Advances in Mathematical Physics
Volume 2015 (2015), Article ID 154915, 8 pages
http://dx.doi.org/10.1155/2015/154915

Research Article

Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation

Binlu Feng¹ and Yufeng Zhang²

¹School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China
²College of Sciences, China University of Mining and Technology, Xuzhou 221116, China

Received 28 October 2014; Revised 25 November 2014; Accepted 17 December 2014

Academic Editor: Stephen C. Anco

Copyright © 2015 Binlu Feng and Yufeng Zhang. 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.

Based on some known loop algebras with finite dimensions, two different negative-order integrable couplings of the negative-order Korteweg-de Vries (KdV) hierarchy of evolution equations are generated by making use of the Tu scheme, from which the corresponding negative-order integrable couplings of the negative-order KdV equations are followed to be obtained. The resulting Hamiltonian structure of one negative integrable coupling is derived from the variational identity.

1. Introduction

Since the concept on integrable couplings was proposed [1], some integrable couplings of the known integrable systems, such as the AKNS system and the KN system, were obtained. Ma [2] made use of the perturbation method to obtain the integrable couplings of the KdV equation as well as their some algebraic and geometric properties. Reference [3] employed a simple finite-dimensional Lie algebra to present a method for generating integrable couplings of integrable hierarchies of evolution equations. Later, Guo and Zhang [4] proposed the quadratic-form identity for which Hamiltonian structures of some integrable couplings were generated. Ma and Chen [5] further generalized the quadratic-form identity and completely improved it to obtain the variational identity for deducing the Hamiltonian structures of integrable couplings which is more convenient. Actually, there are lots of related references in integrable-coupling area. The adopted algebras were studied and applied in other existing literature. One of them is so-called perturbation algebra, which was even used to present 2 + 1-dimensional local bi-Hamiltonian systems by Ma and Fuchssteiner in 1996, and whose systematical presentation was presented in [6]. In addition, Ma et al. [2, 7–9] introduced different Lie algebras to obtain some new and interesting integrable couplings. By following the methods, many interesting integrable couplings of some known integrable hierarchies were obtained, such as the works in [10–12]. However, to our best knowledge, the negative-order integrable couplings of the negative-order integrable hierarchies have not been discussed systematically. Up to now, we know that some interesting related negative-order integrable equations including the negative-order KdV equation and some associated properties were obtained, such as the results in [13–18]. But their negative-order integrable couplings have not been discussed. In the paper, we want to discuss the problem. Qiao and Fan [18] employed the Lenard sequence method to obtain the negative-order KdV equation and then represented it by its Lax pair. Enlightened by this work, we will generate the negative-order KdV hierarchy and its integrable couplings by enlarged Lie algebras and the enlarged Lax pairs. It is remarkablethat the integrable couplings obtained in the paper include the linear and nonlinear integrable couplings. In the aspect of nonlinear integrable couplings, Ma and Zhu [19] proposed some semisimple Lie algebras to present a way for generating nonlinear discrete integrable couplings. Then [20–22] also reveal some nonlinear integrable couplings by constructing various Lie algebras. As reduced cases, two negative-order integrable couplings of the negative-order KdV equation are obtained. Finally, the Hamiltonian structure of one negative-order integrable coupling in the negative-order KdV hierarchy is obtained by the variational identity.

2. Two Different Negative-Order Integrable Couplings

In the section, we adopt the Tu scheme [23] which was proposed by Ma [24] and the known finite-dimensional Lie algebras to deduce two integrable couplings of the negative-order KdV hierarchy. As reduced cases, two negative-order integrable couplings of the negative-order KdV equation are also followed to present.In [25, 26], a 6-dimensional Lie algebra reads

where

here , , and .

In [26], another 6-dimensional Lie algebra was given by

where

The resulting loop algebras can be defined as , where or . The loop algebras of the Lie algebras and are denoted by and , respectively. According to the Tu scheme, we first solve a stationary 0-curvature equation

where the Lax matrices and belong to the loop algebra or ; and can be represented by the loop algebra or :

where and belong to or and require ; here stands for the degree of the elements in the loop algebras; are the derivative functions in and . Then choose a suitable modified term so that

where

satisfies the 0-curvature equation

which can derive an integrable hierarchy of evolution equations. In order to deduce negative-order integrable hierarchies, we need to change (7) to the following form:

Qiao and Li [17] once obtained the negative-order KdV equation through the Lenard recursive sequence

which is equivalent to a nonlinear quartic integrable system

The Lax pair of (12) was presented by a proposition in [18]. Furthermore, Qiao and Fan produced the Hamiltonian structure, Darboux transformations, and some exact solutions. Enlightened by their results, we will deduce the negative-order KdV hierarchy by employing the Tu scheme which can reduce to the negative-order KdV equation (12).

Denote where ; then is an obvious loop algebra. Employing the loop algebra, we introduce a pair of Lax matrices

A solution to (5) is presented as

which is equivalent to the following relation:

where

Equation (16) is called the Lenard sequence [18]. Denote

Equation (5) can be decomposed into

With the help of (14) and (19), we get that

Take

we have that

Therefore, the 0-curvature equation

admits the following integrable hierarchy:

which is called the negative-order KdV hierarchy. Obviously, we have

When , (16) gives , ; here let .

When , (16) reduces to

Assume ; then (26) becomes

Combined with (25), we obtain the negative-order KdV equation when :

which is the same as (12) except for a constant in the first equation. Qiao and Fan [18] established a Lax pair of the negative-order KdV equation by employing the Lenard sequence method.

In what follows, we can directly single out the Lax pair of (28). It is easy to see that

Due to , we can get . In terms of (15), we have , . Thus, one infers that

Hence, the compatibility condition of the Lax pair

gives rise to the negative-order KdV equation (28); that is, the 0-curvature equation

admits (28). The Hamiltonian structure of (28) can be given similar to that in [18]; here we do not repeat it again.

3. Two Negative-Order Integrable Couplings

We first make use of the loop algebra to deduce the first negative-order integrable couplings of the negative-order KdV hierarchy and then employ the second loop algebra to derive the second negative-order integrable couplings.

3.1. The First Negative-Order Integrable Coupling

Set

then (5) gives the following recursive relations among :

here (34) is just the same as (15). Denote

after tedious calculation, one infers that

Therefore, the 0-curvature equation (10) is equivalent to

which is the negative-order integrable couplings of the negative-order KdV hierarchy. In what follows, we want to consider its reduction. First of all, we have a recursive relation from (35):

When , we take . When , we have in terms of (39) that

Let ; we find from (40) that

Assume ; since , thus we have

The first equation in (42) is the negative-order KdV equation, while the second equation is just coupled part of integrable couplings. Therefore, (42) is an integrable coupling of the negative-order KdV equation.

3.2. The Second Negative-Order Integrable Coupling

Set

Similar to the previous case, we can get (34) and the following equations which are different from (35):

from which we have

Take ; we have

Set ; one gets from (46) that

Denote

one infers that

Thus, the 0-curvature equation

gives that

We call (51) a negative-order integrable coupling of the negative-order KdV hierarchy. A reduced case of (51) presents that when ,

which is the second negative-order integrable coupling of the negative-order KdV equation. As comparison, we rewrite the second equation in (42) as follows:

which is linear with respect to the dependent variable . The second coupled part in (52) presents again

which is nonlinear in the variable . Therfore, we obtain the linear integrable coupling of the negative-order KdV equation (28):

and the nonlinear integrable coupling:

Remark 1. As previous statement, Qiao and Fan [18] have obtained the Hamiltonian structure of the negative-order KdV equation by using the trace identity proposed by Tu [23]. In what follows, we will deduce the Hamiltonian structure of the whole negative-order integrable coupling (51) by employing the variational identity.

3.3. The Hamiltonian Structure of the Nonlinear Integrable Couplings

Rewrite (43) as the forms of square matrices

where .

In order to employ the variational identity, we should establish a Lie algebra with collumn-vector elements. For that sake, we consider the linear space equipped with a kind of commutator as follows:

where

The commutator (59) can be written as

where

A direct verification indicates that the linear space becomes a Lie algebra with the commutator (59). It is obvious that the linear map

is an isomorphism between the Lie algebras and , where the matrix is presented in (58). Therefore, we can rewrite the Lax matrices (57) and (58) as follows:

where .

The compatibility condition of (64) under the frame of the Lie algebra is equivalent to that of (43) under the Lie algebra .

According to the scheme for generating Hamiltonian structure of integrable couplings by employing the variational identity, a constant symmetric matrix should be obtained which satisfies the matrix equation

A direct verification shows that

satisfies (65), where , constants. Define a linear functional

With the help of (64), (66), and (67), we get that

Substituting the above consequences into the variational identity reads

where .

Inserting , into (69) gives

If taking , we have . Hence, we get from (70) that

where are the conserved densities of the negative-order integrable coupling (51). Therefore, (51) can be written as the Hamiltonian structure form

where is a Hamiltonian operator.

Remark 2. Two negative-order integrable couplings of the negative-order KdV equation were obtained. One of them is linear; the other is nonlinear. We can only generate the Hamiltonian structure of the nonlinear integrable coupling (51) by employing the variational identity; however, up to now, we have not found a suitable method to deduce the Hamiltonian structure of the linear negative-order integrable coupling (38). May its Hamiltonian structure not exist? which is worth of further discussing in forthcoming days.Besides, we observe that the recurrence relations in negative-order integrable couplings are all nonlocal. In addition, [27] presented some higher dimensional Lie algebras and weredevoted to discussion of 2 + 1-dimensional integrable systems, which may be used to produce more interesting negative-order integrable hierarchies.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11371361), the Natural Science Foundation of Shandong Province (Grant no. ZR2013AL016), and the Fundamental Research Funds for the Central Universities (2013XK03).

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