Integrable Couplings: Generation, Hamiltonian Structures, Conservation Laws, and ApplicationsView this Special Issue
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The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy
By use of the loop algebra , integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.
Integrable coupling is a new topic of the Soliton theory; especially, looking for the new Hamiltonian structure of integrable coupling is more important. The integrable coupling of some known integrable hierarchies is obtained. But their Hamiltonian structure has not been presented because there exists a limitation in trace identity till the quadratic-form identity  and the variational identity  are proposed. In this paper, a higher-dimensional Lie algebra and the loop algebra are constructed [3, 4]. With the help of Tu scheme  and the quadratic-form identity, the integrable coupling of C-KdV hierarchy as well as its bi-Hamiltonian structures is produced.
2. Basic Principle of the Semidirect Sum of Lie Algebras
Let be a linear space over real or complex number field together with multiplication, for any , if satisfy(1)distributive law (2)multiplication commutativity Then, is called algebra.
Lie algebra is an algebra over number field , if its multiplication satisfies the following:(1)bilinearity (2)anticommutative (3)the Jacobi identity where denote the multiplication of , . The multiplication of Lie algebra is called Lie product. One kind of the most important Lie algebras on integrable systems is , where denote matrix order over number field .
satisfies for arbitrary Lie algebra ; then is called Lie ideal.
Lie algebra is called simple Lie algebra if has and 0 as Lie ideal and without other Lie ideal. Semisimple Lie algebra can be written as where is simple Lie algebra. We have already known that , , and are all semisimple Lie algebras which has been studied by Cartan long ago . We also know that Lie algebra can be written as where is semisimple Lie algebras and is solvable Lie algebras [3, 6, 7] and denote the semidirect sum. So we can apply the above basic principle to integrable coupling systems.
3. C-KdV Hierarchy
The C-KdV spectral problem reads as Upon setting , solving the stationary zero curvature equation, engenders The compatibility conditions of the spectral problems determine the C-KdV hierarchy of Soliton equations where
4. A New Integrable Coupling of the C-KdV Hierarchy
In what follows, we expand Lie algebra into a bigger one as the following Lie algebra : We do this along with the following commutative relations: Taking and , it is easy to verify that where is semisimple Lie algebras and is solvable Lie algebras [3, 6, 7].
In terms of the Lie algebra , we constructed the loop algebra as follows [4, 10], with the following commutative relations: , , when , and , when . With the help of above equations, we consider an isospectral problem: Set Solving the stationary zero curvature equation (10) permits that where , , , and are nonzero constants.
Assume that ; then (10) may be written as A direct calculation reads Take ; then the zero curvature equation is equivalent towhere and are Hamiltonian operators.
From (22), a recurrence operator is obtained, which satisfies where It is easy to verify that Therefore, the hierarchy (26) is Liouville integrable. Taking , , and , (26) reduces to (13). According to the integrable theory, the hierarchy (26) is the integrable coupling of the C-KdV hierarchy. Furthermore, in the following part we will point out that there exist bi-Hamiltonian structures from constructing of Lie loop algebras.
5. The Bi-Hamiltonian Structures of the Hierarchy (26)
Let We have =.
In what follows, from , we get Solving the matrix equation for gives rise to So we have .
A direct calculation reads where and .
Substituting the above formulas into the quadratic-form identity yields Comparison of coefficients of of both sides of the above equations leads to To fix the we take into the above equation and find .
So Comparison of coefficients of of both sides of the above equations gives In this situation, we have .
On the one hand, we obtain a new integrable coupling of C-KdV hierarchy by expanding a bigger Lie algebra. On the other hand, the bi-Hamiltonian structures of the integrable coupling of C-KdV hierarchy are observed by use of the quadratic-form identity.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project is in part supported by the Natural Science Foundation of China (Grant no. 11271008), the First-class Discipline of University in Shanghai, and the Shanghai University.
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