Integrable Couplings: Generation, Hamiltonian Structures, Conservation Laws, and ApplicationsView this Special Issue
Tri-Integrable Couplings of the Giachetti-Johnson Soliton Hierarchy as well as Their Hamiltonian Structure
Based on zero curvature equations from semidirect sums of Lie algebras, we construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy of soliton equations and establish Hamiltonian structures of the resulting tri-integrable couplings by the variational identity.
Soliton theory is a power tool in expanding and describing the nonlinear phenomena in the fields of nonlinear optics, plasma physics, magnetic fluid, and so on. Searching for new integrable systems is an interesting and significant event and the subject of the integrable coupling is a new and important direction in soliton theory. Recently, various examples of bi-integrable couplings and tri-integrable couplings were introduced, which bring us inspiring thoughts and ideas to classify integrable systems with multicomponents and can generate even more diverse recursion operators in block matrix form.
For a given integrable system of evolution type : where is a column vector of dependent variables. It has an integrable coupling as follows: where and denote two column vectors of additional dependent variables. The earliest paper on integrable couplings obtained by Lie algebras and the Tu scheme is the , which gave a direct method for establishing integrable couplings and the integrable couplings of TD hierarchy. Many papers have been dedicated to this topic [3–10]. And there are other ways to construct integrable couplings such as by using perturbations , enlarging spectral problems , and creating new loop algebras . Professor Yu, especially, shows that the Kronecker product is an important and effective method to construct the discrete integrable couplings in  and presents a scheme for constructing real nonlinear integrable couplings of continuous soliton hierarchy in . In 2012, we know that bi-integrable couplings were introduced and developed in . Recently, bi-integrable couplings were further extended to tri-integrable couplings. The following enlarged triangular integrable system: is called a tri-integrable coupling of the system (1) in [17, 18]. If at least one of , , and is nonlinear with respect to any subvectors , , and of new dependent variables, we call this system (3) a nonlinear integrable coupling.
To construct tri-integrable couplings, we need a class of triangular block matrices with being square matrices of the same order. Therefore the Lie algebra has a semidirect sum decomposition: in which , . is non-semisimple because of being a nontrivial ideal of . The block corresponds to the original integrable system, and the other three blocks , , and are used to generate the supplementary vector fields , , and in (3) that we are looking for. Such presented Lie algebras establish a basis for generating nonlinear Hamiltonian tri-integrable couplings, while many other existing Lie algebras lead to linear Hamiltonian integrable couplings [5, 19–22].
Four classes of block matrices were introduced in  and the Hamiltonian tri-integrable couplings of the AKNS hierarchy were constructed based on one of the four triangular block matrices. While in this paper, we would like to construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy based on other triangular block matrices as follows: where are square matrices of the same order and , , are arbitrary constants. Moreover, we also hope to generate the Hamiltonian structure of the resulting tri-integrable couplings.
The rest of the paper is organized as follows. In the next section, we first recall the GJ soliton hierarchy; then we construct a kind of tri-integrable couplings of the Giachetti-Johnson (GJ) soliton hierarchy and furnish Hamiltonian structures for the resulting tri-integrable couplings by the corresponding variational identity. Moreover, we will show that the resulting tri-integrable couplings have a recursion relation. In the final section, conclusions will be given.
2. Tri-Integrable Couplings of the Giachetti-Johnson (GJ) Hierarchy
2.1. The Giachetti-Johnson (GJ) Hierarchy
We first recall the GJ soliton hierarchy as follows : where is the spectral parameter, , , and are three dependent variables. Upon setting and choosing the initial data , , the stationary zero curvature equation generates
Using the compatibility conditions with we have the GJ hierarchy of soliton equations: The Hamiltonian operator , the recursion operator , and the Hamiltonian functionals in (11) are given by Note is not antisymmetric; therefore, the system (11) does not possess bi-Hamiltonian structures (the method of the verification is the same as the Appendix of ) and is not Liouville integrable.
2.2. Tri-Integrable Couplings
Based on the special non-semisimple Lie algebra , we choose the enlarged spectral matrix with being defined as in (6) and where , are new dependent variables.
To solve the enlarged stationary zero curvature equation we take a solution of the following type: where is defined by (7), and Then, from (17), we immediately get with the help of Maple, which leads to The corresponding recursion relations are together with (8), where . We select the initial data to be Then the recursion relations (22) uniquely determine the sequence of , , , , , , , , , , recursively. It is direct to compute the first two sets of functions by Maple:
For each integer , let us further introduce the enlarged Lax matrices: with being defined as in (10), and in which Then the enlarged zero curvature equation generates together with (9). This presents the supplementary systems: where In this way, the hierarchy from enlarged zero curvature equations can be written as for the given hierarchy (11).
2.3. Hamiltonian Structures
As we all know, when an integrable system is generated, one of our primary tasks is to construct Hamiltonian structures of the resulting integrable system. In this subsection, we will generate Hamiltonian structures for the tri-integrable couplings (34) by applying the associated variational identity : For the sake of convenience, we transform the Lie algebra into a vector form by the mapping: The mapping induces a Lie algebraic structure and the commutator on reads where A bilinear form on can be defined by , where is a constant matrix. The symmetric property and the Lie product mean that and for all . This matrix equation leads to a linear system of equations on the elements of . Solving the resulting system by Maple yields where , , are arbitrary constants. Therefore, a bilinear form on the semidirect sum of Lie algebras can be determined by with . It is easy to compute ; obviously, when and are nonzero constants, the bilinear form (40) is nondegenerate. But , , can be arbitrary constants. Simply, we take ; therefore, to apply the variational identity (35), we compute that
Thus by (35), we obtain
Therefore, the tri-integrable couplings of the GJ hierarchy in (34) possess the following Hamiltonian structures: where the Hamiltonian operator is given by with being the same as (12), and and the Hamiltonian functionals are determined by
The hierarchy (43) can be rewritten as where the recursion operator is given by with being the same as (13), andwith Therefore, the hierarchy (34) possesses a recursion relation: where . But is not antisymmetric; therefore, the system (11) does not have bi-Hamiltonian structures (the way of the verification is the same as the Appendix in ) and is not Liouville integrable.
In this paper, tri-integrable couplings for the Giachetti-Johnson hierarchy of continuous soliton equations were generated by using semidirect sums of Lie algebras. Moreover, we established their Hamiltonian structures through the variational identities. Clearly, mathematical structures behind integrable couplings are indeed rich and interesting, though complicated. It is worthy to mention that the method proposed in this paper can also be applied to other soliton hierarchy.
Note that we can generate more diverse tri-integrable couplings because the enlarged spectral matrix has more other forms. For instance, we can specify it in either one of the following forms: which were introduced in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the National Natural Science Foundation of China (Grant no. 11202161) and the Basic Research Fund of the Northwestern Polytechnical University, China (Grant no. GBKY1034).
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