Journal of Nonlinear Mathematical Physics

Volume 28, Issue 1, March 2021, Pages 53 - 67

The Miura Links of the Symmetries in the q-Deformed Case

Authors
Siqi Jian1, Jipeng Cheng2, *
1School of Statistics, Capital University of Economics and Business, Beijing 100070, P. R. China
2School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, P. R. China
*Corresponding author. Email: chengjp@cumt.edu.cn
Corresponding Author
Jipeng Cheng
Received 29 April 2019, Accepted 27 April 2020, Available Online 10 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.200922.005How to use a DOI?
Keywords
Squared eigenfunction symmetry, additional symmetry, Miura transformation, the q-KP hierarchies, the q-mKP hierarchies
Abstract

In this paper, we first construct the squared eigenfunction symmetries for the q-deformed Kadomtsev–Petviashvili (KP) and q-deformed modified KP hierarchies, including the unconstrained and constrained cases. Then the Miura links of the squared eigenfunction symmetries are investigated. At last, we also discuss the Miura links of the additional symmetries, since the additional symmetries are closely related with the squared eigenfunction symmetries.

Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The q-deformed integrable systems [1,5,913,16,1820,2832] play important roles in the theoretical physics and mathematics, which are usually defined by replacing the usual derivative with respect to the space variable x in the classical integrable systems with the q-derivative q [see (2) in Section 2]. Among them, the q-deformed Kadomtsev–Petviashvili (q-KP) [1,9,11,12,30,31] and q-deformed modified Kadomtsev–Petviashvili (q-mKP) hierarchies [5,19,28] are two typical ones, which are connected with each other by the Miura links [2]. Here, Miura transformation means the one from the solutions of the q-mKP hierarchy to the solutions of the q-KP hierarchy, while the reverse one from the q-KP hierarchy to the q-mKP hierarchy is called reverse-Miura transformation. There are many integrable properties, such as the Hamiltonian structures [15,25,27] and the solution structures [14,21], which can coincide with the Miura links. In this paper, we will invesigate the changes of the Squared Eigenfunction (SE) symmetries and the additional symmetries under the Miura links in the q-deformed case.

The SE symmetry [3,2224], also called the “ghost” symmetry [3], is a kind of symmetry generated by the squared eigenfunctions, i.e. the product of the eigenfunctions and their adjoints. The SE symmetry plays an important role in various aspects of the integrable systems, and thus is extensively studied recently [4,6,7,17]. In this paper, we construct the SE symmetries for the q-KP and q-mKP hierarchies, including the unconstrained and constrained cases. And further, the Miura links of the SE symmetries are considered. We find that the Miura links can keep the structures of the SE symmetries in the q-deformed case. Another important kind of symmetries which are closely related with the SE symmetry is the additional symmetries [2,3,6,8,26], depending explicitly on the space and time variables. The additional symmetries of the q-KP and q-mKP hierarchies are investigated in Tian et al. [29], Tu [31]. In this paper, we also consider the Miura links of the additional symmetries by direct computations, which can confirm the results of the SE symmetry.

The paper is organized as follows. In Section 2, the Miura links between the q-KP and q-mKP hierarchies are reviewed. In Section 3, the SE symmetries are constructed and their Miura links are investigated in the unconstrained case. The corresponding constrained case is discussed in Section 4. In Section 5, we investigate the Miura links of the additional symmetries. Finally, Section 6 is devoted to the conclusions and discussions.

2. q-KP AND q-mKP HIERARCHIES

In this section, we will review and revise the results in Cheng [5] for the further study. Let us firstly review the q-pseudo difference operators [1113]. Denote θ and q be the q-shift operator and q-difference operator acting on a function f(x):

[θf](x)=f(qx),[qf](x)=f(qx)-f(x)x(q-1).

The commutative law between q and θ is as follows,

[qθkf]=qk[θkqf],k𝕑. (2.1)

For any q-pseudo-differential operator A=iAiqi and any function f(x), the symbol Af denotes the operator product of A and f, which obeys the q-deformed Leibniz rule

qnf=k=0(nk)q[θn-kqkf]qn-k,n𝕑, (2.2)
where the q-number and the q-binomial are defined by
(n)q=qn-1q-1,(nk)q=(n)q(n-1)q(n-k+1)q(1)q(2)q(k)q,(n0)q=1. (2.3)

The symbol [Af] or fA denotes the action of A on f. Some simple notations of operations on the operator A are defined as follows,

Ak=ikAiqi,A<k=i<kAiqi,(A)k=Akqk,res(A)=A-1. (2.4)

Define θ * and q* as the adjoints of the q-shift operator θ and the q-difference operator q respectively:

θ*=θ-1,q*=-qθ-1=-1q1q. (2.5)

And the adjoint operator * satisfies (AB)* = B*A* for any q-difference operators A and B. For the q-pseudo-differential operator A=iAiqi , we use the notation A|x/q=i[θ-1Ai]qiqi .

Then the q-KP and q-mKP hierarchies [5,11,12,19] are defined as the following Lax equations,

tnL=[(Ln)k,L]. (2.6)

Here the Lax operator L is a general q-pseudo difference operator,

L={q+u+u1q-1+u2q-2+u3q-3+,k=0(L=LqKP),vq+v0+v1q-1+v2q-2+v3q-3+,k=1(L=LqmKP), (2.7)
which can be expressed in a terms of a dressing operator
L={WqW-1,k=0(L=LqKP),ZqZ-1,k=1(L=LqmKP), (2.8)
with
W=1+w1q-1+w2q-2+, (2.9)
Z=z0+z1q-1+z2q-2+(z0-1exists). (2.10)

Then the Lax equations are equivalent to

tnW=-(LqKPn)<0W=-(WqnW-1)<0W, (2.11)
tnZ=-(LqmKPn)0Z=-(ZqnZ-1)0Z. (2.12)

With the dressing operators W and Z, the q-wave function ω (x, t, z) and q-adjoint wave function ω*(x, t, z) are defined by

ωq(x,t,z)=[Weq(xz)expξ(t,z)](k=0), (2.13)
ωq(x,t,z)=[Zeq(xz)expξ(t,z)](k=1), (2.14)
and
ωq*(x,t,z)=[(W*)-1|x/qe1/q(-xz)exp(-ξ(t,z))](k=0), (2.15)
ωq*(x,t,z)=[(Z*)-1|x/qe1/q(-xz)exp(-ξ(t,z))](k=1). (2.16)

Here eq(x)=i=0xi(i)q!=exp[k=0(1-q)kk(1-qk)xk] and ξ(t,z)=i=1tizi . It can be proved that

[Lωq]=zωq,tnωq=[(Ln)kωq],(k=0,1), (2.17)
[L*|x/qωq*]=zωq*,tnωq*=-[((Ln)k|x/q)*ωq*],(k=0,1). (2.18)

The construction of the Miura transformations depends on the eigenfunction ϕ and adjoint eigenfunction ψ, which are defined as follows

ϕtn=[(Ln)kϕ],ψtn=-[(qk(Ln)kq-k)*ψ],k=0,1. (2.19)

Note that in Cheng [5], the definition of the adjoint eigenfunction ψ (in the case of k = 1) is different from the one here. Another important objects are the SE potentials Ω(ψ (k), ϕ) and Ω^(ψ,ϕ(k)) ,

Ω(ψ(k),ϕ)q=ψ(k)ϕ,Ω(ψ(k),ϕ)tn=res(q-1ψ(k)(Ln)kϕq-1),k=0,1, (2.20)
where, in turn for k = 0, 1, ϕ(k) denotes ϕ or ϕq , and ψ(k) denotes ψ or ψq* . The potential Ω^ is defined by
Ω^(ψ,ϕ(k))={Ω(ψ,ϕ),k=0,Ω(ψ(k),ϕ)+[θ-1ψ]ϕ,k=1. (2.21)

From the q-KP hierarchy to the q-mKP hierarchy, there are two different types of reverse-Miura transformations

Tα(ϕ)=ϕ-1,Tβ(ψ)=q-1ψ, (2.22)
where ϕ and ψ are the eigenfunction and the adjoint eigenfunction of the q-KP hierarchy respectively. As for Miura transformations, there are also two different types from the q-mKP hierarchy to the q-KP hierarchy
Tμ(z0)=z0-1,Tν(z0)=[θz0]-1q. (2.23)

Here z0 is the coefficient of q0 -term in the dressing operator of the q-mKP hierarchy [see (2.10)].

Next, we review the results in Cheng [5] of the Miura links in the q-deformed cases without constraints.

Theorem 2.1.

(Reverse-Miura transformation: q-KP → q-mKP) [5] Let Ω(.,.) and Ω^(.,.) be defined in (2.20) and (2.21). Let L = LqKP be the solution of the q-KP hierarchy Ltn=[(Ln)0,L] with (adjoint) eigenfunctions ϕ, ϕ1,,ϕm1 , ψ, ψ1,,ψm1 defined in (2.19). Then under the reverse-Miura transformation Tα or Tβ,

  1. (i)

    LαL˜=ϕ-1Lϕ , WαZ=ϕ-1W , ϕiαϕ˜i=ϕ-1ϕi , ψiαψ˜i=-[θΩ(ψi,ϕ)] ;

  2. (ii)

    LβL˜=q-1ψLψ-1q , WβZ=q-1ψW , ϕiβϕ˜i=Ω(ψ,ϕi) , ψiβψ˜i=ψ-1ψi ,

L˜ satisfies the q-mKP hierarchy L˜tn=[(L˜n)1,L˜] and ϕ˜i,   ψ˜i,  i=1,,m1 are (adjoint) eigenfunctions of L˜ .

Theorem 2.2.

(Miura transformation: q-mKP → q-KP) [5] Let L = LqmKP be the solution of the q-mKP hierarchy Ltn=[(Ln)1,L] with (adjoint) eigenfunctions ϕ1,,ϕm1 , ψ1,,ψm1 defined in (2.19). Let z0 be the zero term in the dressing operator Z such that L = Z∂qZ–1. Then under the Miura transformation Tμ or Tν,

  1. (i)

    LμL˜=z0-1Lz0 , ZμW=z0-1Z , ϕiμϕ˜i=z0-1ϕi , ψiμψ˜i=z0[q*ψi] ;

  2. (ii)

    LνL˜=[θz0]-1qLq-1[θz0] , ZνW=[θz0]-1qZ , ϕiνϕ˜i=[θz0]-1[qϕi] , ψiνψ˜i=[θz0]ψi ,

L˜ satisfies the q-KP hierarchy L˜tn=[(L˜n)0,L˜] and ϕ˜i,   ψ˜i,  i=1,,m1 are (adjoint) eigenfunctions of L˜ .

Remark:

Please note that in Cheng [5], for k = 1, the definition of the adjoint eigenfunction ψ is different from the one here. Thus, our results here are slightly different from the ones in Cheng [5].

It is also given in Cheng [5] the results of the constrained case. Here we will revise the corresponding results. The constrained case means the imposition of the constraints on the Lax operator L in (2.7),

(LN)<k=j=1mqjq-1rjqk,k=0,1, (2.24)
for some N𝕅 and some functions q1, …, qm, r1, …, rm, where
Ltn=[(Ln)k,L],  qjtn=[(Ltn)kqj],  rjtn=-[q-k*(Ltn)k*qk*rj]. (2.25)

Theorem 2.3.

(Reverse-Miura transformation: constrained q-KPconstrained q-mKP) Let Ω(.,.) and Ω^(.,.) be defined in (2.20) and (2.21). Let L = LqKP satisfy the constraint (LN)<0=j=1mqjq-1rj . Then for any function ϕ or ψ, under the reverse-Miura transformation Tα or Tβ,

  1. (i)

    LαL˜=ϕ-1Lϕ , qjαq˜j=ϕ-1qj , rjαr˜j=-[θΩ^(rj,ϕ)],  j=1,,m , and q˜m+1=ϕ-1[(LN)0ϕ]+ϕ-1j=1mqjΩ^(rj,ϕ)=ϕ-1[LNϕ],  r˜m+1=1 ;

  2. (ii)

    LβL˜=q-1ψLψ-1q , qjβq˜j=Ω(ψ,qj) , and rjβr˜j=ψ-1rj,  j=1,,m q˜m+1=1,  r˜m+1=-j=1m[θΩ(ψ,qj)]rjψ-1+[(LN)0*ψ]ψ-1=ψ-1[(LN)*ψ] ,

the transformed operator L˜ satisfies the constraint,
(L˜N)<1=j=1m+1q˜jq-1r˜jq.

If L, qj, rj satisfy the constrained q-KP dynamics (2.25, k = 0) and ϕ or ψ is an (adjoint) eigenfunction (i.e. ϕtn=[(Ln)0ϕ] or ψtn=-[(Ln)0*ϕ] ), according to Theorem 2.1, the transformed Lax operator L˜ satisfies the q-mKP dynamics L˜tn=[(L˜n)1,L˜] . Further, q˜j , r˜j are the new (adjoint) eigenfunctions.

Theorem 2.4.

(Miura transformation: constrained q-mKP → constrained q-KP) Let L = LqmKP satisfy the constraint (LN)<1=j=1mqjq-1rjq . Then for any function z0, under the Miura transformation Tμ or Tν,

  1. (i)

    LμL˜=z0-1Lz0 , qjμq˜j=z0-1qj , rjμr˜j=-z0[θ-1qrj] ;

  2. (ii)

    LνL˜=[θz0]-1qLq-1[θz0] , qjνq˜j=[θz0]-1[qqj] , rjνr˜j=[θz0]rj ,

the transformed operator L˜ satisfies the constraint,
(L˜N)<0=j=1mq˜jq-1r˜j.

If L, qj, rj satisfies the constrained q-mKP dynamics (2.25, k = 1) and z0 is the zero term in the dressing operator Z such that L = Z∂qZ–1, according to Theorem 2.2, the transformed Lax operator L˜ satisfies the q-KP dynamics L˜tn=[(L˜n)0,L˜] . Further, q˜j , r˜j are the new (adjoint) eigenfunctions.

3. SE SYMMETRIES AND MIURA LINKS: UNCONSTRAINED CASE

The SE symmetry flow [3,2224] is a one parameter symmetry group of the Lax equations generated by the squared eigenfunctions, i.e. the product of the eigenfunctions and their adjoints. We denote the corresponding group parameter by s and prove that the flow is the symmetry of q-KP and q-mKP hierarchies via its infinitesimal generator.

Theorem 3.1.

For k = 0 or 1, let L satisfy the Lax equations Ltn=[(Ln)k,L] , and ϕ1,,ϕm1 and ψ1,,ψm1 be the (adjoint) eigenfunctions defined in (2.19). Then the SE symmetry flow defined by

Ls=[i=1m1ϕiq-1ψiqk,L] (3.1)
can commute with the q-deformed Lax hierarchy, i.e. [s,tn]L=0 , which is equivalent to the zero curvature equation
Mns-Mtn=[M,Mn], (3.2)
where Mn=(Ln)k and M=i=1m1ϕiq-1ψiqk .

Proof. The Lax equation Ls = [M′, L] leads to Lsn=[M,Ln] , then Mns=(Lsn)k=[M,Ln]k=[M,(Ln)k]k+[M,(Ln)<k]k=[M,Mn]k . We obtain

Mns=[M,Mn]k. (3.3)

On the other hand, from the formula

(q-1A)<0=q-1(A*)0+q-1(A)<0, (3.4)
we have
[Mn,M]<k=i(Mnϕiq-1ψiqk)<k-i(ϕiq-1ψiqkMn)<k=i(Mnϕiq-1)<0ψiqk-iϕi(q-1ψiqkMnq-k)<0qk=(3.4)i(Mnϕi)0q-1ψiqk-iϕiq-1((ψiqkMnq-k)*)0qk=i[Mnϕi]q-1ψiqk-iϕiq-1[q-k*(Ln)1*qk*ψi]qk=i(ϕiq-1ψiqk)tn,
which equals to
Mtn=[Mn,M]<k. (3.5)

Equations (3.3) and (3.5) lead to the zero curvature equation (3.2) and

Lstn-Ltns=[M,L]tn-[Mn,L]s=[Mtn-Mns-[M,Mn],L]=0.

As in Theorem 3.1, s is the group parameter of the SE symmetry flows. We will introduce conditions of ϕ and ψ corresponding to the parameter s,

ϕs=iϕiΩ^(ψi,ϕ(k)),ψs=iψi[θΩ(ψ(k),ϕi)],k=0,1. (3.6)

Further,

Lemma 3.1.

The SE symmetry in (3.1) is the compatible condition of linear problems:

ϕtn=[(Ln)kϕ],ϕs=iϕiΩ^(ψi,ϕ(k)), (3.7)
and
ψtn=-[q*(-k)(Ln)k*q*kψ],ψs=iψi[θΩ(ψ(k),ϕi)],k=0,1, (3.8)
i.e. ϕtns=ϕstn and ψtns=ψstn .

When k = 1, one can deduce by Ls=[iϕiq-1ψiq,L] ,

z0s=iϕi[θ-1ψi]z0, (3.9)
where z0 is the zero order term of the dressing operator Z.

Next we discuss the Miura links of the SE symmetries, which are given in the next two theorems.

Theorem 3.2.

(Reverse-Miura transformation: q-KPq-mKP) Let ϕ and ψ be the (adjoint) eigenfunctions also satisfying (3.6, k = 0), i.e. ϕs=iϕiΩ^(ψi,ϕ) and ψs=iψi[θΩ(ψ,ϕi)] . And the SE symmetry of the q-KP hierarchy is given by Ls=[iϕiq-1ψi,L] . Then under the reverse-Miura transformation Tα or Tβ in Theorem 2.1, L˜ also satifies the SE symmetry

L˜s=[iϕ˜iq-1ψ˜iq,L˜],
where L˜ , ϕ˜i and ψ˜i are defined in Theorem 2.1.

Proof. For the case Tα, by using Theorem 3.1, we have

L˜s=[ϕ-1Lϕ]s=-[ϕ-1ϕs,L˜]+ϕ-1Lsϕ=[iϕ-1ϕi[θ-1ψ˜i],L˜]+ϕ-1[iϕiq-1ψi,L]ϕ=[iϕ-1ϕi[θ-1ψ˜i]+ϕ-1ϕiq-1ψiϕ,L˜]=[iϕ˜i[θ-1ψ˜i]-ϕ˜iq-1[θ-1ψ˜i]q,L˜]=[iϕ˜iq-1(q[θ-1ψ˜i]-[θ-1ψ˜i]q),L˜]=[iϕ˜iq-1ψ˜iq,L˜].

For the case Tβ, also by using Theorem 3.1, we have

L˜s=(q-1ψLnψ-1q)s=q-1(ψLψ-1)sq=q-1[ψ-1ψs,ψLψ-1]q+q-1ψLsψ-1q=iq-1[ψ-1ψi[θΩ(ψ,ϕi)],ψLψ-1]q+iq-1ψ[ϕiq-1ψi,L]ψ-1q=i[q-1ψ˜i[θϕ˜i]q,L˜]+i[q-1ψϕiq-1ψ˜iq,L˜]=i[q-1([θϕ˜i]+[qϕ˜i]q-1)ψ˜iq,L˜]=i[q-1qϕ˜iq-1ψ˜iq,L˜]=i[ϕ˜iq-1ψ˜iq,L˜].

Theorem 3.3.

(Miura transformation: q-mKPq-KP) Let z0 be a function satisfying (3.9), i.e., z0s=iϕi[θ-1ψi]z0 . The SE symmetry of the q-mKP hierarchy is given by Ls=[iϕiq-1ψiq,L] . Then under the Miura transformation Tμ or Tν defined in Theorem 2.2, L˜ also satifies the SE symmetry

L˜s=[iϕ˜iq-1ψ˜i,L˜].
Here L˜ , ϕ˜i and ψ˜i are defined in Theorem 2.2.

Proof. For the case Tμ, by using Theorem 3.1, we have

L˜s=(z0-1Lz0)s=-[z0-1z0s,L˜]+z0-1Lsz0=i[-z0-1ϕi[θ-1ψi]z0,L˜]+iz0-1[ϕiq-1ψiq,L]z0=i[z0-1ϕi(-[θ-1ψi]+q-1ψiq)z0,L˜]=i[z0-1ϕiq-1[q*ψi]z0,L˜]=i[ϕ˜iq-1ψ˜i.L˜].

For the case Tν, also by using Theorem 3.1, we have

L˜s=([θz0]-1qLq-1[θz0])s=-[[θz0]-1[θz0]s,L˜]+[θz0]-1qLsq-1[θz0]=[-[θz0]-1[θz0]s,L˜]+i[θz0]-1q[ϕiq-1ψiq,L]q-1[θz0]=i[-[θz0]-1[θϕi]ψi[θz0],L˜]+i[[θz0]-1qϕiq-1ψi[θz0],L˜]=i[[θz0]-1(-[θϕi]+qϕiq-1)ψi[θz0],L˜]=i[[θz0]-1[qϕi]q-1ψi[θz0],L˜]=i[ϕ˜iq-1ψ˜i,L˜].

4. SE SYMMETRIES AND MIURA LINKS: CONSTRAINED CASE

The SE symmetries for the constrained q-KP and q-mKP hierarchies are given in the next theorem.

Theorem 4.1.

Let Ω(.,.) and Ω^(.,.) be defined in (2.20) and (2.21). For k = 0 or 1, assuming the (adjoint) eigenfunctions ϕi, ψi, i = 1, …, m1 be defined in (2.19). If ϕ1,,ϕm1 , and ψ1,,ψm1 satisfy

[(LN)0ϕi]+j=1mqjΩ^(rj,ϕi(k))=λiϕi,[q-k*(LN)0*qk*ψi]-j=1m[θΩ(ψi(k),qj)]rj=λiψi, (4.1)
where λi𝔺 , and in turn for k = 0, 1, ϕi(k) denotes ϕi or ϕiq , ψi(k) denotes ψi or ψiq* , qj(k) denotes qj or qjq and rj(k) denotes rj or rjq* . The constraint (2.24) is invariant under the SE symmetry flow,
Ls=[i=1m1ϕiq-1ψiqk,L],qjs=i=1m1ϕiΩ^(ψi,qj(k)),rjs=i=1m1[θΩ(rj(k),ϕi)]ψi. (4.2)

Proof. By the direct computation,

q-1ψqkϕq-1=Ω^(ψ,ϕ(k))q-1-q-1[θΩ(ψ(k),ϕ)],k=0,1. (4.3)

With

((LN)<k)s=[i=1m1ϕiq-1ψiqk,LN]<k=[i=1m1ϕiq-1ψiqk,(LN)k]<k+[i=1m1ϕiq-1ψiqk,(LN)<k]<k,
[i=1m1ϕiq-1ψiqk,(LN)k]<k=(i=1m1ϕiq-1ψiqk(LN)k-i=1m1(LN)kϕiq-1ψiqk)<k=i=1m1ϕi(q-1ψiqk(LN)kq-k)<0qk-i=1m1((LN)kϕiq-1ψi)<0qk=(3.4)i=1m1ϕiq-1(ψiqk(LN)kq-k)0*qk-i=1m1[(LN)kϕi]q-1ψiqk=i=1m1ϕiq-1[q-k*(LN)k*qk*ψi]qk-i=1m1[(LN)kϕi]q-1ψiqk
and
[i=1m1ϕiq-1ψiqk,(LN)<k]<k=(2.24)[i=1m1ϕiq-1ψiqk,j=1mqjq-1rjqk]<k=(i,jϕiq-1ψiqkqjq-1rjqk-i,jqjq-1rjqkϕiq-1ψiqk)<k=(4.3)(i,jϕiΩ^(ψi,qj(k))q-1rjqk-i,jϕiq-1[θΩ(ψi(k),qj)]rjqk)<k-(i,jqjΩ^(rj,ϕi(k))q-1ψiqk-i,jqjq-1[θΩ(rj(k),ϕi)]ψiqk)<k=i,jϕiΩ^(ψi,qj(k))q-1rjqk-i,jϕiq-1[θΩ(ψi(k),qj)]rjqk-i,jqjΩ^(rj,ϕi(k))q-1ψiqk+i,jqjq-1[θΩ(rj(k),ϕi)]ψiqk,
by the Eq. (4.1), we have
((LN)<k)s=i,jϕiΩ^(ψi,qj(k))q-1rjqk+i,jqjq-1[θΩ(rj(k),ϕi)]ψiqk=jqjsq-1rjqk+jqjq-1rjsqk=(jqjq-1rjqk)s.

The Miura links of the SE symmetries in the constrained case are listed in the two theorems below.

Theorem 4.2.

(Reverse-Miura transformation: constrained q-KPconstrained q-mKP) Let L = LqKP satisfy the constraint (LN)<0=j=1mqjq-1rj . Then for any functions ϕ and ψ satisfying (3.6, k = 0), under the reverse-Miura transformation Tα or Tβ defined in Theorems 2.1 and 2.3, if ϕ1,,ϕm1 , ψ1,,ψm1 , q1, ..., qm and r1, ..., rm satisfy

[(LN)0ϕi]+j=1mqjΩ^(rj,ϕi)=λiϕi,[(LN)0*ψi]-j=1m[θΩ(ψi,qj)]rj=λiψi, (4.4)
Ls=[i=1m1ϕiq-1ψi,L],qjs=i=1m1ϕiΩ^(ψi,qj),rjs=i=1m1[θΩ(rj,ϕi)]ψi, (4.5)
then these two conditions keep invariant,
[(L˜N)1ϕ˜i]+j=1m+1q˜jΩ^(r˜j,ϕ˜iq)=λiϕ˜i,[q-1*(L˜N)1*q*ψ˜i]-j=1m+1[θΩ(ψ˜iq*,q˜j)]r˜j=λiψ˜i, (4.6)
L˜s=[i=1m1ϕ˜iq-1ψ˜iq,L˜],q˜js=i=1m1ϕ˜iΩ^(ψ˜i,q˜jq),r˜js=i=1m1[θΩ(r˜jq*,ϕ˜i)]ψ˜i. (4.7)

Proof. We give a proof of the case Tα, and the proof of the case Tβ is similar. With

j=1mq˜jΩ^(r˜j,ϕ˜iq)=j=1mϕ-1qjΩ^(-[θΩ^(rj,ϕ)],ϕ˜iq)
=(2.21)-j=1mϕ-1qjΩ(rj,ϕ)ϕ-1ϕi+j=1mϕ-1qjΩ(rjϕ,ϕ-1ϕi)
=-j=1mϕ-2qjΩ(rj,ϕ)ϕi+j=1mϕ-1qjΩ(rj,ϕi),
and
q˜m+1Ω^(r˜m+1,ϕ˜iq)=ϕ-1[(LN)0ϕ]ϕ-1ϕi+ϕ-1j=1mqjΩ^(rj,ϕ)ϕ-1ϕi,
we have
j=1m+1q˜jΩ^(r˜j,ϕ˜iq)=ϕ-1[(LN)0ϕ]ϕ-1ϕi+j=1mϕ-1qjΩ(rj,ϕi).

On the other side,

[(L˜N)1ϕ˜i]=[(ϕ-1LNϕ)1ϕ-1ϕi]=ϕ-1[(LN)0ϕi]-ϕ-1[(LN)0ϕ]ϕ-1ϕi,
we have the first equation in (4.6)
[(L˜N)1ϕ˜i]+j=1m+1q˜jΩ^(r˜j,ϕ˜iq)=ϕ-1[(LN)0ϕi]+j=1mϕ-1qjΩ(rj,ϕi)=λiϕ˜i.

We now prove the second equation in (4.6). With

[q-1*(L˜N)1*q*ψ˜i]=[q-1*(ϕ-1LNϕ)1*ϕψi]=[q-1*ϕ(LN)0*ϕ-1ϕψi]-[q-1*[(LN)0ϕ]ϕ-1ϕψi]=[q-1*ϕj=1m[θΩ(ψi,qj)]rj]+[q-1*ϕλiψi]-[q-1*[(LN)0ϕ]ψi]
and
-j=1m+1[θΩ(ψ˜iq*,q˜j)]r˜j=j=1m[θΩ(ψiϕ,ϕ-1qj)][θΩ^(rj,ϕ)]-[θΩ(ψiϕ,ϕ-1[(LN)0ϕ]+ϕ-1j=1mqjΩ^(rj,ϕ))]=j=1m[θΩ(ψi,qj)][θΩ^(rj,ϕ)]-[θΩ(ψi,[(LN)0ϕ])]-[θΩ(ψi,j=1mqjΩ^(rj,ϕ))],
we have the second equation in (4.6)
[q-1*(L˜N)1*q*ψ˜i]-j=1m+1[θΩ(ψ˜iq*,q˜j)]r˜j=[q-1*ϕλiψi]=λiψ˜i,
by using the equality
j=1m[θΩ(ψi,qj)][θΩ^(rj,ϕ)]+[q-1*ϕj=1m[θΩ(ψi,qj)]rj]-[θΩ(ψi,j=1mqjΩ^(rj,ϕ))]=0.

The first equation in (4.7) has been proved in Theorem 3.2. We now prove the second equation in (4.7). For j = 1, ..., m, with

q˜js=-ϕ-2qjϕs+ϕ-1qjs=-i=1m1ϕ-2qjϕiΩ^(ψi,ϕ)+ϕ-1i=1m1ϕiΩ^(ψi,qj)
and
i=1m1ϕ˜iΩ^(ψ˜i,q˜jq)=i=1m1ϕ-1ϕiΩ^(-[θΩ(ψi,ϕ)],(ϕ-1qj)q)=(2.21)i=1m1ϕ-1ϕiΩ^(ψi,qj)-i=1m1ϕ-1ϕiΩ(ψi,ϕ)ϕ-1qj,
we have the second equation in (4.7).

For j = m + 1, with

q˜(m+1)s=-ϕ-2ϕs[Lnϕ]+ϕ-1[Lsnϕ]+ϕ-1[Lnϕs]=-ϕ-2i=1m1ϕiΩ^(ψi,ϕ)[Lnϕ]+ϕ-1i=1m1[ϕiq-1ψiLnϕ]-ϕ-1i=1m1[Lnϕiq-1ψiϕ]+ϕ-1i=1m1[LnϕiΩ^(ψi,ϕ)]=-ϕ-2i=1m1ϕiΩ^(ψi,ϕ)[Lnϕ]+ϕ-1i=1m1[ϕiq-1ψiLnϕ]
and
i=1m1ϕ˜iΩ^(ψ˜i,q˜(m+1)q)=i=1m1ϕ-1ϕiΩ^(-[θΩ(ψi,ϕ)],(ϕ-1[Lnϕ])q)=(2.21)i=1m1ϕ-1ϕiΩ^(ψi,[Lnϕ])-i=1m1ϕ-1ϕiΩ(ψi,ϕ)ϕ-1[Lnϕ],
we have the second equation in (4.7).

For j = 1, ..., m, with

r˜js=-[θΩ^(rj,ϕ)]s=-[θΩ^(rjs,ϕ)]-[θΩ^(rj,ϕs)]=-i=1m1[θΩ^([θΩ(rj,ϕi)]ψi,ϕ)]-i=1m1[θΩ^(rj,ϕiΩ^(ψi,ϕ))]
and
i=1m1[θΩ(r˜jq*,ϕ˜i)]ψ˜i=-i=1m1[θΩ(rjϕ,ϕ-1ϕi)][θΩ(ψi,ϕ)],
by taking the adjoint q-difference operator q* , we have the third equation in (4.7).

Since rm+1 = 1, it is trivial that the third equation in (4.7) holds.

Theorem 4.3.

(Miura transformation: constrained q-KPconstrained q-mKP) Let L = LqmKP satisfy the constraint (LN)<1=j=1mqjq-1rjq . Then for any function z0 satisfying (3.9), under the Miura transformation Tμ or Tν defined in Theorems 2.2 and 2.4, if ϕ1,,ϕm1 , ψ1,,ψm1 , q1, ..., qm and r1, ..., rm satisfy

[(LN)1ϕi]+j=1mqjΩ^(rj,ϕiq)=λiϕi,[q-1*(LN)1*q*ψi]-j=1m[θΩ(ψiq*,qj)]rj=λiψi, (4.8)
Ls=[i=1m1ϕiq-1ψiq,L],qjs=i=1m1ϕiΩ^(ψi,qjq),rjs=i=1m1[θΩ(rjq*,ϕi)]ψi, (4.9)
then these two conditions keep invariant,
[(L˜N)0ϕ˜i]+j=1mq˜jΩ^(r˜j,ϕ˜i)=λiϕ˜i,[(L˜N)0*ψ˜i]-j=1m[θΩ(ψ˜i,q˜j)]r˜j=λiψ˜i, (4.10)
L˜s=[i=1m1ϕ˜iq-1ψ˜i,L˜],q˜js=i=1m1ϕ˜iΩ^(ψ˜i,q˜j),r˜js=i=1m1[θΩ(r˜j,ϕ˜i)]ψ˜i. (4.11)

Proof. We give a proof of the case Tμ, and the proof of the case Tν is similar. With

j=1mq˜jΩ^(r˜j,ϕ˜i)=j=1mz0-1qjΩ^(z0[q*rj],z0-1ϕi)=j=1mz0-1qjΩ^([q*rj],ϕi)
and
[(L˜N)0ϕ˜i]=[(z0-1LNz0)0z0-1ϕi]=[z0-1(LN)0z0z0-1ϕi]=[z0-1(LN)1ϕi]+[z0-1(LN)0ϕi]=-z0-1j=1mqjΩ^(rj,ϕiq)+λiz0-1ϕi+z0-1j=1mqj[θ-1rj]ϕi,
we have the first equation in (4.10) by
([θ-1rj]ϕi)q=rjϕiq+[θ-1qrj]ϕi.

We now prove the second equation in (4.10). With

[(L˜N)0*ψ˜i]=[(z0-1LNz0)0*z0[q*ψi]]=[(z0(LN)1*z0-1+z0(LN)0*z0-1)z0[q*ψi]]=z0[(LN)1*[q*ψi]]+z0j=1mqj[θ-1rj][q*ψi]=z0[q*q-1*(LN)1*q*ψi]+z0j=1mqj[θ-1rj][q*ψi]=z0j=1m[q*[θΩ(ψiq*,qj)]rj]+λiz0[q*ψi]+z0j=1mqj[θ-1rj][q*ψi]=z0j=1m[θΩ(ψiq*,qj)][q*rj]+λiz0[q*ψi]
and
-j=1m[θΩ(ψ˜i,q˜j)]r˜j=-j=1m[θΩ(z0[q*ψ],z0-1qj)]z0[q*rj]=-z0j=1m[θΩ([q*ψ],qj)][q*rj],
we have the second equation in (4.10).

The first equation in (4.11) has been proved in Theorem 3.3. We now prove the second equation in (4.11). For j = 1, ..., m, with

q˜js=(z0-1qj)s=-z0-2qjz0s+z0-1qjs=-z0-1qji=1m1ϕi[θ-1ψi]+z0-1i=1m1ϕiΩ^(ψi,qjq)
and
i=1m1ϕ˜iΩ^(ψ˜i,q˜j)=i=1m1z0-1ϕiΩ(z0[q*ψi],z0-1qj)=(2.21)z0-1i=1m1ϕiΩ(ψi,qjq)-z0-1i=1m1ϕi[θ-1ψi]qj,
we have the second equation in (4.11).

For j = 1, ..., m, with

r˜js=(z0[q*rj])s=z0s[q*rj]+z0[q*rjs]=i=1m1ϕi[θ-1ψi]z0[q*rj]+z0i=1m1[q*[θΩ(rjq*,ϕi)]ψi]=z0i=1m1[θΩ(rjq*,ϕi)][q*ψi]
and
i=1m1[θΩ(r˜j,ϕ˜i)]ψ˜i=i=1m1[θΩ(z0[q*rj],z0-1ϕi)]z0[q*ψi]=z0i=1m1[θΩ([q*rj],ϕi)][q*ψi],
we have the third equation in (4.11).

Remark:

Theorems 4.2 and 4.3 show that the additional conditions (4.1) and (4.2) that keep the constaint (2.24) invariant under the SE symmetries are consistent with the Miura and the reverse Miura transformations. By considering Theorems 3.2 and 3.3, we can further find that the SE symmetries are consistent with the Miura and reverse Miura transformations, for the constrained case. In the meanwhile, the conditions (3.6), (3.9), (4.1) and (4.2) are required in these constructions.

5. ADDITIONAL SYMMETRIES AND MIURA LINKS

It is proved [3,6] that the SE symmetries defined by the (adjoint) wave functions in the q-deformed case can be viewed as the generating functions of the additional symmetries. Thus in this section, we will compute the Miura links directly, which can confirm the results of Miura links of the SE symmetries.

The additional symmetries [29,31] of the q-KP and q-mKP hierarchies are defined as follows. For each pair integers m and l, the additional symmetry flows of q-KP and q-mKP hierarchies are defined respectively,

{ml*W=-(MqmLl)<0W,q-KP,ml*Z=-(MqmLl)<1Z,q-mKP. (5.1)
with some additional variable tml* and its derivative ml*tml* . Here the Orlov-Schulman operator Mq [26,29,31] of the q-KP and q-mKP hierarchies is
Mq={WΓqW-1,q-KP,ZΓqZ-1,q-mKP, (5.2)
where the operator Γq is defined by
Γq=i=1[iti+(1-q)i1-qixi]. (5.3)
Γq commutes with the operator tm-qm , i.e. [tm-qm,Γq]=0 . By dressing it in q-KP and q-mKP hierarchies respectively, we have the equation [tm-(Lm)k,Mq]=0 or
tmMq=[(Lm)k,Mq],k=0,1. (5.4)

The additional symmetry flow ml* acts on the (adjoint) eigenfunctions of the q-KP hierarchy are defined as follows

[ml*ϕ]=[(MmLl)0ϕ],[ml*ψ]=-[(MmLl)0*ψ], (5.5)
and for z0 is the zero order term in the dressing operator Z of q-mKP hierarchy,
[ml*z0]=-(MqmLl)0z0. (5.6)

Then the Miura links of the additional symmetries are given in the next two theorems.

Theorem 5.1.

(Reverse-Miura transformation with additional symmetries: q-KP → q-mKP) Let L = LqKP have the dressing form L = W∂qW–1. Let ϕ and ψ be the (adjoint) eigenfuntions respectively. Then under the reverse-Miura transformation Tα or Tβ defined in Theorem 2.1,

  1. (i)

    WαZ=ϕ-1W , MqαM˜q=ϕ-1Mqϕ ;

  2. (ii)

    WβZ=q-1ψW , MqβM˜q=q-1ψMqψ-1q ,

one can find

tmM˜q=[(L˜m)1,M˜q].

If ϕ or ψ satisfies (5.5), and the dressing operator W satisfies the additional flows (5.1, k = 0), i.e. ml*W=-(MmLl)<0W , then in any case Tα or Tβ, the tranformed dressing operator Z also satisfies the additional flows (5.1, k = 1), i.e.,

ml*Z=-(M˜qmL˜l)0Z.

Proof. From the definition of M˜q , it can be proved that tmM˜q=[(L˜m)1,M˜q] holds.

Now we prove the additional flows (5.1, k = 1) for the transformed operator Z.

Case Tα.

ml*Z=[ml*ϕ-1W]=[ml*ϕ-1]W+ϕ-1[ml*W]=-ϕ-1[ml*ϕ]ϕ-1W-ϕ-1(MqmLl)<0W=(5.5)-ϕ-1[(MqmLl)0ϕ]ϕ-1W-ϕ-1(MqmLlϕ)<0ϕ-1W=-(ϕ-1MqmLlϕ)0ϕ-1W=-(M˜qmL˜l)0Z.

Case Tβ.

Note that

-(M˜qmL˜l)0Z=-(q-1ψMqmLlψ-1q)0Z=-q-1ψ(MqmLl)<0ψ-1qZ-q-1[(MqmLl)0*ψ]ψ-1qZ=-q-1ψ(MqmLl)<0W-q-1[(MqmLl)0*ψ]W,
then we have
ml*Z=ml*(q-1ψW)=q-1[ml*ψ]W+q-1ψ[ml*W]=q-1[ml*ψ]W-q-1ψ(MqmLl)<0W=(5.5)-q-1[(MqmLl)0*ψ]W-q-1ψ(MqmLl)<0W=-(M˜qmL˜l)0Z.

Theorem 5.2.

(Miura transformation with additional symmetries: q-mKP → q-KP) Let L = LqmKP have the dressing form L = ∂qZ–1. Let z0 be the zero term in the dressing operator Z. Then under the Miura transformation Tμ or Tν defined in Theorem 2.2,

  1. (i)

    ZμW=z0-1Z , MqμM˜q=z0-1Mqz0 ;

  2. (ii)

    ZνW=[θz0]-1qZ , MqνM˜q=[θz0]-1qMqq-1[θz0] ,

one can obtain

tmM˜q=[(L˜m)0,M˜q].

If the dressing operator Z satisfies the additional flows (5.1, k = 1), i.e. ml*Z=-(MmLl)0Z , then in any case Tμ or Tν, the tranformed dressing operator W also satisfies the additional flows (5.1, k = 0), i.e.,

ml*W=-(M˜qmL˜l)<0W.

Proof. From the definition of M˜q , we have tmM˜q=[(L˜m)0,M˜q] .

Now we prove the additional flows (5.1, k = 0) for the transformed operator W.

Case Tμ.

ml*W=ml*(z0-1Z)=[ml*z0-1]Z+z0-1[ml*Z]=-z0-1[ml*z0]z0-1Z-z0-1(MqmLl)0Z=(5.6)z0-1(MqmLl)=0z0z0-1Z-z0-1(MqmLl)0Z=-z0-1(MqmLl)<0Z=-(z0-1MqmLlz0)<0z0-1Z=-(M˜qmL˜l)<0W.

Case Tν.

Note that

-(M˜qmL˜l)<0W=-([θz0]-1qMqmLlq-1[θz0])<0[θz0]-1qZ=-[θz0]-1(qMqmLlq-1)<0qZ=-[θz0]-1[q(MqmLl)=0]Z-[θz0]-1q(MqmLl)<0Z,
then we have
ml*W=ml*([θz0]-1qZ)=ml*([θz0]-1)qZ+[θz0]-1q[ml*Z]=-[θz0]-1[ml*θz0][θz0]-1qZ-[θz0]-1q(MqmLl)0Z=(5.6)[θz0]-1[θ(MqmLl)=0][θz0][θz0]-1qZ-[θz0]-1q(MqmLl)=0Z-[θz0]-1q(MqmLl)<0Z=-[θz0]-1[q(MqmLl)=0]Z-[θz0]-1q(MqmLl)<0Z=-(M˜qmL˜l)<0W.

6. CONCLUSION AND DISCUSSION

We summarize the main results of the paper. In Theorem 3.1, we construct the SE symmetry for the q-KP and q-mKP hierarchies. In Theorems 3.2 and 3.3, with the conditions (3.6) and (3.9), we investigate the Miura links of the SE symmetries for the unconstrained case. For the constrained q-KP and q-mKP hierarchies with the condition (2.24) on the Lax operators, the conditions (4.1) and (4.2) are required to ensure the SE flows are consistent with the constraint (2.24) in Theorem 4.1. Theorems 4.2 and 4.3 reveal that with the previous conditions (3.6) and (3.9), the additional conditions (4.1) and (4.2) are maintained under the Miura links in the constrained case. Thus for the constrained case, as in the unconstrained case, the SE symmetries are consistent with the Miura and reverse-Miura transformations. Finally in Theorems 5.1 and 5.2, we show that additional symmetries are invariant under the Miura and reverse-Miura transformations, which confirm the previous results of the Miura links of the SE symmetries.

Our results show that for the q-KP and q-mKP hierarchies, there exists the SE symmetries and the Miura links of the SE symmetries, both in the unconstrained and constrained cases. The additional symmetries also coincide with the Miura links. These results indicate that the structures of the Miura links are closely related to the SE symmetries and additional symmetries in the integrable systems.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENT

The first author is supported by Scientific Research Level Improvement Quota Project of Capital University of Economics and Business.

REFERENCES

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[20]S Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.
[22]W Oevel, Darboux theorems and Wronskian formulas for integrable systems: I. constrained KP flows, Phys. A, Vol. 195, 1993, pp. 533-576.
[29]K Tian, Y Ge, and X Zhu, On the q-deformed modified Kadomtsev-Petviashvili hierarchy and its additional symmetries, Roman. Rep. Phys., Vol. 69, 2017, pp. 110.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 1
Pages
53 - 67
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.200922.005How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Siqi Jian
AU  - Jipeng Cheng
PY  - 2020
DA  - 2020/12
TI  - The Miura Links of the Symmetries in the q-Deformed Case
JO  - Journal of Nonlinear Mathematical Physics
SP  - 53
EP  - 67
VL  - 28
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.200922.005
DO  - https://doi.org/10.2991/jnmp.k.200922.005
ID  - Jian2020
ER  -