Advances in Mathematical Physics
Volume 2015 (2015), Article ID 392723, 11 pages
http://dx.doi.org/10.1155/2015/392723

Research Article

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

Yehui Huang,¹ Yuqin Yao,² and Yunbo Zeng³

¹School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
²Department of Applied Mathematics, China Agricultural University, Beijing 100083, China
³Department of Mathematical Science, Tsinghua University, Beijing 100084, China

Received 12 July 2015; Accepted 16 November 2015

Academic Editor: Boris G.  Konopelchenko

Copyright © 2015 Yehui Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

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 [12–14]. The exKPH consists of -flow of KP hierarchy, -flow, and -evolution equations of eigenfunctions. To make difference, we may call the exKPH as KPH.

Recently, we generalize the KPH to KPH by introducing two new time series and with two parameters and [15]:

where the pseudodifferential operator with potential functions is defined as

and denotes the adjoint operator. The compatibility of -flow (1a) and -flow (1b) under (1c) gives rise to the zero-curvature representation for (1a), (1b), and (1c),

with the Lax representation

The -KPH consists of -flow, -flow, and one mixed and evolution equation of eigenfunctions. The -KPH can be reduced to the KPH and -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 -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)--Harry Dym hierarchy ((2+1)--HDH), which consists of -flow, -flow, and one mixed and evolution equation of eigenfunctions. The (2+1)--HDH can be reduced to the (2+1)-Harry Dym hierarchy ((2+1)-HDH) and -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 -mKPH is proposed. The generalized gauge transformations and generalized reciprocal links between these three kinds of hierarchies are studied. Further, based on these transformations and the solutions of the -KPH [15], the solutions of -mKPH and (2+1)--HDH are obtained, respectively.

The paper is organized as follows. In Section 2, we propose a new -HDH and present its reduction. A new -mKPH is proposed in Section 3. Section 4 is devoted to studying the links between the three kinds of hierarchies. Section 5 presents the soliton solutions and a conclusion is given in the last section.

2. A New (2+1)--Harry Dym Hierarchy

2.1. A New (2+1)--Harry Dym Hierarchy

It is well known that the pseudodifferential operator for (2+1)-HDH with potential functions is defined as

The (2+1)-HDH is given by [17]

where ,  .

The compatibility of the -flow and -flow of (6) leads to the zero-curvature representation of (2+1)-HDH:

In particular, when taking and   and setting ,  , and , (7) yields the (2+1)-HD equation

which is reduced to the Harry Dym equation. Consider

by dropping the -dependence.

Based on the squared eigenfunction symmetry constraint

which is compatible with (2+1)-HDH [17], we propose the following generalized (2+1)-HDH with two generalized time series and :

We will prove the compatibility of (11a) and (11b) under (11c) in the following theorem.

Theorem 1. The -flow (11a) and -flow (11b) under (11c) are compatible.

Proof. Denote

In order to prove , that is,

we only need to prove

For convenience, we omit . We can find that

and, similarly,

Moreover, on the other hand, under the formula [18]

we have

Then (15), (16), and (18) under (11c) yield

The compatibility of -flow (11a) and -flow (11b) under (11c) gives rise to the zero-curvature representation for (11a), (11b), and (11c):

which 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:

with the Lax representation

We briefly call (11a), (11b), and (11c) and (21a) and (21b) as (2+1)--HDH. It is easy to see that (2+1)--HDH ((11a), (11b), and (11c) and (21a) and (21b)) for reduces to (2+1)-HDH ((6) and (7)) and (2+1)--HDH for ,   reduces to -HDH [18]. So (2+1)--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)--HDH as the special cases.

Example 3. Let us take and , and set ,  . Then (21a) and (21b) become

which gives the following nonlinear equation:

with the Lax representation as follows:

Particularly, when taking ; , ; , ; and , , 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

Then (11b) yields

which imply that , , , and under (27) are independent of . Subsequently, and in (11c) should be replaced by and as in the case of constrained flow of KP [19, 20]; namely, (11c) under the constraint (27) should be replaced by

We will show that constraint (27) is invariant under -flow (11a) and (31). In fact, making use of (11a), (17), and (31), we have

Then

This means that the submanifold determined by -constraint (27) is invariant under the -flow (11a) and (31).

Therefore, the constrained flow of (2+1)--HDH ((11a), (11b), and (11c) and (21a) and (21b)) under (27) reads

Example 4. When , , , (34a), (34b), and (34c) give

which is the -constraint (2+1)--Harry Dym equation.

3. The -mKPH

In the same way, the -mKPH can be formulated. The operator of mKP hierarchy is defined by

The Lax equation of mKP hierarchy is given by

The commutativity of and flows gives the zero-curvature equation (7).

Since the squared eigenfunction symmetry constraint, given by

is compatible with mKP hierarchy [21, 22], we have the following.

Definition 5. The -mKPH is defined by

(39a), (39b), and (39c) have the Lax representation

In the same way as in Section 2, we can verify the compatibility of (39a) and (39b) under (39c).

By using the formula

the zero-curvature equation for -mKPH can be written as

It is easy to see that -mKPH ((39a), (39b), and (39c) and (42a) and (42b)) for reduces to mKPH (37) and -mKPH for , reduces to exmKPH [23].

Example 6. Let us take and , and set , , and . Then (42a) and (42b) become

which gives the following nonlinear equation:

Particularly, when taking ; , ; , ; and , , 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 -KPH and -mKPH and -mKPH and (2+1)--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 -KPH, -mKPH, and (2+1)--HDH. The answer is positive and the main results are as follows.

Theorem 7. (a) Suppose satisfy -KPH (1a), (1b), and (1c) and and are independent eigenfunctions for Lax pair (4); then

satisfy the -mKPH ((39a), (39b), and (39c)) and its Lax pair (40).
(b) Suppose ,  ,   satisfy the -mKPH ((39a), (39b), and (39c)); is eigenfunction for Lax pair (40). After the transformation ,  ,  ; then

satisfy the (2+1)--HDH ((11a), (11b), and (11c)).

Proof. For convenience, we omit .

 (a) Consider

 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

 Similarly, we have

 Thus, we have

 In the following, we will prove that satisfies Lax pair (40). Consider

 Similarly, we can prove that satisfies the second equation in (40). (b) Since the first order coefficient of is given by , we have

  yields

 which implies that (11a) holds. In the same way, we can prove (11b). Consider

 Similarly

 So

In this way, we present the connection between the solutions of -KPH, -mKPH, and -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 -mKPH and -HDH from the solutions of -KPH.

5. Solutions for -mKPH and (2+1)--HDH

We first briefly recall the generalized dressing method for the -KPH proposed in [15]. Let , satisfy

and let be the linear combination of and :

with being a differentiable function of , . The dressing operator is defined by

where

is the Wronskian determinant. Define

where the hat  means ruling out this term from the Wronskian determinant, . Then we have the following.

Theorem 8 (see [15]). Let be defined by (59) and (58), let , and let and be given by (61); then , , , satisfy -KPH ((1a), (1b), and (1c) and (3a) and (3b)).

Choose

as the particular eigenfunction for (4); based on Theorem 7, we have the Wronskian solution for -mKPH:

Choose

as another eigenfunction for (4). Based on Theorem 7, we have

Using ,  , based on Theorem 7, we have

where the elements and denotes . Similarly, we have

where . Equations (67) and (68a) and (68b) give the Wronskian solution for (2+1)--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:

Example 9 (solutions for the mixed type of mKPESCS ((44a) and (44b))). Since , then it is easy to find

For , we have

The one-soliton solution for (44a) and (44b) with is as follows:

where .

Example 10 (solutions for the mixed type of HDESCS ((24a), (24b), and (24c))). Based on Theorem 7, we have that

is the eigenfunction of the mixed type of mKPESCS and we have ; that is,

From the above equation, we obtain

The implicit solutions for the mixed type of HDESCS ((24a), (24b), and (24c)) with are

where ,  ,  ,  , and the relation between and is given by (77a).

6. Conclusion

In this paper, a new (2+1)--HDH is proposed by introducing new time series and and adding eigenfunctions as components. The (2+1)--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)--HDH is studied and the -constrained (2+1)--HDH is given. In the same way as (2+1)--HDH, the -mKPH is formulated and its zero-curvature equation and Lax representation are presented.

The gauge transformations and reciprocal transformations between -KPH, -mKPH, and (2+1)--HDH are given. They help us to obtain solutions of -mKPH and (2+1)--HDH. By making use of the transformations, we find the soliton solutions of the -mKPH and (2+1)--HDH. Particularly, the soliton solutions for the mixed type of mKPESCS and HDESCS are obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (11171175, 11201477, and 11301179) and the Fundamental Research Funds for the Central Universities (2014ZZD10).

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