A fractional quadratic-form identity is derived from a general isospectral problem of fractional order, which is devoted to constructing the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy. The method can be generalized to other fractional integrable couplings.

1. Introduction

The theory of derivatives of noninteger order can go back to Leibniz, Liouville, Grunwald, Letnikov, and Riemann. And the fractional analysis has attracted increasing interest of many researchers, because fractional analysis has numerous applications: kinetic theories [13], such as statistical mechanics [46], dynamics in complex media [7, 8], and many others [916]. In recent studies in physics, the researchers have found many applications of the derivatives and integrals of fractional order [16, 17]. They also pointed out that fractional-order models are more appropriate than integer-order models for various real materials. The main advantage of fractional derivative in comparison with classical integer-order models is that it provides an effective instrument for the description of memory and hereditary properties of various materials and progress. Also, the advantages of the fractional derivatives become apparent in modeling mechanical and electrical properties of real materials and in the description of rheological properties of rocks, as well as in many other fields.

The fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order [1720]. Since Riewe [4, 21] presented a concept of nonconservation mechanics, fractional conservation laws [22], Lie symmetries [9], and fractional Hamiltonian systems [2333] have been receiving more and more attention.

It is an important and interesting topic to search for new Hamiltonian hierarchies of soliton equations and their integrable couplings in soliton theory. Tu once proposed a simple and efficient method to construct the integrable systems and Hamiltonian structures [34], which was called the Tu scheme by Ma [35]. Later, many integrable systems and their Hamiltonian structures were worked out [3639]. Recently, Wu and Zhang proposed the generalized Tu formula and searched for the Hamiltonian structure of fractional AKNS hierarchy [40]. In [41], a generalized Hamiltonian structure of the fractional soliton equation hierarchy was presented. Very recently, Wang and Xia obtained the fractional supersoliton hierarchies and their super-Hamiltonian structures by using fractional supertrace identity [42, 43]. Then, how to generate integrable coupling system and Hamiltonian structure of fractional soliton equation?

In this paper, begining with a general isospectral problem of fractional order, we propose a fractional quadratic-form identity, from which the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy is derived.

2. Brief Overview of Fractional Differentiable Functions

Several local versions have been presented [4452], among which Jumarie’s derivative is defined as follows [52]: some properties of the fractional differentiable functions are given as follows.

(a) The Leibniz product law.

Assuming that is an order differentiable function in the area of point , from the Jumarie-Kolwankar’s Taylor series [5254], we can have

If is a differentiable function of order, the Leibniz product law can hold for the nondifferentiable functions [39, 44, 45]

(b) Denoting as the Riemann-Liouville integration in the following form: we can have a generalized Newton-Leibniz formulation

(c) With the properties (a) and (b), integration by parts for order differentiable functions and can be generated as

(d) From [31, 32, 55], the fractional variational derivative is written as where is a positive integer. In this paper, we propose a generalized quadratic-form identity for fractional soliton hierarchy from (7).

3. Fractional Exterior Differential and Hamiltonian Equations

Since Adda proposed the fractional generalization of differential forms [56, 57], several versions of fractional exterior differential approaches and applications related to different forms of fractional derivatives appeared in some parts of the open literature [58, 59]. The properties of fractional derivatives are discussed in [60].

The exterior derivative is defined as The exterior derivative map forms into forms and has the following algebraic results. Let and be forms, and let be an form; we have The last identity is called the Poincaré lemma. A form is called closed if . A form is called exact if there exists a form such that . The order of is one less than the order of . Exact forms are always closed, closed forms are not always exact.

Next, we introduce the fractional exterior derivative A differential 1-form is defined by with the vector field that can be represented as and is a continuously differentiable function. Using (10), the exact fractional form can be expressed as Note that (11) is a fractional generalization of the differential form (8). It is easy to find that fractional 1-form can be closed when the differential 1-form is not closed.

Then, we define the fractional functional hence, we can readily derive the generalized Poincare-Cartan 1-form, which reads

From (14), one has In the previous derivation, and are fractional differentiable functions with respect to .

The fractional closed condition admits the fractional Hamilton’s equations [40] which can be generalized to the following case [31]:

4. The Fractional Quadratic-Form Identity

Guo and Zhang once proposed quadratic-form identity [61], which is very efficient tool to systematically generate integrable couplings and their Hamiltonian structures. In the following, the fractional quadratic-form identity is presented.

Set to be an -dimensional Lie algebra with the basis whose corresponding loop algebra possesses the following basis: In terms of , we construct the following isospectral problem: The compatibility condition of (20) gives rise to the generalized zero curvature equation: Taking (21) reduces to the classical zero curvature equation. For and in , defining , then , can be presented. If the ranks of are taken as , , then each term in has the homogeneous rank which is denoted by

Set , , as a solution of the stationary zero curvature equation and rank is assumed to be given so that rank, ; each team in has the same rank as follows: Let the two arbitrary solutions and of (23) with the same rank be linearly related by In the following, relation (25) will be used when deducing the fractional quadratic-form identity. For , the -order matrix is determined by and constant matrix is determined by Defining functional satisfies the symmetry and the bilinear relation In the sense of the local fractional derivative, the gradient of the functional is defined by where is variational derivative with respect to . With the fractional variational derivative (7), one can have where is a positive integer and . The communication relationship of can be given as

Introduce a functional where , meet (23), while is to be determined; using (7), we can obtain the following fractional variation constraint conditions: according to the Jacobi identity and the previous equations, we can have and are solutions of (23); using (25) and , due to satisfying (34), we can have . From (23) and (33), a fractional quadratic-form identity is firstly presented as follows:

5. Application of the Fractional Quadratic-Form Identity

Introduce a loop algebra , with the commuting relations Consider the following spectral problem: Solving equation leads to

Set then the generalized zero curvature equation, , gives rise to a system where is a Hamiltonian operator. From (40), we have a recurrence operatorwhich meets . Hence, expression (42) can be written as From expression (37), we have Solving the matrix equation (27) for leads to Let we have Substituting the previous results into the fractional quadratic-form identity (36) gives Comparing the coefficients of on both sides of (49) yields It is easy to find that ; then we obtain the fractional Hamiltonian structure of (42) where is the fractional Hamiltonian function. When taking , we have an integrable coupling of a fractional BK hierarchy

Reduction Cases

Case 1. When , , ; (52) reduces to the BK hierarchy

Case 2. Let , , (53) is transformed to the classical Boussinesq equation

6. Conclusion

A way to construct the Hamiltonian structure of integrable coupling of fractional soliton equation hierarchy is presented. As an application, the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy is obtained by use of the fractional quadratic-form identity. The method can be generalized to other fractional integrable couplings.


The Project is in part supported by the Natural Science Foundation of China under Grants nos. 11271008, 61072147, and 11071159, by a Grant of “The First-class Discipline of Universities in Shanghai” and the Shanghai Universty Leading Academic Discipline Project (A.13-0101-12-004), the Natural Science Foundation of Shandong Province (Grant Nos. ZR2012AM021, and ZR2012AQ011), and Scientific Research Reward Fund for Excellent Middle-Aged and Young Scientists in Shandong Province (Grants no. BS2011DX038).