Kelei TIAN Jingsong HE Yucai SU

1 Introduction

The origin ofq-calculus(quantum calculus)(see[1–2])traces back to the early 20th century.Many mathematicians have important works in the area ofq-calculus,q-hypergeometric series and quantum group.There exist two different forms ofq-derivative operators,which are defined respectively by

and

The so-calledq-deformation of the integrable system(orq-deformed integrable system)started in the 1990s by means of the firstq-derivativeDqin(1.1)instead of the usual derivative∂with respect toxin the classical system.As we know,theq-deformed integrable system reduces to a classical integrable system asqgoes to 1.Severalq-deformed integrable systems have been presented,for example,theq-deformation of the KdV hierarchy(see[3–6]),theq-Toda equation(see[7]),theq-Calogero-Moser equation(see[8])and so on.Theq-deformed Kadomtsev-Petviashvili(q-KP for short)hierarchy is also a subject of intensive study in the literature[9–17].Indeed,it is worth pointing out that there exist two variants of theq-deformed integrable system,one belonging to Frenkel[3]and the other to Zhang et al.[4–17].

It has been known for some time that different sub-hierarchies of the KP hierarchy can be obtained by adding different reduction conditions on Lax operatorL.Two important subhierarchies of the KP hierarchy are CKP hierarchy(see[18])through a restrictionL∗=−Land BKP hierarchy(see[19])through a restrictionHowever,to the best of our knowledge,there has been no result on theq-deformed CKP hierarchy and theq-deformed BKP hierarchy so far.The difficulty to define them is the conjugate operation “∗” ofq-derivativeDqin(1.1).In fact,This paper shows a quite interesting fact as,where the symmetricq-derivative operator∂qis defined by(1.2).In what follows,we shall fill the gap by constructing the new symmetricq-deformed KP hierarchy based on the symmetricq-derivative operator∂q.

This paper is organized as follows.Some basic results of the symmetricq-derivative operator∂qare given in Section 2,and one formula for the symmetricq-exponenteq(x)is established.Then a new symmetricq-KP hierarchy is stated in Sections 3 similar to the classical KP hierarchy(see[20]),and also a symmetricq-CKP hierarchy and a symmetricq-BKP hierarchy are given in this section.We further study the additional symmetries for the symmetricq-KP hierarchy in Section 4.Section 5 is devoted to conclusions and discussions.

2 Symmetric Quantum Calculus

We give some useful facts about the symmetricq-derivative operator∂qin the form of(1.2)based on the literature[2].We work in an associative ring of functions which includes aq-variablexand infinite time variablesti∈R,

Theq-shift operator is defined by

Note thatθdoes not commute with∂q.Indeed,the relation

holds.The limit of∂q(f(x))asqapproaches to 1 is the ordinary differentiation∂x(f(x)).We denote the formal inverse of∂qas

Theorem 2.1The conjugate of∂qcan be defined as

ProofThe first step is to proveAccording to the definition,we have

Letg→θ−2gin(2.3),and it now yields

Comparing it with(2.2),the above equation becomes

It can now be written in the form

By lettingandin the above equation,we find that

so one can choose

We will now proceed to proveLetandin(2.2),and it now reads

This implies

According to the equationwe get

The followingq-deformed Leibnitz rule holds:

where theq-number

and theq-binomial is introduced as

To illustrate theq-deformed Leibnitz rule,the following examples are given:

Using the Taylor’s formula,we can get the following proposition for the symmetricqexponenteq(x),which is crucial to developing the tau function of the symmetricq-KP hierarchy and to researching the interaction ofq-solitons in the future.

Theorem 2.2The q-exponent eq(x)is defined as

where

and then the formula

holds,where

ProofFrom the definition ofeq(x)and Taylor’s formula,it follows that

whereckis given by(2.7).

Several explicit forms ofq-exponenteq(x)can be written out as follows:

where

Recall that theq-exponent functioneq(x)is the eigenfunction of the operator∂q,i.e.,

Furthermore,from

one obtains immediately the formula

which is useful for defining theq-wave function of the symmetricq-KP hierarchy in the following section.

3 Symmetric q-Deformed KP Hierarchy

Similar to the classical KP hierarchy(see[19–20]),we will define a new symmetricqdeformed KP hierarchy.The Lax operatorLof the symmetricq-KP hierarchy is given by

whereThe corresponding Lax equation of the symmetricq-KP hierarchy is defined by

The first fewBnand flow equations in(3.2)for dynamical variables{u1,u2,u3,···}can be written out as follows:

where·and

The first flow equations are

HereSis called a dressing operator or a wave operator of the symmetricq-KP hierarchy.

Theorem 3.1The dressing operator S of the symmetric q-KP hierarchy satisfies the Sato equation

ProofFrom the Lax equation,which is followed by

On the other hand,

and then

The above equation implies that

which ends the proof.

De fi nition 3.1The q-wave function wq(x,t;z)for the symmetric q-KP hierarchy(3.2)with the wave operator S in(3.3)is given by

where t=(t1,t2,t3,···).

Theorem 3.2The q-wave function wq(x,t;z)of the symmetric q-KP hierarchy satisfies the following linear q-differential equations:

where

ProofUsing the equationthen

From the Sato equation,it follows that

Furthermore,we would like to give the definitions of the symmetricq-CKP hierarchy and the symmetricq-BKP hierarchy respectively to answer the previous question proposed in the introduction.

Definition 3.2Let the operator L in(3.1)be the Lax operator for the symmetric q-KP hierarchy associated with(3.2),if L satisfies the reduction condition L∗=−L,and then we call it the symmetric q-CKP hierarchy.

Definition 3.3Let the operator L in(3.1)be the Lax operator for the symmetric q-KPhierarchy associated with(3.2),if L satisfies the reduction condition,and then it is the symmetric q-BKP hierarchy.

4 Additional Symmetries of the Symmetric q-KP Hierarchy

Another main goal of this paper is to consider the additional symmetries of the symmetricq-KP hierarchy.First,let us define Γqand Orlov-Shulman’sMoperator as

respectively,whereciis given by(2.7).Then the additional flows of the symmetricq-KP hierarchy for each pair{m,n}are defined by

Theorem 4.1The additional flows act on L and M of the symmetric q-KP hierarchy as

ProofBy performing the derivativeand using(4.1),we observe that

For the action onthere exists a similar derivation as,and then

In the above calculation,the fact that Γqdoes not depend on the additional flow variableshas been used.

Theorem 4.2

ProofWe present only the proof of the first equation here.The others can be proved in a similar way.

where we have used the formulain Theorem 4.1.

Theorem 4.3The additional flowscommute with the hierarchyi.e.,

and thus we call them additional symmetries of the symmetric q-KP hierarchy.

ProofAccording to the definition and Theorem 4.2,it equals

andhave been used in the above derivation.

5 Conclusions and Discussions

To summarize,we have derived the antisymmetric property of∂qin Theorem 2.1 and a crucial expression ofeq(x)by the usual exponential in Theorem 2.2.The analytic property of symmetriceq(x)in Theorem 2.2 is used to define the wave function of the symmetricq-KP hierarchy.After introducing the dressing operator and theq-wave function of the symmetricq-KP hierarchy in Section 3,we also give the definitions of the symmetricq-CKP hierarchy and the symmetricq-BKP hierarchy.The additional symmetries of the symmetricq-KP hierarchy are obtained in Section 4.The above results of this paper show obviously that the symmetricq-KP hierarchy is different from theq-KP hierarchy(see[8–17])based onDq(f(x)).

In comparison with the known interesting results of the KP hierarchy(see[18–20])and theq-KP hierarchy based on theDq(f(x))(see[8–17]),the symmetricq-KP hierarchy defined in this paper deserves further study from several aspects including the tau function and its Hirota bilinear identity,the Hamiltonian structure,the gauge transformation,the symmetry analysis and the interaction ofq-solitons.Furthermore,it is highly nontrivial to consider the above topics of the symmetricq-CKP(orq-BKP)hierarchy because of the reduction conditionand the complexity of the∂q.

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