The Fractional Quadratic-Form Identity and Hamiltonian Structure of an Integrable Coupling of the Fractional Broer-Kaup Hierarchy

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.


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 [1][2][3], such as statistical mechanics [4][5][6], dynamics in complex media [7,8], and many others [9][10][11][12][13][14][15][16].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.
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 [36][37][38][39].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 quadraticform identity, from which the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy is derived.
(a) The Leibniz product law.
Assuming that () is an  order differentiable function in the area of point , from the Jumarie-Kolwankar's Taylor series [52][53][54], we can have If () is a differentiable function of  order, the Leibniz product law can hold for the nondifferentiable functions [39,44,45]    ( ()  ()) =  ()     () +  ()     () .(3) (b) Denoting 0    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).

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  + 1 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  = 0.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  =  1 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    = 0 admits the fractional Hamilton's equations [40] which can be generalized to the following case [31]:

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 G possesses the following basis: () =     ,  = 1, 2, . . ., ,  = 0, ±1, ±2, . . ., In terms of G, we construct the following isospectral problem: The compatibility condition of (20) gives rise to the generalized zero curvature equation: Taking  =  = 1 Set  = ∑ ≥0    − ,   = Σ  =1     ∈ , as a solution of the stationary zero curvature equation − ()   + [, ] = 0, and rank(  )  is assumed to be given so that rank(  )  = ,  ≥ 0; each team in  has the same rank as follows: Let the two arbitrary solutions  1 and  2 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 ,  ∈ G, 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 Introduce a functional where ,  meet (23), while Λ(∈ G) 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 = [Λ, ] −   and / are solutions of (23); using (25) and rank() = rank(  ) = rank(/), due to / satisfying (34), we can have  = (/).From ( 23) and ( 33), a fractional quadratic-form identity is firstly presented as follows:

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.