Integrable Couplings: Generation, Hamiltonian Structures, Conservation Laws, and ApplicationsView this Special Issue
Nonlinear Integrable Couplings of Levi Hierarchy and WKI Hierarchy
With the help of the known Lie algebra, a type of new 8-dimensional matrix Lie algebra is constructed in the paper. By using the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the Wadati-Konno-Ichikawa (WKI) hierarchy are worked out, which are different from the linear integrable couplings. Based on the variational identity, the Hamiltonian structures of the above hierarchies are derived.
The notion of integrable couplings was introduced when the study of Virasoro symmetric algebras [1, 2]. To find as many new integrable systems and their integrable couplings as possible and to elucidate in depth their algebraic and geometric properties are of both theoretical and practical value. During the past few years, some interesting integrable couplings and associated properties of some known interesting integrable hierarchies, such as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the Kaup-Newell (KN) hierarchy, were obtained [3–13]. Here it is necessary to point out that the above mentioned integrable couplings are linear for the supplementary variable, so they are called linear integrable couplings.
Recently, Professor Ma proposed the notion of nonlinear integrable couplings and gave the general scheme to construct nonlinear integrable couplings of hierarchies . Based on the general scheme of constructing nonlinear integrable couplings, Professor Zhang introduced some new explicit Lie algebras and obtained the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy, the Yang hierarchy, and the classical Boussinesq-Burgers (CBB) hierarchy [15, 16].
The aim of the paper is to seek the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy as well as their Hamiltonian structures. The plan of the paper is as follows. In Section 2, with the help of the Lie algebra , an 8-dimensional matrix Lie algebra is presented. It is different from the Lie algebras given in [14–16]. By employing the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy are derived in Section 3. Furthermore, the corresponding Hamiltonian structures are worked out by virtue of the variational identity in Section 4. Finally, some conclusions are obtained in Section 5.
2. 8-Dimensional Matrix Lie Algebra
The Lie algebra is presented as with the basis as follows: equipped with the commutators By virtue of the Lie algebra , we construct an 8-dimensional matrix Lie algebra with the basis as follows: which have the commutative relations Denoting and , then we have Here we need to emphasize that the subalgebras and are both nonsemisimple, which is very important for deriving nonlinear integrable couplings of hierarchies. By using the Lie algebra , we can construct a few kinds of loop algebras , and stand for natural numbers. Among these loop algebras, the simplest one is along with the commutators , , , , and .
In this section, by virtue of the Lie algebra , we construct an 8-dimensional matrix Lie algebra and corresponding loop algebra ; in what follows we will generate the nonlinear integrable couplings of hierarchies by using the loop algebra .
3. Nonlinear Integrable Couplings of Hierarchies
In this section, based on the loop algebra , we construct two isospectral problems to generate the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy, respectively.
3.1. Nonlinear Integrable Couplings of Levi Hierarchy
Take the following isospectral problem: Set , where , . Solving the stationary zero curvature equation gives rise to the recursion relation as follows: Denoting and , it is easy to compute Take Thus, the zero curvature equation leads to the following integrable system: where is a Hamiltonian operator and , the recurrence operator is given from (9) by where Therefore, the system (13) can be written as When , the system (13) reduces to the Levi hierarchy; therefore, in terms of the definition of integrable coupling, we conclude that the system (13) is an integrable coupling of the Levi hierarchy. Especially taking , we have the following reduced equations: Obviously, (18) are nonlinear equations in and , so we call (13) the nonlinear integrable coupling of the Levi hierarchy.
3.2. Nonlinear Integrable Couplings of WKI Hierarchy
Consider an isospectral problem Set , where Because every term in includes , is different from the common form and includes potentials , , , and and , , , , and so on. Then the zero curvature equation yields Denoting , then we have . A direct calculation reads Therefore, the zero curvature equation admits Here is a Hamiltonian operator and , the recurrence operator is given from (21) by where Here , are given in (21) and . Hence, the system (24) can be written as When , the system (24) is just the WKI hierarchy. Taking , the system (24) reduces the following equations: It is easy to find that (28) are nonlinear equations in and , so we call (24) the nonlinear integrable coupling of the WKI hierarchy.
4. Hamiltonian Structures
In this section, we will seek the Hamiltonian structures of the nonlinear integrable couplings of the Levi hierarchy (13) and the WKI hierarchy (24) by virtue of the variational identity. First, we construct a linear map , , . We can conclude that the linear map is an isomorphism from to . Let ; matrix is determined by  where and . From (29), we have
Solving the matrix equation for the constant matrix , , , Then in terms of , define a linear functional in the It is easy to find that satisfies the variational identity Rewrite the Lax pair of nonlinear integrable coupling of the Levi hierarchy as follows: By using (32), we have According the variational identity (33), we have Comparing the coefficients of yields Taking gives rise to . Therefore, Hence, the nonlinear integrable coupling of the Levi hierarchy has the following Hamiltonian structure: Similar to (34), in order to deduce to the Hamiltonian structure of the nonlinear integrable coupling of the WKI hierarchy, we rewrite the Lax pair as follows: Repeat the above procedure; we have Taking in above equation gives . Therefore, , , where Hence, the nonlinear integrable coupling of the WKI hierarchy has the following Hamiltonian structure:
In this paper, we presented a set of new 8-dimensional matrix Lie algebra by virtue of the Lie algebra given in [14–16]. With the help of the Lie algebra, we obtain the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy. Their Hamiltonian structures are also worked out by the variational identity. The Lie algebra constructed in this paper can be used to generate the nonlinear integrable couplings of other hierarchies. We will study these problems in the future.
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 Strategic Pioneering Program of Chinese Academy of Sciences (no. XDA 10020104), the Global Change and Air-Sea Interaction (no. GASI-03-01-01-02), the Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), the Science and Technology Plan Project of Qingdao (no. 14-2-4-77-jch), the Open Fund of the Key Laboratory of Ocean Circulation and Waves, the Chinese Academy of Science (no. KLOCAW1401), the Open Fund of the Key Laboratory of Data Analysis and Application, and the State Oceanic Administration (no. LDAA-2013-04).
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