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

Abstract

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 [1–3], such as statistical mechanics [4–6], dynamics in complex media [7, 8], and many others [9–16]. 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.

The fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order [17–20]. Since Riewe [4, 21] presented a concept of nonconservation mechanics, fractional conservation laws [22], Lie symmetries [9], and fractional Hamiltonian systems [23–33] 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 [36–39]. 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 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 [44–52], among which Jumarie’s derivative is defined as follows [52]: 𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑1−𝛼𝑑𝑥^𝑥₀(𝑥−𝜉)^−𝛼𝑓(𝜉)(0)();−𝑓𝑑𝜉,0<𝛼<1 (1) some properties of the fractional differentiable functions are given as follows.

(a) The Leibniz product law.

Assuming that 𝑓(𝑥) is an 𝛼 order differentiable function in the area of point 𝑥, from the Jumarie-Kolwankar’s Taylor series [52–54], we can have 𝐷^𝛼_𝑥𝑓(𝑥)=lim_𝑦→𝑥+Γ(1+𝛼)𝑓𝑦−𝑓(𝑥)𝑦−𝑥^𝛼,(0<𝛼≤1). (2)

If 𝑔(𝑥) is a differentiable function of 𝛼 order, the Leibniz product law can hold for the nondifferentiable functions [39, 44, 45] 𝐷^𝛼_𝑥𝑓(𝑥)𝑔(𝑥)=𝑔(𝑥)𝐷^𝛼_𝑥𝑓(𝑥)+𝑓(𝑥)𝐷^𝛼_𝑥𝑔(𝑥). (3)

(b) Denoting ₀𝐼^𝛼_𝑥 as the Riemann-Liouville integration in the following form: ₀𝐼^𝛼_𝑥𝑓(𝑥)=𝐷_𝑥^−𝛼𝑓(𝑥)=1Γ(𝛼+1)^𝑥₀𝑓(𝜉)(𝑑𝜉)^𝛼,(0<𝛼≤1), (4) we can have a generalized Newton-Leibniz formulation 1Γ(1+𝛼)¹₀𝐷^𝛼_𝑥𝑓(𝑥)(𝑑𝑥)^𝛼(1)(0),1=𝑓−𝑓Γ(1+𝛼)^𝑥₀𝐷^𝜉_𝑥𝑓(𝜉)(𝑑𝜉)^𝛼(𝑥)(0),𝐷=𝑓−𝑓_𝛼𝑥Γ(1+𝛼)^𝑥₀𝑓(𝜉)(𝑑𝜉)^𝛼=𝑓(𝑥). (5)

(c) With the properties (a) and (b), integration by parts for 𝛼 order differentiable functions 𝑓(𝑥) and 𝑔(𝑥) can be generated as 1Γ(1+𝛼)^𝑏_𝑎𝑔(𝑥)𝐷^𝛼_𝑥𝑓(𝑥)(𝑑𝑥)^𝛼=𝑔(𝑥)𝑓(𝑥)^𝑏_𝑎−1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑥)𝐷^𝛼_𝑥𝑔(𝑥)(𝑑𝑥)^𝛼. (6)

(d) From [31, 32, 55], the fractional variational derivative is written as 𝛿𝐿=𝛿𝑦𝜕𝐿𝜕𝑦+_𝑘=1(−1)^𝑘𝐷^𝛼_𝑥^𝑘𝜕𝐿𝜕𝐷^𝛼_𝑥^𝑘𝑦, (7) where 𝑘 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 𝑑=^𝑛_𝑚=1𝑑𝑥_𝑚𝜕𝜕𝑥_𝑚. (8) The exterior derivative map 𝑘 forms into 𝑘+1 forms and has the following algebraic results. Let 𝛾 and 𝜆 be 𝑘 forms, and let 𝜇 be an 𝑚 form; we have 𝑑𝑑=()𝛾+𝜆=𝑑𝛾+𝑑𝜆,𝛾∧𝜇𝑑𝛾∧𝜇+−1^𝑘𝑑𝛾∧𝑑𝜇,𝑑𝛾=0. (9) The last identity is called the Poincaré lemma. A form 𝛾 is called closed if 𝑑𝛾=0. A form 𝛾 is called exact if there exists a form 𝜇 such that 𝑑𝜇=𝛾. The order of 𝜇 is one less than the order of 𝛾. Exact forms are always closed, closed forms are not always exact.

Next, we introduce the fractional exterior derivative 𝑑^𝛼=𝑑𝑥_𝑖^𝛼𝐷^𝛼_𝑥𝑖. (10) A differential 1-form is defined by 𝜔_𝛼=𝐹^𝑖(𝑥)𝑑𝑥_𝑖^𝛼, (11) with the vector field 𝐹^𝑖(𝑥) that can be represented as 𝐹^𝑖(𝑥)=−𝐷^𝛼_𝑥𝑖𝑉 and 𝑉(𝑥) is a continuously differentiable function. Using (10), the exact fractional form can be expressed as 𝜔_𝛼=−𝑑^𝛼𝑉=−𝑑𝑥_𝑖^𝛼𝐷^𝛼_𝑥𝑖𝑉. (12) Note that (11) is a fractional generalization of the differential form (8). It is easy to find that fractional 1-form 𝜔_𝛼 can be closed when the differential 1-form 𝜔=𝜔₁ is not closed.

Then, we define the fractional functional 𝐽=1𝑝,𝑞Γ(1+𝛼)𝑝𝐷^𝛼_𝑡𝑞−𝐻𝑡,𝑝,𝑞(𝑑𝑡)^𝛼; (13) hence, we can readily derive the generalized Poincare-Cartan 1-form, which reads 𝜔=𝑝𝑑^𝛼𝑞−𝐻(𝑑𝑡)^𝛼. (14)

From (14), one has 𝑑^𝛼𝜔=𝑝^𝛼_𝑡(𝑑𝑡)^𝛼∧𝑑^𝛼𝑞+𝑑^𝛼𝑝∧𝑑^𝛼𝑞−𝜕𝐻𝑑𝜕𝑝^𝛼𝑝∧(𝑑𝑡)^𝛼−𝜕𝐻𝑑𝜕𝑞^𝛼𝑞∧(𝑑𝑡)^𝛼=𝑝^𝛼_𝑡+𝜕𝐻)𝜕𝑞(𝑑𝑡^𝛼∧𝑑^𝛼𝑞+𝜕𝐻()𝜕𝑝𝑑𝑡^𝛼−𝑑^𝛼𝑞∧𝑑^𝛼𝑝. (15) In the previous derivation, 𝑝 and 𝑞 are fractional differentiable functions with respect to 𝑡.

The fractional closed condition 𝑑^𝛼𝜔=0 admits the fractional Hamilton’s equations [40] 𝑞_𝑡^(𝛼)=𝜕𝐻𝜕𝑝,𝑝_𝑡^(𝛼)=−𝜕𝐻,𝜕𝑞 (16) which can be generalized to the following case [31]: 𝑞_𝑖^(𝛼)(𝑡)=𝜕𝐻𝜕𝑝_𝑖,𝑝_𝑖^(𝛼)(𝑡)=−𝜕𝐻𝜕𝑞_𝑖, (17)

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 𝐺 to be an 𝑠-dimensional Lie algebra with the basis 𝑒₁,𝑒₂,…,𝑒_𝑠, (18) whose corresponding loop algebra 𝐺 possesses the following basis: 𝑒_𝑖(𝑚)=𝑒_𝑖𝜆^𝑚𝑒,𝑖=1,2,…,𝑠,𝑚=0,±1,±2,…,_𝑖(𝑚),𝑒_𝑗(𝑚)=𝑒_𝑖,𝑒_𝑗𝜆^𝑚+𝑛. (19) In terms of 𝐺, we construct the following isospectral problem: 𝜓_𝑥^(𝛼)=𝑈,𝜓,𝜓_𝑡^(𝛽)=𝑉,𝜓. (20) The compatibility condition of (20) gives rise to the generalized zero curvature equation: 𝑈_𝑡^(𝛽)−𝑉_𝑥^(𝛼)+[𝑈,𝑉]=0. (21) Taking 𝛼=𝛽=1 (21) reduces to the classical zero curvature equation. For 𝜆 and 𝑢_𝑖(𝑖=1,2,…,𝑝) in 𝑈=𝑈(𝜆,𝑢)=Σ^𝑠_𝑖=1𝑈_𝑖𝑒_𝑖, defining rank(𝜆)=deg(𝜆), then rank(𝑒_𝑖(𝜆))=𝛼_𝑖, 0≤𝑖≤𝑠 can be presented. If the ranks of 𝑢_𝑖 are taken as 𝜁−𝛼_𝑖, 1≤𝑖≤𝑠, then each term in 𝑈 has the homogeneous rank 𝛼 which is denoted by (𝑈)𝜕rank=rank^𝛼𝜕𝑥^𝛼=𝜁. (22)

Set 𝑉=∑_𝑚≥0𝑉_𝑚𝜆^−𝑚, 𝑉_𝑚=Σ^𝑠_𝑖=1𝑉_𝑚𝑖𝑒_𝑖∈𝐺, as a solution of the stationary zero curvature equation −𝑉_𝑥^(𝛼)+[𝑈,𝑉]=0, (23) and rank(𝑉_𝑚)_𝜆 is assumed to be given so that rank(𝑉_𝑚)_𝜆=𝜉, 𝑚≥0; each team in 𝑉 has the same rank as follows: (𝑉)𝜕rank=rank^𝛽𝜕𝑡^𝛽=𝜉. (24) Let the two arbitrary solutions 𝑉₁ and 𝑉₂ of (23) with the same rank be linearly related by 𝑉=𝛾𝑉,𝛾=const. (25) In the following, relation (25) will be used when deducing the fractional quadratic-form identity. For 𝑎,𝑏∈𝐺, the 𝑠-order matrix 𝑅(𝑏) is determined by [𝑎,𝑏]^𝑇=𝑎^𝑇𝑅(𝑏), (26) and constant matrix 𝐹=(𝑓_𝑖𝑗)_𝑠×𝑠 is determined by 𝐹=𝐹^𝑇,𝑅(𝑏)𝐹=−(𝑅(𝑏)𝐹)^𝑇. (27) Defining functional {𝑎,𝑏}=𝑎^𝑇𝐹𝑏 satisfies the symmetry {𝑎,𝑏}={𝑏,𝑎}, (28) and the bilinear relation 𝑐₁𝑎₁+𝑐₂𝑎₂,𝑏=𝑐₁𝑎₁,𝑏+𝑐₂𝑎₂,𝑏. (29) In the sense of the local fractional derivative, the gradient ∇_𝑏{𝑎,𝑏} of the functional {𝑎,𝑏} is defined by 𝜕{}=𝛿𝜕𝜖𝑎,𝑏+𝜖𝑉_𝑏{𝑎,𝑏},𝑉,𝑎,𝑏,𝑉∈𝐺, (30) where 𝛿_𝑏 is variational derivative with respect to 𝑏. With the fractional variational derivative (7), one can have 𝛿_𝑏𝑎,𝑏_𝑥^(𝑘𝛼)=(−1)^𝑘𝑎_𝑥^(𝑘𝛼), (31) where 𝑘 is a positive integer and 𝐷_𝑥^𝑘𝛼=𝐷^𝛼_𝑥⋯𝐷^𝛼_𝑥𝑘. The communication relationship of {𝑎,𝑏} can be given as {[𝑎,𝑏],𝑐}={𝑎,[𝑏,𝑐]},𝑎,𝑏,𝑐∈𝐺. (32)

Introduce a functional 𝑊=𝑉,𝑈_𝜆+Λ,𝑉_𝑥^(𝛼)−[𝑈,𝑉], (33) where 𝑈, 𝑉 meet (23), while Λ(∈𝐺) is to be determined; using (7), we can obtain the following fractional variation constraint conditions: 𝛿𝑊𝛿Λ=𝑉_𝑥^(𝛼)−[𝑈,𝑉],𝛿𝑊𝛿𝑉=𝑈_𝜆−Λ_𝑥^(𝛼)+[𝑈,Λ]; (34) according to the Jacobi identity and the previous equations, we can have [Λ,𝑉]^𝛼_𝑥=𝑈_𝜆,𝑉+[𝑈,[Λ,𝑉]], (35)𝑍=[Λ,𝑉]−𝑉_𝜆 and 𝑉/𝜆 are solutions of (23); using (25) and rank(𝑍)=rank(𝑉_𝜆)=rank(𝑉/𝜆), due to 𝑉/𝜆 satisfying (34), we can have 𝑍=(𝛾/𝜆)𝑉. From (23) and (33), a fractional quadratic-form identity is firstly presented as follows: 𝛿𝛿𝑢_𝑖𝑉,𝑈_𝜆=𝑉,𝜕𝑈_𝜆𝜕𝑢_𝑖+[Λ,𝑉],𝜕𝑈𝜕𝑢_𝑖=𝑉,𝜕𝑈_𝜆𝜕𝑢_𝑖+𝑉_𝜆,𝜕𝑈𝜕𝑢_𝑖+𝛾𝜆𝑉,𝜕𝑈𝜕𝑢_𝑖=𝜕𝜕𝜆𝑉,𝜕𝑈𝜕𝑢_𝑖+𝜆^−𝛾𝜕𝜆𝜕𝜆^𝛾𝑉,𝜕𝑈𝜕𝑢_𝑖=𝜆^−𝛾𝜕𝜆𝜕𝜆^𝛾𝑉,𝜕𝑈𝜕𝑢_𝑖,1≤𝑖≤𝑝. (36)

5. Application of the Fractional Quadratic-Form Identity

Introduce a loop algebra 𝐺₆={𝑎=(𝑎₁,𝑎₂,…,𝑎₆)^𝑇,𝑎_𝑘=∑_𝑚𝜆^𝑚}, with the commuting relations [𝑎,𝑏]=𝑎₂𝑏₃−𝑎₃𝑏₂,2𝑎₁𝑏₂−2𝑎₂𝑏₁,2𝑎₃𝑏₁−2𝑎₁𝑏₃,𝑎₂𝑏₆−𝑎₆𝑏₂+𝑎₅𝑏₃−𝑎₃𝑏₅,2𝑎₁𝑏₅−2𝑎₅𝑏₁+2𝑎₄𝑏₂−2𝑎₂𝑏₄,2𝑎₆𝑏₁−2𝑎₁𝑏₆+2𝑎₃𝑏₄−2𝑎₄𝑏₃^𝑇. (37) Consider the following spectral problem: 𝜓_𝑥^(𝛼)=,𝑣𝑈,𝜓𝑈=−𝜆+2,1,−𝑤,𝑢₁,0,𝑢₂^𝑇𝑉=𝑎,𝑏,𝑐,𝑑,𝑒,𝑓^𝑇. (38) Solving equation −𝑉_𝑥^(𝛼)+[𝑈,𝑉]=0 (39) leads to 𝑎^(𝛼)_𝑚𝑥=𝑐_𝑚+𝑤𝑏_𝑚,𝑏^(𝛼)_𝑚𝑥=−2𝑏_𝑚+1+𝑣𝑏_𝑚−2𝑎_𝑚,𝑐^(𝛼)_𝑚𝑥=2𝑐_𝑚+1−𝑣𝑐_𝑚−2𝑤𝑎_𝑚,𝑑^(𝛼)_𝑚𝑥=𝑓_𝑚+𝑤𝑒_𝑚−𝑢₂𝑏_𝑚,𝑒^(𝛼)_𝑚𝑥=−2𝑒_𝑚+1+𝑣𝑒_𝑚−2𝑑_𝑚+2𝑢₁𝑏_𝑚,𝑓^(𝛼)_𝑚𝑥=2𝑓_𝑚+1−𝑣𝑓_𝑚−2𝑤𝑑_𝑚−2𝑢₁𝑐_𝑚+2𝑢₂𝑎_𝑚,𝑎₁=𝑑₁=𝑒₁=𝑐₂=0,𝑏₁=1,𝑐₁𝑓=−𝑤,₁=𝑢₂,𝑎₂1=−2𝑏𝑤,₂=12𝑣,𝑑₂=12𝑢₂,𝑒₂=𝑢₁,𝑓₂=12𝑢^(𝛼)_2𝑥+12𝑢₂𝑣−𝑤𝑢₁,…. (40)

Set 𝑉^(𝑛)=^𝑛_𝑚=0𝑎_𝑚,𝑏_𝑚,𝑐_𝑚,𝑑_𝑚,𝑒_𝑚,𝑓_𝑚^𝑇𝜆^𝑛−𝑚+𝑏_𝑛+1,0,0,0,0,0^𝑇; (41) then the generalized zero curvature equation, 𝐷^𝛽_𝑡𝑈−𝐷^𝛼_𝑥𝑉^(𝑛)+[𝑈,𝑉^(𝑛)]=0, gives rise to a system 𝑢_𝑡^(𝛽)_𝑛=𝑣_𝑡^(𝛽)_𝑛𝑤_𝑡^(𝛽)_𝑛𝑢^(𝛽)_1𝑡𝑛𝑢^(𝛽)_2𝑡𝑛=2𝑏^(𝛼)_𝑛+1,𝑥−2𝑎^(𝛼)_𝑛+1,𝑥𝑒^(𝛼)_𝑛+1,𝑥2𝑑^(𝛼)_𝑛+1,𝑥=0002𝐷^𝛼_𝑥00−𝐷^𝛼_𝑥00−𝐷^𝛼_𝑥0−𝐷^𝛼_𝑥2𝐷^𝛼_𝑥0−𝐷^𝛼_𝑥0×𝑎_𝑛+1+𝑑_𝑛+1−𝑏_𝑛+1−𝑒_𝑛+12𝑎_𝑛+1𝑏_𝑛+1=𝐽𝑃_𝑛+1, (42) where 𝐽 is a Hamiltonian operator. From (40), we have a recurrence operator1𝐿=2𝐷^𝛼_𝑥+12𝐷_𝑥^−𝛼𝑣𝐷^𝛼_𝑥12𝑤+𝐷_𝑥^−𝛼𝑤𝐷^𝛼_𝑥12𝐷_𝑥^−𝛼𝑢₁𝐷^𝛼_𝑥12𝑢₂+𝐷_𝑥^−𝛼𝑢₂𝐷^𝛼_𝑥112𝑣−𝐷^𝛼_𝑥0−𝑢₁1002𝐷^𝛼_𝑥+𝐷_𝑥^−𝛼𝑣𝐷^𝛼_𝑥−𝑤−𝐷_𝑥^−𝛼𝑤𝐷^𝛼_𝑥100−212𝑣−𝐷^𝛼_𝑥 (43)which meets 𝑃_𝑛+1=𝐿𝑃_𝑛. Hence, expression (42) can be written as 𝑢_𝑡^(𝛽)_𝑛=𝑣_𝑡^(𝛽)_𝑛𝑤_𝑡^(𝛽)_𝑛𝑢^(𝛽)_1𝑡𝑛𝑢^(𝛽)_2𝑡𝑛=𝐽𝐿^𝑛001−1. (44) From expression (37), we have [𝑎,𝑏]^𝑇=𝑎^𝑇02𝑏₂−2𝑏₃02𝑏₅−2𝑏₆𝑏₃−2𝑏₁0𝑏₆−2𝑏₄0−𝑏₂02𝑏₁−𝑏₅02𝑏₄00002𝑏₂−2𝑏₃000𝑏₃−2𝑏₁0000−𝑏₂02𝑏₁=𝑎^𝑇𝑅(𝑏). (45) Solving the matrix equation (27) for 𝐹 leads to 𝐹=200200001001010010200000001000010000. (46) Let {𝑎,𝑏}=2𝑎₁+𝑎₄𝑏₁+𝑎₃+𝑎₆𝑏₂+𝑎₂+𝑎₅𝑏₃+2𝑎₁𝑏₄+𝑎₃𝑏₅+𝑎₂𝑏₆; (47) we have 𝑉,𝜕𝑈𝜕𝑣=𝑎+𝑑,𝑉,𝜕𝑈𝜕𝑤=−𝑏−𝑒,𝑉,𝜕𝑈𝜕𝑢₁=2𝑎,𝑉,𝜕𝑈𝜕𝑢₂=𝑏,𝑉,𝜕𝑈𝜕𝜆=−2𝑎−2𝑑. (48) Substituting the previous results into the fractional quadratic-form identity (36) gives 𝛿()𝛿𝑢−2𝑎−2𝑑=𝜆^−𝛾𝜕𝜆𝜕𝜆^𝛾𝑏𝑎+𝑑−𝑏−𝑒2𝑎. (49) Comparing the coefficients of 𝜆^−𝑛−1 on both sides of (49) yields 𝛿𝛿𝑢−2𝑎_𝑛+1−2𝑑_𝑛+1=𝑎𝛾−𝑛^𝑛+𝑑_𝑛−𝑏_𝑛−𝑒_𝑛2𝑎_𝑛𝑏_𝑛. (50) It is easy to find that 𝛾=0; then we obtain the fractional Hamiltonian structure of (42) 𝑢_𝑡^(𝛽)_𝑛=𝑣_𝑡^(𝛽)_𝑛𝑤_𝑡^(𝛽)_𝑛𝑢^(𝛽)_1𝑡𝑛𝑢^(𝛽)_2𝑡𝑛𝑎=𝐽^𝑛+1+𝑑_𝑛+1−𝑏_𝑛+1−𝑒_𝑛+12𝑎_𝑛+1𝑏_𝑛+1=𝐽𝛿𝐻_𝑛𝛿𝑢, (51) where 𝐻_𝑛=(2𝑎_𝑛+1+2𝑑_𝑛+1)/𝑛and(𝑛=0,1,2,….) is the fractional Hamiltonian function. When taking 𝑛=2, we have an integrable coupling of a fractional BK hierarchy 𝐷^𝛽_𝑡21𝑣=−2𝐷^𝛼_𝑥𝐷^𝛼_𝑥𝑣+𝑣𝐷^𝛼_𝑥𝑣+𝐷^𝛼_𝑥𝐷𝑤,_𝛽𝑡21𝑤=2𝐷^𝛼_𝑥𝐷^𝛼_𝑥𝑤+𝐷^𝛼_𝑥(),𝐷𝑤𝑣_𝛽𝑡2𝑢₁1=−2𝐷^𝛼_𝑥𝐷^𝛼_𝑥𝑢₁+𝐷^𝛼_𝑥𝑢₁𝑣−12𝐷^𝛼_𝑥𝑢₂,𝐷^𝛽_𝑡2𝑢₂=12𝐷^𝛼_𝑥𝐷^𝛼_𝑥𝑢₂+𝐷^𝛼_𝑥𝑢₂𝑣−2𝐷^𝛼_𝑥𝑤𝑢₁. (52)

Reduction Cases

Case 1. When 𝛼=𝛽=1, 𝑢₁=𝑢₂=0, 𝑡₂=𝑡; (52) reduces to the BK hierarchy 𝑣_𝑡1=−2𝑣_𝑥𝑥+𝑣𝑣_𝑥+𝑤_𝑥,𝑤_𝑡=1𝑣𝑤+2𝑤_𝑥𝑥. (53)

Case 2. Let 𝑣=−𝑞, 𝑤=𝑟+1+(1/2)𝑣_𝑥, (53) is transformed to the classical Boussinesq equation 𝑞_𝑡=−𝑞𝑞_𝑥−𝑟_𝑥,𝑟_𝑡1=−4𝑞_𝑥𝑥𝑥−𝑞(𝑟+1)_𝑥. (54)

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.

Acknowledgments

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).