The Fractional Super Broer-Kaup-Kupershmidt Hierarchy and its Nonlinear Integrable Couplings?

1 Introduction

The theory of integrals and derivatives of non-integer order goes back to Leibniz,Liouville,Riemann,Grunwald and Letnikov.The fractional analysis has attracted interest of many researchers due to its numerous applications:kinetic theories[1,2],dynamics in complex media[3,4],statistical mechanics[5,6],and many others[7-9].In recent studies in physics,the researchers found many applications of the derivatives and integrals of fractional order[10].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 ef f ective instrument for the description of memory and hereditary properties of various materials and progresses.Also,the advantages of the fractional derivatives become apparent in modeling mechanical and electrical properties of real materials,as well as in the description of rheological properties of rocks,and in many other fields.

Searching for new integrable systems and super integrable systems is an important and interesting work in soliton theory.Tu[11]has proposed a Lie algebra and trace identity to construct the integrable system and Hamiltonian structure,which was called the Tu scheme by Ma[12].From then on,lots of researchers have done research on this topic and have got some integrable and super integrable results[13-16].Meanwhile,constructing nonlinear super integrable couplings is one of the interesting topics in super model theory.There are much richer mathematical structures behind nonlinear super integrable couplings than scalar super integrable equations.Which fractional evolution equations admit fractional integrable and super integrable systems?In[17],Yu got the fractional-order coupled Boussinesq and KdV equation.In[18],Wu proposed the generalized Tu formula and searched for the Hamiltonian structure of fractional AKNS hierarchy.In[19],we establish the fractional supertrace identity and fractional super Hamiltonian structure using the fractional theory.

This paper is organized as follows:a brief review of the fractional derivatives and integrals is given.Based on the fractional Hamiltonian structure and the fractional supertrace identity,we get the fractional super Broer-Kaup-Kupershmidt(BKK)hierarchy as well as its super Hamiltonian structure.Then,we establish nonlinear integrable couplings of the fractional super BKK hierarchy.As its reduction,we present the fractional nonlinear super integrable couplings of the fractional super BKK equations.The solution of reduced equations is a very important and difficult work,and we will take great ef f orts in our next work.

2 Brief overview of fractional derivatives and integrals

Kolwankar and Gangal[20,21]defined the local fractional derivative as

Chenet al[22]gave the necessary conditions for the following relationship

We adopt derivative(2)for simplicity.Some fractional derivative properties are proposed as follows:

(i) The generalized Leibniz product law:

Iff(x),g(x)areαorder dif f erentiable functions,one can have

This result can be proved as follows:by the simplicity definition(2),we have

the proof is completed.

(ii) The Leibniz formula for fractional dif f erentiable functions reads

wheredenotes the Riemann-Liouville integration,which is defined as

Therefore,from the defined fractional integration,the properties(i)and(ii),the integration by parts can be used during the fractional calculus

(iii) Fractional variational derivative

wherekis a positive integer.The properties(ii)and(iii)can be proved similarly,and we omit these proofs in this paper.

3 Fractional exterior dif f erential and Hamiltonian equations

Since Adda proposed the fractional generalization of dif f erential forms[23,24],several versions of fractional exterior dif f erential approaches and applications related to dif f erent forms of fractional derivatives appeared in open literature[25],and the properties of fractional derivatives are discussed[26].

The exterior derivative is defined as

The exterior derivative mapskforms intok+1 forms and has the following algebraic results.Letγandλbekforms,andμbe anmform,then

The last identity is called the Poincar′e lemma.A formγis called closed ifdγ=0.A formγis called exact if there exists a formμsuch thatdμ=γ.The order ofμis one less than the order ofγ.Exact forms are always closed,and closed forms are not always exact.

Next,we introduce the fractional exterior derivative

A dif f erential 1-form is defined by

where the vector fieldFi(x)can be represented asis a continuously dif f erentiable function.Using(13)the exact fractional form can be represented as

Note that(14)is a fractional generalization of the dif f erential form(9).Obviously that fractional 1-formωpcan be closed when the dif f erential 1-formω=ω1is not closed.

Now,we define the fractional functional

and then,we can readily derive the generalized Poincare-Cartan 1-form,which reads

From(17),we have

In the above derivation,pandqare fractional dif f erentiable functions with respect tot.

The fractional closed conditiondαω=0 allows the following fractional Hamiltonian equations[18]

4 Fractional supertrace identity

Let us introduce the supertrace on a given matrix loop superalgebraG,which satisies:

(a) The symmetric property

(b) The invariance property under the Lie product

andadadenotes the adjoint action ofa∈GonG,which is defined by

where the bracket[·,·]is the Lie super bracket ofG.

Constructing a functional

whereandU,Vsatisfying fractional matrix spectral problems

are to be determined.

In the sense of local fractional derivative,the directional derivative?αR∈Gof a functionalRwith respect toa∈Gis defined by

Based on the nondegenerate property of the Killing form,we have

So,we give rise the following constrained conditions for the variational calculation of functionW,

Which determineVand Λ,and imply thatVand Λ are relatedU,and potentialu.

Moreover,it follows that

whereis variational derivative with respect to the potentialu.Obviously,only the dependence ofuinUneeds to be considered in computingand the property that if=0.So,based on the invariance property(21),we get

Using(26)and the Jacobi identity,we have

and from(26),we can get

Thus,Z=[Λ,V]?Vλsatisfies

Because rank(Z)=rank(Vλ)=there exists a constantγsatisfy

Finally,(28)is derived as follows

Lemma(The fractional supertrace identity)[19]LetU=U(u,λ)∈Gbe homogeneous in rank.Assume that the stationary zero curvature equation has a unique solutionV∈Gof a fixed rank up to a constant multiplier.Then,for any homogeneous rank solutionV∈G,we have the following fractional supertrace identity

whereγis a constant.

5 The fractional super BKK hierarchy and its super Hamiltonian structure

Based on a Lie superalgebraG={e1,e2,e3,e4,e5},wheree1,e2,e3are even,e4,e5are odd.The commutation and anti-commutation relations are defined by

along with the communicative operation

Let us now consider the super BKK spectral problem as follows

and

Asλis the spectral parameter,randsare commuting variables,andu1andu2are anti-commuting variables.An,Bn,Cnare commutation fields,andρn,σnare anticommuting fields.

From the stationary zero curvature equationDαxV=[U,V],we get

Then we consider the auxiliary spectral problem

Setting

Substituting(40)into the fractional zero curvature equation

give rise to the fractional super BKK hierarchy

where

In what follows,we will construct the fractional super Hamiltonian structure.A direct calculation reads

According to the fractional supertrace identity(34),we have

Comparing the coefficients ofλ?n?1yields

Since str(adVadV)=const?=0,we haveγ=0.Therefore,The fractional super BKK hierarchy possesses the following super Hamiltonian structure

The super Hamiltonian structure also implies that each super BKK system in the hierarchy possesses infinitely many commuting conserved quantitiesand infinitely many commuting symmetries

When taken=2,we obtain the fractional nonlinear super BKK system

Forα=β=1,the system(47)can be reduced the classical nonlinear super BKK system[27]

If we takeu1=u2=0,the system(48)can be reduced the classical BKK system[27]

6 Nonlinear integrable couplings for fractional super BKK hierarchy

Let us introduce a new Lie superalgebrawhere{E1,E2,E3,E4,E5,E6}are even,and{E7,E8}are odd.The generators of the Lie superalgebraEi,0≤i≤8,satisfy the following(anti)commutation relations

Now,let us start from an enlarged spectral matrix as follows

Taking

solving the enlarged stationary equationwe obtain

Setting

Then,from the generalized zero curvature equationwe can obtain a hierarchy of nonlinear super integrable couplings

for the fractional super BKK hierarchy(42).

According to the fractional supertrace identity(34),through tedious calculation reads,we can obtain the fractional super Hamiltonian structure of the nonlinear super integrable couplings(54)

where

is the super fractional Hamiltonian function,andˉJis the super fractional Hamiltonian operator

From(38)and(52),we obtain a recurrence operatorwhich meets

Whenn=2 in(55),we obtain the nonlinear super integrable couplings of the secondorder fractional super BKK(47)

7 Conclusions

A way to construct the fractional super integrable systems and its fractional super Hamiltonian structure is proposed.An application to the super BKK hierarchy yields a hierarchy of nonlinear super integrable couplings.How to obtain the solution of reduced equations is an interesting and challenging work,and we plan to study and discuss this problem in the near future.In fact,the proposed method can be generalized to get more fractional super hierarchy.

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