-1r~ix¯=-∂x¯-1Wrh¯1′,…,h¯^i′,…,h¯N′Wrh¯1,…,h¯N,where g¯i=gi(x¯). Equations (67) and (68a) and (68b) give the Wronskian solution for (2+1)-(γn,σk)-HDH.Let us illustrate it by solving (24a), (24b), and (24c) and (44a) and (44b). We take the solution of (57a) and (57b) as follows: (69)fi≔exp⁡λix+λi2y+λi3t=eξi,gi≔exp⁡μix+μi2y+μi3t=eηi,(70)hi≔fi+Fiα2y+β3tgi=2Fiexp⁡ξi+ηi2cosh⁡Ωi,Ωi=12ξi-ηi-ln⁡Fi.Example 9 (solutions for the mixed type of mKPESCS ((44a) and (44b))).
SinceL~=q-1Lq, then it is easy to find (71)v0=qxq.For N=1, we have (72)q=-λ1+μ12+μ1-λ12tanh⁡Ω1.The one-soliton solution for (44a) and (44b) with N=1 is as follows: (73)v=v0=λ1-μ12tanh⁡Ω1+θ1-tanh⁡Ω1,q~1=F˙12F1λ1μ1μ1-λ1eξ1+η1sech⁡Ω1+θ1,r~1=-12F1e-ξ1+η1sech⁡Ω1,where θ1=ln⁡λ1/μ1.Example 10 (solutions for the mixed type of HDESCS ((24a), (24b), and (24c))).
Based on Theorem7, we have that (74)q~=-1NeηNWrh1′,…,hN′h1h2⋯hN1h1′h2′⋯hN′μN⋮⋮⋮⋮⋮h1Nh2N⋯hNNμNNis the eigenfunction of the mixed type of mKPESCS and we have L¯=L~; that is, (75)w∂x¯+w0+w1∂x¯-1+⋯=∂x+v0+v1∂x-1+⋯=q~x∂x¯+v0+⋯.From the above equation, we obtain (76)w=q~x,w0=v0.The implicit solutions for the mixed type of HDESCS ((24a), (24b), and (24c)) with N=1 are (77a)x¯=q~=λ1-μ12λ1μ1eη1+Ω1sech⁡Ω1+θ1,(77b)w=λ1-μ12eη1+Ω1sech2⁡Ω1+θ1cosh⁡Ω1,(77c)w0=λ1-μ12tanh⁡Ω1+θ1-tanh⁡Ω1,(77d)q¯1=F˙12F1λ1μ1μ1-λ1eξ1+η1sech⁡Ω1+θ1,(77e)r¯1=12F1∂x¯-1e-ξ1+η1sech⁡Ω1,where θ1=ln⁡λ1/μ1,  ξ1=λ1x+λ12y+λ13t,  η1=μ1x+μ12y+μ13t,  Ω1=(1/2)(ξ1-η1-ln⁡F1), and the relation between x and x¯ is given by (77a).


## 6. Conclusion

In this paper, a new (2+1)-(γn,σk)-HDH is proposed by introducing new time series γn and σk and adding eigenfunctions as components. The (2+1)-(γn,σk)-HDH includes (2+1)-HD hierarchy and extended HD hierarchy and contains first type and second type as well as mixed type of (2+1)-HD equation with self-consistent sources as special cases. The reduction of the (2+1)-(γn,σk)-HDH is studied and the k-constrained (2+1)-(γn,σk)-HDH is given. In the same way as (2+1)-(γn,σk)-HDH, the (γn,σk)-mKPH is formulated and its zero-curvature equation and Lax representation are presented.The gauge transformations and reciprocal transformations between(γn,σk)-KPH, (γn,σk)-mKPH, and (2+1)-(γn,σk)-HDH are given. They help us to obtain solutions of (γn,σk)-mKPH and (2+1)-(γn,σk)-HDH. By making use of the transformations, we find the soliton solutions of the (γn,σk)-mKPH and (2+1)-(γn,σk)-HDH. Particularly, the soliton solutions for the mixed type of mKPESCS and HDESCS are obtained.



---

*Source: 392723-2015-12-14.xml*