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

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 representationThe -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. Considerby dropping the -dependence.

Based on the squared eigenfunction symmetry constraintwhich 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 proveFor 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 byThen (11b) yieldswhich 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 byThe 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 representationIn the same way as in Section 2, we can verify the compatibility of (39a) and (39b) under (39c).

By using the formulathe 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); thensatisfy 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 ,  ,  ; thensatisfy 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 bywhere is the Wronskian determinant. Definewhere 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).