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

Authors: Chao Yue; Tiecheng Xia; Guijuan Liu; Jianbo Liu
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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]:
(1)Dxαf(x)=1Γ(1-α)ddx∫0x‍(x-ξ)-α(f(ξ)-f(0))dξ,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(0<α<1);
some properties of the fractional differentiable functions are given as follows.(a) The Leibniz product law.Assuming thatf(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
(2)Dxαf(x)=limy→x+Γ(1+α)(f(y)-f(x))(y-x)α,(0<α≤1).Ifg(x) is a differentiable function of α order, the Leibniz product law can hold for the nondifferentiable functions [39, 44, 45]
(3)Dxα(f(x)g(x))=g(x)Dxαf(x)+f(x)Dxαg(x).(b) DenotingI0xα as the Riemann-Liouville integration in the following form:
(4)0Ixαf(x)=Dx-αf(x)=1Γ(α+1)∫0x‍f(ξ)(dξ)α,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii(0<α≤1),
we can have a generalized Newton-Leibniz formulation
(5)1Γ(1+α)∫01‍Dxαf(x)(dx)α=f(1)-f(0),1Γ(1+α)∫0x‍Dxξf(ξ)(dξ)α=f(x)-f(0),DxαΓ(1+α)∫0x‍f(ξ)(dξ)α=f(x).(c) With the properties (a) and (b), integration by parts forα order differentiable functions f(x) and g(x) can be generated as
(6)1Γ(1+α)∫ab‍g(x)Dxαf(x)(dx)α=g(x)f(x)|ab-1Γ(1+α)∫ab‍f(x)Dxαg(x)(dx)α.(d) From [31, 32, 55], the fractional variational derivative is written as
(7)δLδy=∂L∂y+∑k=1‍(-1)k(Dxα)k(∂L∂(Dxα)ky),
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(8)d=∑m=1n‍dxm∂∂xm.
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
(9)d(γ+λ)=dγ+dλ,d(γ∧μ)=(dγ)∧μ+(-1)kγ∧dμ,d(dγ)=0.
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(10)dα=(dxi)αDxiα.
A differential 1-form is defined by
(11)ωα=Fi(x)(dxi)α,
with the vector field Fi(x) that can be represented as Fi(x)=-DxiαV and V(x) is a continuously differentiable function. Using (10), the exact fractional form can be expressed as
(12)ωα=-dαV=-(dxi)αDxiαV.
Note that (11) is a fractional generalization of the differential form (8). It is easy to find that fractional 1-form ωα can be closed when the differential 1-form ω=ω1 is not closed.Then, we define the fractional functional(13)J[p,q]=1Γ(1+α)∫‍[pDtαq-H(t,p,q)](dt)α;
hence, we can readily derive the generalized Poincare-Cartan 1-form, which reads
(14)ω=pdαq-H(dt)α.From (14), one has
(15)dαω=ptα(dt)α∧dαq+dαp∧dαq-∂H∂pdαp∧(dt)α-∂H∂qdαq∧(dt)α=[ptα+∂H∂q](dt)α∧dαq+[∂H∂p(dt)α-dαq]∧dαp.
In the previous derivation, p and q are fractional differentiable functions with respect to t.The fractional closed conditiondαω=0 admits the fractional Hamilton’s equations [40]
(16)qt(α)=∂H∂p,pt(α)=-∂H∂q,
which can be generalized to the following case [31]:
(17)qi(α)(t)=∂H∂pi,pi(α)(t)=-∂H∂qi,


## 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.SetG to be an s-dimensional Lie algebra with the basis
(18)e1,e2,…,es,
whose corresponding loop algebra G~ possesses the following basis:
(19)ei(m)=eiλm,i=1,2,…,s,m=0,±1,±2,…,[ei(m),ej(m)]=[ei,ej]λm+n.
In terms of G~, we construct the following isospectral problem:
(20)ψx(α)=[U,ψ],ψt(β)=[V,ψ].
The compatibility condition of (20) gives rise to the generalized zero curvature equation:
(21)Ut(β)-Vx(α)+[U,V]=0.
Taking α=β=1 (21) reduces to the classical zero curvature equation. For λ and ui(i=1,2,…,p) in U=U(λ,u)=Σi=1sUiei, defining rank(λ)=deg(λ), then rank(ei(λ))=αi, 0≤i≤s can be presented. If the ranks of ui are taken as ζ-αi, 1≤i≤s, then each term in U has the homogeneous rank α which is denoted by
(22)rank(U)=rank(∂α∂xα)=ζ.SetV=∑m≥0‍Vmλ-m, Vm=Σi=1sVmiei∈G, as a solution of the stationary zero curvature equation
(23)-Vx(α)+[U,V]=0,
and rank(Vm)λ is assumed to be given so that rank(Vm)λ=ξ, m≥0; each team in V has the same rank as follows:
(24)rank(V)=rank(∂β∂tβ)=ξ.
Let the two arbitrary solutions V1 and V2 of (23) with the same rank be linearly related by
(25)V-=γV,γ=const.
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
(26)[a,b]T=aTR(b),
and constant matrix F=(fij)s×s is determined by
(27)F=FT,R(b)F=-(R(b)F)T.
Defining functional {a,b}=aTFb satisfies the symmetry
(28){a,b}={b,a},
and the bilinear relation
(29){c1a1+c2a2,b}=c1{a1,b}+c2{a2,b}.
In the sense of the local fractional derivative, the gradient ∇b{a,b} of the functional {a,b} is defined by
(30)∂∂ϵ{a,b+ϵV}=(δb{a,b},V),a,b,V∈G~,
where δb is variational derivative with respect to b. With the fractional variational derivative (7), one can have
(31)δb{a,bx(kα)}=(-1)kax(kα),
where k is a positive integer and Dxkα=Dxα⋯Dxα︸k. The communication relationship of {a,b} can be given as
(32){[a,b],c}={a,[b,c]},a,b,c∈G~.Introduce a functional(33)W={V,Uλ}+{Λ,Vx(α)-[U,V]},
where U, V meet (23), while Λ(∈G~) is to be determined; using (7), we can obtain the following fractional variation constraint conditions:
(34)δWδΛ=Vx(α)-[U,V],δWδV=Uλ-Λx(α)+[U,Λ];
according to the Jacobi identity and the previous equations, we can have
(35)[Λ,V]xα=[Uλ,V]+[U,[Λ,V]],Z=[Λ,V]-Vλ and V/λ are solutions of (23); using (25) and rank(Z)=rank(Vλ)=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:
(36)δδui{V,Uλ}={V,∂Uλ∂ui}+{[Λ,V],∂U∂ui}={V,∂Uλ∂ui}+{Vλ,∂U∂ui}+γλ{V,∂U∂ui}=∂∂λ{V,∂U∂ui}+(λ-γ∂∂λλγ){V,∂U∂ui}=λ-γ∂∂λ(λγ{V,∂U∂ui}),1≤i≤p.


## 5. Application of the Fractional Quadratic-Form Identity

Introduce a loop algebraG~6={a=(a1,a2,…,a6)T,ak=∑m‍λm}, with the commuting relations
(37)[a,b]=(a2b3-a3b2,2a1b2-2a2b1,2a3b1-2a1b3,a2b6-a6b2+a5b3-a3b5,2a1b5-2a5b1+2a4b2-2a2b4,2a6b1-2a1b6+2a3b4-2a4b3)T.
Consider the following spectral problem:
(38)ψx(α)=[U,ψ],U=(-λ+v2,1,-w,u1,0,u2)TV=(a,b,c,d,e,f)T.
Solving equation
(39)-Vx(α)+[U,V]=0
leads to
(40)amx(α)=cm+wbm,bmx(α)=-2bm+1+vbm-2am,cmx(α)=2cm+1-vcm-2wam,dmx(α)=fm+wem-u2bm,emx(α)=-2em+1+vem-2dm+2u1bm,fmx(α)=2fm+1-vfm-2wdm-2u1cm+2u2am,a1=d1=e1=c2=0,b1=1,c1=-w,f1=u2,a2=-12w,b2=12v,d2=12u2,e2=u1,f2=12u2x(α)+12u2v-wu1,….Set(41)V(n)=∑m=0n‍(am,bm,cm,dm,em,fm)Tλn-m+(bn+1,0,0,0,0,0)T;
then the generalized zero curvature equation, DtβU-DxαV(n)+[U,V(n)]=0, gives rise to a system
(42)utn(β)=(vtn(β)wtn(β)u1tn(β)u2tn(β))=(2bn+1,x(α)-2an+1,x(α)en+1,x(α)2dn+1,x(α))=(0002Dxα00-Dxα00-Dxα0-Dxα2Dxα0-Dxα0)×(an+1+dn+1-bn+1-en+12an+1bn+1)=JPn+1,
where J is a Hamiltonian operator. From (40), we have a recurrence operator(43)L=(12Dxα+12Dx-αvDxα12(w+Dx-αwDxα)12Dx-αu1Dxα12(u2+Dx-αu2Dxα)112(v-Dxα)0-u10012(Dxα+Dx-αvDxα)-w-Dx-αwDxα00-1212(v-Dxα))which meets Pn+1=LPn. Hence, expression (42) can be written as
(44)utn(β)=(vtn(β)wtn(β)u1tn(β)u2tn(β))=JLn(0-101).
From expression (37), we have
(45)[a,b]T=aT(02b2-2b302b5-2b6b3-2b10b6-2b40-b202b1-b502b400002b2-2b3000b3-2b10000-b202b1)=aTR(b).
Solving the matrix equation (27) for F leads to
(46)F=(200200001001010010200000001000010000).
Let
(47){a,b}=2(a1+a4)b1+(a3+a6)b2+(a2+a5)b3+2a1b4+a3b5+a2b6;
we have
(48){V,∂U∂v}=a+d,{V,∂U∂w}=-b-e,{V,∂U∂u1}=2a,{V,∂U∂u2}=b,{V,∂U∂λ}=-2a-2d.
Substituting the previous results into the fractional quadratic-form identity (36) gives
(49)δδu(-2a-2d)=λ-γ∂∂λλγ(a+d-b-e2ab).
Comparing the coefficients of λ-n-1 on both sides of (49) yields
(50)δδu(-2an+1-2dn+1)=(γ-n)(an+dn-bn-en2anbn).
It is easy to find that γ=0; then we obtain the fractional Hamiltonian structure of (42)
(51)utn(β)=(vtn(β)wtn(β)u1tn(β)u2tn(β))=J(an+1+dn+1-bn+1-en+12an+1bn+1)=JδHnδu,
where Hn=(2an+1+2dn+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
(52)Dt2βv=-12DxαDxαv+vDxαv+Dxαw,Dt2βw=12DxαDxαw+Dxα(wv),Dt2βu1=-12DxαDxαu1+Dxα(u1v)-12Dxαu2,Dt2βu2=12DxαDxαu2+Dxα(u2v)-2Dxα(wu1).Reduction CasesCase 1.
Whenα=β=1, u1=u2=0, t2=t; (52) reduces to the BK hierarchy
(53)vt=-12vxx+vvx+wx,wt=(vw+12wx)x.Case 2.
Letv=-q, w=r+1+(1/2)vx, (53) is transformed to the classical Boussinesq equation
(54)qt=-qqx-rx,rt=-14qxxx-(q(r+1))x.


## 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.



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