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

Siqi Jian; Jipeng Cheng

doi:10.2991/jnmp.k.200922.005

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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: 10.2991/jnmp.k.200922.005 How 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 [1,5,9–13,16,18–20,28–32] 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 [3,22–24], 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 [11–13]. 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[θk∂qf],k∈𝕑. (2.1)

For any q-pseudo-differential operator A=∑i≪∞Ai∂qi 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-k∂qkf]∂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 f_A denotes the action of A on f. Some simple notations of operations on the operator A are defined as follows,

A≥k=∑i≥kAi∂qi,A<k=∑i<kAi∂qi,(A)k=Ak∂qk,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=-1q∂1q. (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=∑i≪∞Ai∂qi , we use the notation A|x/q=∑i≪∞[θ-1Ai]qi∂qi .

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+u1∂q-1+u2∂q-2+u3∂q-3+⋯,k=0(L=LqKP),v∂q+v0+v1∂q-1+v2∂q-2+v3∂q-3+⋯,k=1(L=LqmKP), (2.7)

which can be expressed in a terms of a dressing operator

L={W∂qW-1,k=0(L=LqKP),Z∂qZ-1,k=1(L=LqmKP), (2.8)

with

W=1+w1∂q-1+w2∂q-2+⋯, (2.9)

Z=z0+z1∂q-1+z2∂q-2+⋯(z0-1exists). (2.10)

Then the Lax equations are equivalent to

∂tnW=-(LqKPn)<0W=-(W∂qnW-1)<0W, (2.11)

∂tnZ=-(LqmKPn)≤0Z=-(Z∂qnZ-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=0∞xi(i)q!=exp[∑k=0∞(1-q)kk(1-qk)xk] and ξ(t,z)=∑i=1∞tizi . 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)≥k∂q-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)≥kϕ∂q-1), k=0,1, (2.20)

where, in turn for k = 0, 1, ϕ^(k) denotes ϕ or ϕ∂q , and ψ^(k) denotes ψ or ψ∂q* . 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]-1∂q. (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 = L_qKP 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 T_α or T_β,

(i)
L→αL˜=ϕ-1Lϕ , W→αZ=ϕ-1W , ϕi→αϕ˜i=ϕ-1ϕi , ψi→αψ˜i=-[θΩ(ψi,ϕ)] ;
(ii)

L→βL˜=∂q-1ψLψ-1∂q , W→βZ=∂q-1ψW , ϕi→βϕ˜i=Ω(ψ,ϕi) , ψi→βψ˜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 = L_qmKP 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 T_μ or T_ν,

(i)

L→μL˜=z0-1Lz0 , Z→μW=z0-1Z , ϕi→μϕ˜i=z0-1ϕi , ψi→μψ˜i=z0[∂q*ψi] ;
(ii)

L→νL˜=[θz0]-1∂qL∂q-1[θz0] , Z→νW=[θz0]-1∂qZ , ϕi→νϕ˜i=[θz0]-1[∂qϕi] , ψi→νψ˜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=1mqj∂q-1rj∂qk, k=0,1, (2.24)

for some N∈𝕅 and some functions q₁, …, q_m, r₁, …, r_m, 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-KP → constrained q-mKP) Let Ω(.,.) and Ω^(.,.) be defined in (2.20) and (2.21). Let L = L_qKP satisfy the constraint (LN)<0=∑j=1mqj∂q-1rj . Then for any function ϕ or ψ, under the reverse-Miura transformation T_α or T_β,

(i)

L→αL˜=ϕ-1Lϕ , qj→αq˜j=ϕ-1qj , rj→αr˜j=-[θΩ^(rj,ϕ)], j=1,⋯,m , and q˜m+1=ϕ-1[(LN)≥0ϕ]+ϕ-1∑j=1mqjΩ^(rj,ϕ)=ϕ-1[LNϕ], r˜m+1=1 ;
(ii)

L→βL˜=∂q-1ψLψ-1∂q , qj→βq˜j=Ω(ψ,qj) , and rj→βr˜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˜j∂q-1r˜j∂q.

If L, q_j, r_j 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 = L_qmKP satisfy the constraint (LN)<1=∑j=1mqj∂q-1rj∂q . Then for any function z₀, under the Miura transformation T_μ or T_ν,

(i)

L→μL˜=z0-1Lz0 , qj→μq˜j=z0-1qj , rj→μr˜j=-z0[θ-1∂qrj] ;
(ii)

L→νL˜=[θz0]-1∂qL∂q-1[θz0] , qj→νq˜j=[θz0]-1[∂qqj] , rj→νr˜j=[θz0]rj ,

the transformed operator L˜ satisfies the constraint,

(L˜N)<0=∑j=1mq˜j∂q-1r˜j.

If L, q_j, r_j 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 [3,22–24] 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=1m1ϕi∂q-1ψi∂qk,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-M′tn=[M′,Mn], (3.2)

where Mn=(Ln)≥k and M′=∑i=1m1ϕi∂q-1ψi∂qk .

Proof. The Lax equation L_s = [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(Mnϕi∂q-1ψi∂qk)<k-∑i(ϕi∂q-1ψi∂qkMn)<k=∑i(Mnϕi∂q-1)<0ψi∂qk-∑iϕi(∂q-1ψi∂qkMn∂q-k)<0∂qk=(3.4)∑i(Mnϕi)0∂q-1ψi∂qk-∑iϕi∂q-1((ψi∂qkMn∂q-k)*)0∂qk=∑i[Mnϕi]∂q-1ψi∂qk-∑iϕi∂q-1[∂q-k*(Ln)≥1*∂qk*ψi]∂qk=∑i(ϕi∂q-1ψi∂qk)tn,

which equals to

M′tn=[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=[M′tn-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=[∑iϕi∂q-1ψi∂q,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-KP → q-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=[∑iϕi∂q-1ψi,L] . Then under the reverse-Miura transformation T_α or T_β in Theorem 2.1, L˜ also satifies the SE symmetry

L˜s=[∑iϕ˜i∂q-1ψ˜i∂q,L˜],

where L˜ , ϕ˜i and ψ˜i are defined in Theorem 2.1.

Proof. For the case T_α, 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[∑iϕi∂q-1ψi,L]ϕ=[∑iϕ-1ϕi[θ-1ψ˜i]+ϕ-1ϕi∂q-1ψiϕ,L˜]=[∑iϕ˜i[θ-1ψ˜i]-ϕ˜i∂q-1[θ-1ψ˜i]∂q,L˜]=[∑iϕ˜i∂q-1(∂q[θ-1ψ˜i]-[θ-1ψ˜i]∂q),L˜]=[∑iϕ˜i∂q-1ψ˜i∂q,L˜].

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

L˜s=(∂q-1ψLnψ-1∂q)s=∂q-1(ψLψ-1)s∂q=∂q-1[ψ-1ψs,ψLψ-1]∂q+∂q-1ψLsψ-1∂q=∑i∂q-1[ψ-1ψi[θΩ(ψ,ϕi)],ψLψ-1]∂q+∑i∂q-1ψ[ϕi∂q-1ψi,L]ψ-1∂q=∑i[∂q-1ψ˜i[θϕ˜i]∂q,L˜]+∑i[∂q-1ψϕi∂q-1ψ˜i∂q,L˜]=∑i[∂q-1([θϕ˜i]+[∂qϕ˜i]∂q-1)ψ˜i∂q,L˜]=∑i[∂q-1∂qϕ˜i∂q-1ψ˜i∂q,L˜]=∑i[ϕ˜i∂q-1ψ˜i∂q,L˜].

Theorem 3.3.

(Miura transformation: q-mKP → q-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=[∑iϕi∂q-1ψi∂q,L] . Then under the Miura transformation T_μ or T_ν defined in Theorem 2.2, L˜ also satifies the SE symmetry

L˜s=[∑iϕ˜i∂q-1ψ˜i,L˜].

Here L˜ , ϕ˜i and ψ˜i are defined in Theorem 2.2.

Proof. For the case T_μ, 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[ϕi∂q-1ψi∂q,L]z0=∑i[z0-1ϕi(-[θ-1ψi]+∂q-1ψi∂q)z0,L˜]=∑i[z0-1ϕi∂q-1[∂q*ψi]z0,L˜]=∑i[ϕ˜i∂q-1ψ˜i.L˜].

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

L˜s=([θz0]-1∂qL∂q-1[θz0])s=-[[θz0]-1[θz0]s,L˜]+[θz0]-1∂qLs∂q-1[θz0]=[-[θz0]-1[θz0]s,L˜]+∑i[θz0]-1∂q[ϕi∂q-1ψi∂q,L]∂q-1[θz0]=∑i[-[θz0]-1[θϕi]ψi[θz0],L˜]+∑i[[θz0]-1∂qϕi∂q-1ψi[θz0],L˜]=∑i[[θz0]-1(-[θϕi]+∂qϕi∂q-1)ψi[θz0],L˜]=∑i[[θz0]-1[∂qϕi]∂q-1ψi[θz0],L˜]=∑i[ϕ˜i∂q-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 ϕ_i, ψ_i, 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 ϕ_i or ϕi∂q , ψi(k) denotes ψ_i or ψi∂q* , qj(k) denotes q_j or qj∂q and rj(k) denotes r_j or rj∂q* . The constraint (2.24) is invariant under the SE symmetry flow,

Ls=[∑i=1m1ϕi∂q-1ψi∂qk,L], qjs=∑i=1m1ϕiΩ^(ψi,qj(k)), rjs=∑i=1m1[θΩ(rj(k),ϕi)]ψi. (4.2)

Proof. By the direct computation,

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

With

((LN)<k)s=[∑i=1m1ϕi∂q-1ψi∂qk,LN]<k=[∑i=1m1ϕi∂q-1ψi∂qk,(LN)≥k]<k+[∑i=1m1ϕi∂q-1ψi∂qk,(LN)<k]<k,

[∑i=1m1ϕi∂q-1ψi∂qk,(LN)≥k]<k=(∑i=1m1ϕi∂q-1ψi∂qk(LN)≥k-∑i=1m1(LN)≥kϕi∂q-1ψi∂qk)<k=∑i=1m1ϕi(∂q-1ψi∂qk(LN)≥k∂q-k)<0∂qk-∑i=1m1((LN)≥kϕi∂q-1ψi)<0∂qk=(3.4)∑i=1m1ϕi∂q-1(ψi∂qk(LN)≥k∂q-k)0*∂qk-∑i=1m1[(LN)≥kϕi]∂q-1ψi∂qk=∑i=1m1ϕi∂q-1[∂q-k*(LN)≥k*∂qk*ψi]∂qk-∑i=1m1[(LN)≥kϕi]∂q-1ψi∂qk

and

[∑i=1m1ϕi∂q-1ψi∂qk,(LN)<k]<k=(2.24)[∑i=1m1ϕi∂q-1ψi∂qk,∑j=1mqj∂q-1rj∂qk]<k=(∑i,jϕi∂q-1ψi∂qkqj∂q-1rj∂qk-∑i,jqj∂q-1rj∂qkϕi∂q-1ψi∂qk)<k=(4.3)(∑i,jϕiΩ^(ψi,qj(k))∂q-1rj∂qk-∑i,jϕi∂q-1[θΩ(ψi(k),qj)]rj∂qk)<k-(∑i,jqjΩ^(rj,ϕi(k))∂q-1ψi∂qk-∑i,jqj∂q-1[θΩ(rj(k),ϕi)]ψi∂qk)<k=∑i,jϕiΩ^(ψi,qj(k))∂q-1rj∂qk-∑i,jϕi∂q-1[θΩ(ψi(k),qj)]rj∂qk-∑i,jqjΩ^(rj,ϕi(k))∂q-1ψi∂qk+∑i,jqj∂q-1[θΩ(rj(k),ϕi)]ψi∂qk,

by the Eq. (4.1), we have

((LN)<k)s=∑i,jϕiΩ^(ψi,qj(k))∂q-1rj∂qk+∑i,jqj∂q-1[θΩ(rj(k),ϕi)]ψi∂qk=∑jqjs∂q-1rj∂qk+∑jqj∂q-1rjs∂qk=(∑jqj∂q-1rj∂qk)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-KP → constrained q-mKP) Let L = L_qKP satisfy the constraint (LN)<0=∑j=1mqj∂q-1rj . Then for any functions ϕ and ψ satisfying (3.6, k = 0), under the reverse-Miura transformation T_α or T_β defined in Theorems 2.1 and 2.3, if ϕ1,⋯,ϕm1 , ψ1,⋯,ψm1 , q₁, ..., q_m and r₁, ..., r_m satisfy

[(LN)≥0ϕi]+∑j=1mqjΩ^(rj,ϕi)=λiϕi,[(LN)≥0*ψi]-∑j=1m[θΩ(ψi,qj)]rj=λiψi, (4.4)

Ls=[∑i=1m1ϕi∂q-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˜i∂q)=λiϕ˜i, [∂q-1*(L˜N)≥1*∂q*ψ˜i]-∑j=1m+1[θΩ(ψ˜i∂q*,q˜j)]r˜j=λiψ˜i, (4.6)

L˜s=[∑i=1m1ϕ˜i∂q-1ψ˜i∂q,L˜], q˜js=∑i=1m1ϕ˜iΩ^(ψ˜i,q˜j∂q), r˜js=∑i=1m1[θΩ(r˜j∂q*,ϕ˜i)]ψ˜i. (4.7)

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

∑j=1mq˜jΩ^(r˜j,ϕ˜i∂q)=∑j=1mϕ-1qjΩ^(-[θΩ^(rj,ϕ)],ϕ˜i∂q)

=(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˜i∂q)=ϕ-1[(LN)≥0ϕ]ϕ-1ϕi+ϕ-1∑j=1mqjΩ^(rj,ϕ)ϕ-1ϕi,

we have

∑j=1m+1q˜jΩ^(r˜j,ϕ˜i∂q)=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˜i∂q)=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*ϕ∑j=1m[θΩ(ψi,qj)]rj]+[∂q-1*ϕλiψi]-[∂q-1*[(LN)≥0ϕ]ψi]

and

-∑j=1m+1[θΩ(ψ˜i∂q*,q˜j)]r˜j=∑j=1m[θΩ(ψiϕ,ϕ-1qj)][θΩ^(rj,ϕ)]-[θΩ(ψiϕ,ϕ-1[(LN)≥0ϕ]+ϕ-1∑j=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˜i∂q*,q˜j)]r˜j=[∂q-1*ϕλiψi]=λiψ˜i,

by using the equality

∑j=1m[θΩ(ψi,qj)][θΩ^(rj,ϕ)]+[∂q-1*ϕ∑j=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-1∑i=1m1ϕiΩ^(ψi,qj)

and

∑i=1m1ϕ˜iΩ^(ψ˜i,q˜j∂q)=∑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-2∑i=1m1ϕiΩ^(ψi,ϕ)[Lnϕ]+ϕ-1∑i=1m1[ϕi∂q-1ψiLnϕ]-ϕ-1∑i=1m1[Lnϕi∂q-1ψiϕ]+ϕ-1∑i=1m1[LnϕiΩ^(ψi,ϕ)]=-ϕ-2∑i=1m1ϕiΩ^(ψi,ϕ)[Lnϕ]+ϕ-1∑i=1m1[ϕi∂q-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˜j∂q*,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 r_m₊₁ = 1, it is trivial that the third equation in (4.7) holds.

Theorem 4.3.

(Miura transformation: constrained q-KP → constrained q-mKP) Let L = L_qmKP satisfy the constraint (LN)<1=∑j=1mqj∂q-1rj∂q . Then for any function z₀ satisfying (3.9), under the Miura transformation T_μ or T_ν defined in Theorems 2.2 and 2.4, if ϕ1,⋯,ϕm1 , ψ1,⋯,ψm1 , q₁, ..., q_m and r₁, ..., r_m satisfy

[(LN)≥1ϕi]+∑j=1mqjΩ^(rj,ϕi∂q)=λiϕi, [∂q-1*(LN)≥1*∂q*ψi]-∑j=1m[θΩ(ψi∂q*,qj)]rj=λiψi, (4.8)

Ls=[∑i=1m1ϕi∂q-1ψi∂q,L], qjs=∑i=1m1ϕiΩ^(ψi,qj∂q), rjs=∑i=1m1[θΩ(rj∂q*,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˜i∂q-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 T_μ, and the proof of the case T_ν 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]=-z0-1∑j=1mqjΩ^(rj,ϕi∂q)+λiz0-1ϕi+z0-1∑j=1mqj[θ-1rj]ϕi,

we have the first equation in (4.10) by

([θ-1rj]ϕi)∂q=rjϕi∂q+[θ-1∂qrj]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]]+z0∑j=1mqj[θ-1rj][∂q*ψi]=z0[∂q*∂q-1*(LN)≥1*∂q*ψi]+z0∑j=1mqj[θ-1rj][∂q*ψi]=z0∑j=1m[∂q*[θΩ(ψi∂q*,qj)]rj]+λiz0[∂q*ψi]+z0∑j=1mqj[θ-1rj][∂q*ψi]=z0∑j=1m[θΩ(ψi∂q*,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]=-z0∑j=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=-z0-1qj∑i=1m1ϕi[θ-1ψi]+z0-1∑i=1m1ϕiΩ^(ψi,qj∂q)

and

∑i=1m1ϕ˜iΩ^(ψ˜i,q˜j)=∑i=1m1z0-1ϕiΩ(z0[∂q*ψi],z0-1qj)=(2.21)z0-1∑i=1m1ϕiΩ(ψi,qj∂q)-z0-1∑i=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]+z0∑i=1m1[∂q*[θΩ(rj∂q*,ϕi)]ψi]=z0∑i=1m1[θΩ(rj∂q*,ϕi)][∂q*ψi]

and

∑i=1m1[θΩ(r˜j,ϕ˜i)]ψ˜i=∑i=1m1[θΩ(z0[∂q*rj],z0-1ϕi)]z0[∂q*ψi]=z0∑i=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 M_q [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 Γ_q is defined by

Γq=∑i=1∞[iti+(1-q)i1-qixi]. (5.3)

Γ_q 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 = L_qKP have the dressing form L = W∂_qW^–1. Let ϕ and ψ be the (adjoint) eigenfuntions respectively. Then under the reverse-Miura transformation T_α or T_β defined in Theorem 2.1,

(i)

W→αZ=ϕ-1W , Mq→αM˜q=ϕ-1Mqϕ ;
(ii)

W→βZ=∂q-1ψW , Mq→βM˜q=∂q-1ψMqψ-1∂q ,

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 T_α or T_β, 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 T_α.

∂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 T_β.

Note that

-(M˜qmL˜l)≤0Z=-(∂q-1ψMqmLlψ-1∂q)≤0Z=-∂q-1ψ(MqmLl)<0ψ-1∂qZ-∂q-1[(MqmLl)≥0*ψ]ψ-1∂qZ=-∂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 = L_qmKP have the dressing form L = ∂_qZ^–1. Let z₀ be the zero term in the dressing operator Z. Then under the Miura transformation T_μ or T_ν defined in Theorem 2.2,

(i)
Z→μW=z0-1Z , Mq→μM˜q=z0-1Mqz0 ;
(ii)
Z→νW=[θz0]-1∂qZ , Mq→νM˜q=[θz0]-1∂qMq∂q-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 T_μ or T_ν, 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 T_μ.

∂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 T_ν.

Note that

-(M˜qmL˜l)<0W=-([θz0]-1∂qMqmLl∂q-1[θz0])<0[θz0]-1∂qZ=-[θz0]-1(∂qMqmLl∂q-1)<0∂qZ=-[θz0]-1[∂q(MqmLl)=0]Z-[θz0]-1∂q(MqmLl)<0Z,

then we have

∂ml*W=∂ml*([θz0]-1∂qZ)=∂ml*([θz0]-1)∂qZ+[θz0]-1∂q[∂ml*Z]=-[θz0]-1[∂ml*θz0][θz0]-1∂qZ-[θz0]-1∂q(MqmLl)≤0Z=(5.6)[θz0]-1[θ(MqmLl)=0][θz0][θz0]-1∂qZ-[θz0]-1∂q(MqmLl)=0Z-[θz0]-1∂q(MqmLl)<0Z=-[θz0]-1[∂q(MqmLl)=0]Z-[θz0]-1∂q(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

[1]M Adler, E Horozov, and P van Moerbeke, The solution to the q-KdV equation, Phys. Lett. A, Vol. 242, 1998, pp. 139-151.

[2]M Adler, T Shiota, and P van Moerbeke, A Lax representation for the vertex operator and the central extension, Commun. Math. Phys., Vol. 171, 1995, pp. 547-588.

[3]H Aratyn, E Nissimov, and S Pacheva, Method of squared eigenfunction potentials in integrable hierarchies of KP type, Commun. Math. Phys., Vol. 193, 1998, pp. 493-525.

[4]K Chen, X Deng, and Dj Zhang, Symmetry constraint of the differential-difference KP hierarchy and a second discretization of the ZS-AKNS system, J. Nonlin. Math. Phys., Vol. 24, 2017, pp. 18-35.

[5]J Cheng, Miura and auto-Backlund transformations for the q-deformed KP and q-deformed modified KP hierarchies, J. Nonlin. Math. Phys., Vol. 24, 2017, pp. 7-19.

[6]J Cheng and J He, On the squared eigenfunction symmetry of the Toda lattice hierarchy, J. Math. Phys., Vol. 54, 2013. 023511.

[7]J Cheng, J He, and S Hu, The “ghost” symmetry of the BKP hierarchy, J. Math. Phys., Vol. 51, 2010. 053514.

[8]LA Dickey, On additional symmetries of the KP hierarchy and Sato’s Backlund transformation, Commun. Math. Phys., Vol. 167, 1995, pp. 227-233.

[9]E Frenkel, Deformations of the KdV hierarchy and related soliton equations, Int. Math. Res. Not., Vol. 1996, 1996, pp. 55-76.

[10]L Geng, H Chen, N Li, and J Cheng, The gauge transformations of the constrained q-deformed KP hierarchy, Mod. Phys. Lett. B, Vol. 32, 2018. 1850176.

[11]J He, Y Li, and Y Cheng, q-deformed KP hierarchy and q-deformed constrained KP hierarchy, SIGMA, Vol. 2, 2006, pp. 1-32.

[12]P Iliev, Tau function solutions to a q-deformation of the KP hierarchy, Lett. Math. Phys., Vol. 44, 1998, pp. 187-200.

[13]V Kac and P Cheung, Quantum Calculus, Springer-Verlag, New York, 2002.

[14]BG Konopelchenko, On the gauge-invariant description of the evolution equations integrable by Gelfand-Dikij spectral problems, Phys. Lett. A, Vol. 92, 1982, pp. 323-327.

[15]BA Kupershmidt, Canonical property of the Miura maps between the MKP and KP hierarchies, continuous and discrete, Commun. Math. Phys., Vol. 167, 1995, pp. 351-371.

[16]C Li, Sato theory on the q-Toda hierarchy and its extension, Chaos Solitons Fractals, Vol. 76, 2015, pp. 10-23.

[17]C Li, J Cheng, K Tian, M Li, and J He, Ghost symmetry of the discrete KP hierarchy, Monatsh. Math., Vol. 180, 2016, pp. 815-832.

[18]R Lin, X Liu, and Y Zeng, A new extended q-deformed KP hierarchy, J. Nonlin. Math. Phys., Vol. 15, 2008, pp. 333-347.

[19]R Lin, H Peng, and M Mañas, The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons, J. Phys. A Math. Theor., Vol. 43, 2010. 434022.

[20]S Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.

[21]RM Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys., Vol. 9, 1968, pp. 1202-1204.

[22]W Oevel, Darboux theorems and Wronskian formulas for integrable systems: I. constrained KP flows, Phys. A, Vol. 195, 1993, pp. 533-576.

[23]W Oevel and S Carillo, Squared eigenfunction symmetries for soliton equations: part I, J. Math. Anal. Appl., Vol. 217, 1998, pp. 161-178.

[24]W Oevel and S Carillo, Squared eigenfunction symmetries for soliton equations: part II, J. Math. Anal. Appl., Vol. 217, 1998, pp. 179-199.

[25]W Oevel and W Strampp, Constrained KP hierarchy and bi-Hamiltonian structures, Commun, Math. Phys., Vol. 157, 1993, pp. 51-81.

[26]AY Orlov and EI Schulman, Additional symmetries for integrable equations and conformal algebra representation, Lett. Math. Phys., Vol. 12, 1986, pp. 171-179.

[27]JC Shaw and MH Tu, Miura and auto-Bäcklund transformations for the cKP and cmKP hierarchies, J. Math. Phys., Vol. 38, 1997, pp. 5756-5773.

[28]K Takasaki, q-Analogue of modified KP hierarchy and its quasi-classical Limit, Lett. Math. Phys., Vol. 72, 2005, pp. 165-181.

[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.

[30]K Tian, J He, and Y Su, Symmetric q-deformed KP hierarchy, Chin. Ann. Math. Ser. B, Vol. 36, 2015, pp. 1-10.

[31]MH Tu, q-Deformed KP hierarchy: its additional symmetries and infinitesimal Bäcklund transformations, Lett. Math. Phys., Vol. 49, 1999, pp. 95-103.

[32]S Wang, K Wu, X Wu, and D Yu, The q-deformation of AKNS-D hierarchy, J. Phys. A: Math. Gen., Vol. 34, 2001, pp. 9641-9651.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 28 - 1
Pages: 53 - 67
Publication Date: 2020/12/10
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.2991/jnmp.k.200922.005 How to use a DOI?
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ris enw bib

TY  - JOUR
AU  - Siqi Jian
AU  - Jipeng Cheng
PY  - 2020
DA  - 2020/12/10
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  - 10.2991/jnmp.k.200922.005
ID  - Jian2020
ER  -

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