Research Article | Open Access
Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation
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.
Since the concept on integrable couplings was proposed , some integrable couplings of the known integrable systems, such as the AKNS system and the KN system, were obtained. Ma  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  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  proposed the quadratic-form identity for which Hamiltonian structures of some integrable couplings were generated. Ma and Chen  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 . 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  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  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  which was proposed by Ma  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 , 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  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 . 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 . 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  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 ; 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  have obtained the Hamiltonian structure of the negative-order KdV equation by using the trace identity proposed by Tu . 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
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 .
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,  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.
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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