# On the Algebra ofq-Deformed Pseudodifferential Operators

Authors: Abderrahman EL Boukili; Moulay Brahim Sedra
Journal: ISRN High Energy Physics
 (2012)

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## Abstract

While basing on the study that we we achieved on pseudodifferential operators in the works [arXiv:0708.4046 and hep-th/0610056 ], we interest in this paper to the construction of the algebra ofq-deformed pseudodifferential operators. We use this algebraic structure to study in particular q-Burgers and q-KdV differential operators by the Lax generating technique. We give q-deformed Lax equations as well as the report between these equations through the q-deformed Burgers-KdV mapping.


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## Body

## 1. Basic Notions

### 1.1.q-Pseudodifferential Operators

We start this part with defining theq-derivation. For it, we are going to introduce the general case to know the α-derivation that is defined by(1.1)∂α(fg)=α(f)dα(g)+dα(f)g,
where the two functions f and g are polynomials in an indeterminant x and its inverse x-1.In (1.1), α is a linear mapping. An example of the α-derivation is given by Jackson's q-differential operator ∂q, such as [1](1.2)∂q(f)=f(qx)-f(x)(q-1)x,
which gives the following form for (1.1):(1.3)∂q(fg)=ηq(f)⋅∂q(g)+∂q(f)⋅g.Theq-shift operator ηq is given by(1.4)ηq(f(x))=f(qx).One can define the commutation relation as follows:(1.5)[f,g]=f∘g-g∘f,
where the multiplication law “∘” is(1.6)∂q∘f=ηq(f)∂q+∂qf,∂q-1∘f=∑k⪰0(-1)kq-k(k+1)/2ηq-k-1(∂qkf)∂q-k-1.
The last equation are obtained by using the following relation:(1.7)∂q-1∘∂q∘f=∂q∘∂q-1∘f=f,
where ∂q-1 is the formal inverse of ∂q.We should note thatηq does not commute with ∂q,(1.8)∂q(ηqk(f))=qkηqk(∂qf),k∈Z
or in the following general case:(1.9)∂qm(ηqk(f))=qk+mηqk(∂qmf),k,m∈Z.Note that (1.6) can be unified as follows:(1.10)∂qn∘f=∑k≥0(nk)qηqn-k(∂qkf)∂qn-k,
for all n. In the last equation, the q-binomials take the form(1.11)(nk)q=(n)q(n-1)q⋯(n-k+1)q(1)q(2)q⋯(k)q,
and the q-numbers are given by(1.12)(n)q=qn-1q-1,
where the convention(1.13)(n0)q=1,
is taken.We can write out several explicit forms of (1.10) for q-derivative ∂qn and ∂q-n(n≥0) as(1.14)∂q∘f=(∂qf)+ηq(f)∂q,∂q2∘f=(∂q2f)+(q+1)ηq(∂qf)∂q+ηq2(f)∂q2,∂q3∘f=(∂q3f)+(q2+q+1)ηq(∂q2f)∂q+(q2+q+1)ηq2(∂qf)∂q2+ηq3(f)∂q3,∂q-1∘f=ηq-1(f)∂q-1-q-1ηq-2(∂qf)∂q-2+q-3ηq-3(∂q2f)∂q-3-q-6ηq-4(∂q3f)∂q-4+1q10ηq-5(∂q4f)∂q-5+⋯+(-1)kq-(1+2+3+⋯+k)ηq-k-1(∂qkf)∂q-k-1+⋯,∂q-2∘f=ηq-2(f)∂q-2-1q2(2)qηq-3(∂qf)∂q-3+1q(2+3)(3)qηq-4(∂q2f)∂q-4-1q(2+3+4)(4)qηq-5(∂q3f)∂q-5+⋯+(-1)kq(2+3+⋯+k+1)(k+1)qηq-2-k(∂qkf)∂q-2-k+⋯.
We also add that the residue of the symbol ℒ(x,∂q) can be written as(1.15)Res(∑i=-∞Nui(x)∂qi)=u-1(x),
and its Tr-functional is(1.16)Tr(∑i=-∞Nui(x)∂qi)=∫S1u-1(x)dx.


### 1.2. Algebraic Structure ofq-PDO

Now let us introduce theq-pseudodifferential operators algebra q-PDO. The latter is characterized by the relation [1]:(1.17)q-PDO={L(x,∂q)=∑i=-∞Nui(x)∂qi}.We can noted this space in the following way𝔮𝒜≡q-ΨDO is seen as being the algebra of all local and nonlocal q-differential operators of arbitrary conformal spins and arbitrary degrees, this spaces can be seen as being the q-deformation of pseudodifferential algebra 𝒜 that we saw in [2–11]. One may expand 𝔮𝒜 as(1.18)qA=⊕m≤nqA(m,n)=⊕m≤n⊕s∈ZqAs(m,n),m,n,s,∈Z,
where we have denoted by (m,n) the lowest and the highest degrees, respectively, and by s the conformal spin. To be explicit, consider the space 𝔮𝒜s(m,n) of q-differential operators:(1.19)Ls(m,n)=∑i=mnus-i(z)∂qi.The vector space𝔮𝒜(m,n) of q-differential operators with given degrees (m,n) but undefined spin(1.20)qA(m,n)=⊕s∈ZqAs(m,n)
exhibits a Lie algebra's structure with respect to the Lie bracket for m≤n≤1.In fact, It's straightforward to check that the commutator of two operators of𝔮𝒜s(p,q) is an operator of conformal spin 2s and degrees (p,2q-1). Since the Lie bracket [·,·] acts as(1.21)[⋅,⋅]:qAs(m,n)×qAs(m,n)⟶qA2s(m,2n-1),
imposing the closure, one gets strong constraints on the spin s and the degrees parameters (m,n), namely,(1.22)s=0,m≤n≤1.
From these equations, we learn in particular that the spaces 𝔮𝒜0(m,n),m≤n≤1 admit a Lie algebra's structure with respect to the bracket (1.5) provided that the Jacobi identity is fulfilled. This can be ensured by showing that the Leibnitz product is associative.The spaces𝔮𝒜0(m,n),m≤n≤1 as well as the vector space 𝔮𝒜0(0,1) are in fact subalgebra of the Lie algebra 𝔮𝒜0(-∞,1) which can be decomposed as(1.23)qA0(-∞,1)=qA0(-∞,-1)⊕qA0(0,1).𝔮A0(-∞,-1) is nothing but the Lie algebra of Lorentz scalar pure q-pseudodifferential operators of higher degree n=-1 and 𝔮𝒜0(0,1) is the central extension of the Lie algebra 𝔮𝒜0(1,1) of vector fields Diff(S1):(1.24)qA0(0,1)=qA0(0,0)⊕qA0(1,1),
and where 𝔮𝒜0(0,0)≊𝒜0(0,0) is the one dimensional trivial ideal.The infinite dimensional huge space𝔮𝒜 is the algebra of q-differential operators of arbitrary spins and arbitrary degrees. It’s obtained from the space 𝔮𝒜(m,n) by summing over all allowed degrees 𝔮𝒜:(1.25)qA=⊕m≤nqA(m,n)=⊕m∈Z[⊕k∈NqA(m,m+k)]=⊕m∈Z[⊕k∈N[⊕s∈ZqAs(m,m+k)]].
This infinite dimensional space which is the combined conformal spin and degrees tensor algebra is closed under the Lie bracket without any constraint.A remarkable property of𝔮𝒜 is that it can splits into six infinite subalgebras 𝔮𝒜j+ and 𝔮𝒜j-,j=0,±1 related to each others by conjugation of the spin and degrees. Indeed given two integers m and n≥m, it is not difficult to see that the vector spaces 𝔮𝒜(m,n) and 𝔮𝒜(-n-1,-m-1) are dual with respect to the pairing product (·,·) defined as(1.26)(L(m,n),L(α,β))=δ0,1+m+βδ0,1+n+αRes[L(m,n)∘L(α,β)],
where d(α,β) are q-differential operators with fixed degrees (α,β;β≥α) but arbitrary spin and where the residue operation res is defined as:(1.27)Res(∂qi)=δ0,i+1.
This equation shows that the operation res exhibits a conformal spin Δ=1. Using the properties of this operation and the pairing product (1.26), one can decompose 𝔮𝒜 as follows:(1.28)qA=qA+⊕qA-
with(1.29)qA+=⊕m≥0[⊕k∈NqA(m,m+k)],(1.30)qA-=⊕m≥0[⊕k∈NqA(-m-k-1,-m-1)].
The indices + and − carried by 𝔮𝒜+ and 𝔮𝒜- refer to the positive (local) and negative (nonlocal) degrees respectively. On the other hand one can decomposes the space 𝔮𝒜(m,m+k),k≥0 as(1.31)qA(m,m+k)=qΣ-(m,m+k)⊕qΣ0(m,m+k)⊕qΣ+(m,m+k).qΣ-(m,m+k) and qΣ+(m,m+k) denote the spaces of q-differential operators of negative and positive definite spin. They are read as(1.32)qΣ-(m,m+k)=⊕s>0qA-s(m,m+k),(1.33)qΣ0(m,m+k)=qA0(m,m+k),(1.34)qΣ+(m,m+k)=⊕s>0qAs(m,m+k).qΣ0(m,m+k) is just the vector space of Lorenz scalar q-differential operators. Combining (1.28)–(1.34), one sees that 𝔮𝒜 decomposes into 6=3×2 subalgebras(1.35)qA=⊕j=0,+,-[qAj+⊕qAj-]
with(1.36)qAj+=⊕m≥0[⊕k∈NqΣj(m,m+k)],qAj-=⊕m≥0[⊕k∈NqΣj(-m-k-1,-m-1)].
The duality of these 6=3×2 subalgebras is described by the combined scalar product 〈〈·,·〉〉 built out of the product equation(1.26) and conformal spin pairing:(1.37)〈uk,ul〉∶=∫dzuk(z)u1-k(z)δk+l,1,
as follows [2, 3]:(1.38)〈〈Ls(α,β),Lr(m,n)〉〉∶=δ0,r+sδ0,1+n+αδ0,1+m+β∫dzres[Ls(α,β)∘L-s(-β-1,-α-1)]
with respect to this new product, 𝔮𝒜++, 𝔮𝒜0+, and 𝔮𝒜-+ behave as the dual algebras of 𝔮𝒜--, 𝔮𝒜0-, and 𝔮𝒜+-, respectively, while 𝔮𝒜0- is just the algebra of Lorenz scalar pure q-pseudo operators. This algebra and its dual 𝔮𝒜0+, the space of Lorenz scalar local q-differential operators, are very special subalgebras as they are systematically used to construct new realizations of the wi-symmetry, i≥2 by using scalar q-differential operators type(1.39)L(k)(a)=a-k(a)∂qk.We note that the space𝔮𝒜++ is the algebra of local q-differential operators of positive definite spins and positive degrees. 𝔮𝒜--, however, is the Lie algebra of pure q-pseudodifferential operators of negative degrees and spins.


## 1.1.q-Pseudodifferential Operators

We start this part with defining theq-derivation. For it, we are going to introduce the general case to know the α-derivation that is defined by(1.1)∂α(fg)=α(f)dα(g)+dα(f)g,
where the two functions f and g are polynomials in an indeterminant x and its inverse x-1.In (1.1), α is a linear mapping. An example of the α-derivation is given by Jackson's q-differential operator ∂q, such as [1](1.2)∂q(f)=f(qx)-f(x)(q-1)x,
which gives the following form for (1.1):(1.3)∂q(fg)=ηq(f)⋅∂q(g)+∂q(f)⋅g.Theq-shift operator ηq is given by(1.4)ηq(f(x))=f(qx).One can define the commutation relation as follows:(1.5)[f,g]=f∘g-g∘f,
where the multiplication law “∘” is(1.6)∂q∘f=ηq(f)∂q+∂qf,∂q-1∘f=∑k⪰0(-1)kq-k(k+1)/2ηq-k-1(∂qkf)∂q-k-1.
The last equation are obtained by using the following relation:(1.7)∂q-1∘∂q∘f=∂q∘∂q-1∘f=f,
where ∂q-1 is the formal inverse of ∂q.We should note thatηq does not commute with ∂q,(1.8)∂q(ηqk(f))=qkηqk(∂qf),k∈Z
or in the following general case:(1.9)∂qm(ηqk(f))=qk+mηqk(∂qmf),k,m∈Z.Note that (1.6) can be unified as follows:(1.10)∂qn∘f=∑k≥0(nk)qηqn-k(∂qkf)∂qn-k,
for all n. In the last equation, the q-binomials take the form(1.11)(nk)q=(n)q(n-1)q⋯(n-k+1)q(1)q(2)q⋯(k)q,
and the q-numbers are given by(1.12)(n)q=qn-1q-1,
where the convention(1.13)(n0)q=1,
is taken.We can write out several explicit forms of (1.10) for q-derivative ∂qn and ∂q-n(n≥0) as(1.14)∂q∘f=(∂qf)+ηq(f)∂q,∂q2∘f=(∂q2f)+(q+1)ηq(∂qf)∂q+ηq2(f)∂q2,∂q3∘f=(∂q3f)+(q2+q+1)ηq(∂q2f)∂q+(q2+q+1)ηq2(∂qf)∂q2+ηq3(f)∂q3,∂q-1∘f=ηq-1(f)∂q-1-q-1ηq-2(∂qf)∂q-2+q-3ηq-3(∂q2f)∂q-3-q-6ηq-4(∂q3f)∂q-4+1q10ηq-5(∂q4f)∂q-5+⋯+(-1)kq-(1+2+3+⋯+k)ηq-k-1(∂qkf)∂q-k-1+⋯,∂q-2∘f=ηq-2(f)∂q-2-1q2(2)qηq-3(∂qf)∂q-3+1q(2+3)(3)qηq-4(∂q2f)∂q-4-1q(2+3+4)(4)qηq-5(∂q3f)∂q-5+⋯+(-1)kq(2+3+⋯+k+1)(k+1)qηq-2-k(∂qkf)∂q-2-k+⋯.
We also add that the residue of the symbol ℒ(x,∂q) can be written as(1.15)Res(∑i=-∞Nui(x)∂qi)=u-1(x),
and its Tr-functional is(1.16)Tr(∑i=-∞Nui(x)∂qi)=∫S1u-1(x)dx.


## 1.2. Algebraic Structure ofq-PDO

Now let us introduce theq-pseudodifferential operators algebra q-PDO. The latter is characterized by the relation [1]:(1.17)q-PDO={L(x,∂q)=∑i=-∞Nui(x)∂qi}.We can noted this space in the following way𝔮𝒜≡q-ΨDO is seen as being the algebra of all local and nonlocal q-differential operators of arbitrary conformal spins and arbitrary degrees, this spaces can be seen as being the q-deformation of pseudodifferential algebra 𝒜 that we saw in [2–11]. One may expand 𝔮𝒜 as(1.18)qA=⊕m≤nqA(m,n)=⊕m≤n⊕s∈ZqAs(m,n),m,n,s,∈Z,
where we have denoted by (m,n) the lowest and the highest degrees, respectively, and by s the conformal spin. To be explicit, consider the space 𝔮𝒜s(m,n) of q-differential operators:(1.19)Ls(m,n)=∑i=mnus-i(z)∂qi.The vector space𝔮𝒜(m,n) of q-differential operators with given degrees (m,n) but undefined spin(1.20)qA(m,n)=⊕s∈ZqAs(m,n)
exhibits a Lie algebra's structure with respect to the Lie bracket for m≤n≤1.In fact, It's straightforward to check that the commutator of two operators of𝔮𝒜s(p,q) is an operator of conformal spin 2s and degrees (p,2q-1). Since the Lie bracket [·,·] acts as(1.21)[⋅,⋅]:qAs(m,n)×qAs(m,n)⟶qA2s(m,2n-1),
imposing the closure, one gets strong constraints on the spin s and the degrees parameters (m,n), namely,(1.22)s=0,m≤n≤1.
From these equations, we learn in particular that the spaces 𝔮𝒜0(m,n),m≤n≤1 admit a Lie algebra's structure with respect to the bracket (1.5) provided that the Jacobi identity is fulfilled. This can be ensured by showing that the Leibnitz product is associative.The spaces𝔮𝒜0(m,n),m≤n≤1 as well as the vector space 𝔮𝒜0(0,1) are in fact subalgebra of the Lie algebra 𝔮𝒜0(-∞,1) which can be decomposed as(1.23)qA0(-∞,1)=qA0(-∞,-1)⊕qA0(0,1).𝔮A0(-∞,-1) is nothing but the Lie algebra of Lorentz scalar pure q-pseudodifferential operators of higher degree n=-1 and 𝔮𝒜0(0,1) is the central extension of the Lie algebra 𝔮𝒜0(1,1) of vector fields Diff(S1):(1.24)qA0(0,1)=qA0(0,0)⊕qA0(1,1),
and where 𝔮𝒜0(0,0)≊𝒜0(0,0) is the one dimensional trivial ideal.The infinite dimensional huge space𝔮𝒜 is the algebra of q-differential operators of arbitrary spins and arbitrary degrees. It’s obtained from the space 𝔮𝒜(m,n) by summing over all allowed degrees 𝔮𝒜:(1.25)qA=⊕m≤nqA(m,n)=⊕m∈Z[⊕k∈NqA(m,m+k)]=⊕m∈Z[⊕k∈N[⊕s∈ZqAs(m,m+k)]].
This infinite dimensional space which is the combined conformal spin and degrees tensor algebra is closed under the Lie bracket without any constraint.A remarkable property of𝔮𝒜 is that it can splits into six infinite subalgebras 𝔮𝒜j+ and 𝔮𝒜j-,j=0,±1 related to each others by conjugation of the spin and degrees. Indeed given two integers m and n≥m, it is not difficult to see that the vector spaces 𝔮𝒜(m,n) and 𝔮𝒜(-n-1,-m-1) are dual with respect to the pairing product (·,·) defined as(1.26)(L(m,n),L(α,β))=δ0,1+m+βδ0,1+n+αRes[L(m,n)∘L(α,β)],
where d(α,β) are q-differential operators with fixed degrees (α,β;β≥α) but arbitrary spin and where the residue operation res is defined as:(1.27)Res(∂qi)=δ0,i+1.
This equation shows that the operation res exhibits a conformal spin Δ=1. Using the properties of this operation and the pairing product (1.26), one can decompose 𝔮𝒜 as follows:(1.28)qA=qA+⊕qA-
with(1.29)qA+=⊕m≥0[⊕k∈NqA(m,m+k)],(1.30)qA-=⊕m≥0[⊕k∈NqA(-m-k-1,-m-1)].
The indices + and − carried by 𝔮𝒜+ and 𝔮𝒜- refer to the positive (local) and negative (nonlocal) degrees respectively. On the other hand one can decomposes the space 𝔮𝒜(m,m+k),k≥0 as(1.31)qA(m,m+k)=qΣ-(m,m+k)⊕qΣ0(m,m+k)⊕qΣ+(m,m+k).qΣ-(m,m+k) and qΣ+(m,m+k) denote the spaces of q-differential operators of negative and positive definite spin. They are read as(1.32)qΣ-(m,m+k)=⊕s>0qA-s(m,m+k),(1.33)qΣ0(m,m+k)=qA0(m,m+k),(1.34)qΣ+(m,m+k)=⊕s>0qAs(m,m+k).qΣ0(m,m+k) is just the vector space of Lorenz scalar q-differential operators. Combining (1.28)–(1.34), one sees that 𝔮𝒜 decomposes into 6=3×2 subalgebras(1.35)qA=⊕j=0,+,-[qAj+⊕qAj-]
with(1.36)qAj+=⊕m≥0[⊕k∈NqΣj(m,m+k)],qAj-=⊕m≥0[⊕k∈NqΣj(-m-k-1,-m-1)].
The duality of these 6=3×2 subalgebras is described by the combined scalar product 〈〈·,·〉〉 built out of the product equation(1.26) and conformal spin pairing:(1.37)〈uk,ul〉∶=∫dzuk(z)u1-k(z)δk+l,1,
as follows [2, 3]:(1.38)〈〈Ls(α,β),Lr(m,n)〉〉∶=δ0,r+sδ0,1+n+αδ0,1+m+β∫dzres[Ls(α,β)∘L-s(-β-1,-α-1)]
with respect to this new product, 𝔮𝒜++, 𝔮𝒜0+, and 𝔮𝒜-+ behave as the dual algebras of 𝔮𝒜--, 𝔮𝒜0-, and 𝔮𝒜+-, respectively, while 𝔮𝒜0- is just the algebra of Lorenz scalar pure q-pseudo operators. This algebra and its dual 𝔮𝒜0+, the space of Lorenz scalar local q-differential operators, are very special subalgebras as they are systematically used to construct new realizations of the wi-symmetry, i≥2 by using scalar q-differential operators type(1.39)L(k)(a)=a-k(a)∂qk.We note that the space𝔮𝒜++ is the algebra of local q-differential operators of positive definite spins and positive degrees. 𝔮𝒜--, however, is the Lie algebra of pure q-pseudodifferential operators of negative degrees and spins.


## 2.q-Deformed Lax Generating Technique

The aim of this section is to present some results related to the Lax representation in itsq-deformed version. Using the convention notations and the analysis presented previously, we perform consistent algebraic computations, based on the Pseudodifferential analysis, to derive explicit Lax pair operators of some integrable systems in the q-deformation framework.We underline that the present formulation is based on the (q-pseudo) operators ∂qn and ∂q-n instead of the (pseudo) operators ∂n and ∂-n used in several works. We note also that the obtained results are shown to be compatible with the ones already established in literature [12–16] in the case of q=1.The basic idea of the Lax formulation consists first in considering a noncommutative integrable system which possesses the Lax representation:(2.1)[L,∂t-B]q=0,
with ∂t≡∂/∂t et [f,g]q=f∘g-f∘g.Equation (2.1) and the associated pair of operators (ℒ,B) are called the Lax q-differential equation and the Lax pair, respectively. The q-differential operator ℒ defines the integrable system which we should fix from the beginning.Note that thesln-KdV hierarchy in the q-deformed version is defined as:(2.2)∂L∂tk=[(Lk/2)+,L]q,
and the way with which ones to writes the Lax q-differential equation as in (2.1) is equivalent to the following equation:(2.3)[L,∂t-B]q≡[L,∂tk-(Lk/2)+]q=0,
where the operator B is the analogue of (ℒk/2)+ describing then an q-differential operator of conformal spin k.Now, let us apply theq-deformation Lax-pair generating technique. We need to find an appropriate operator B which satisfies (2.1), for this we have to make some constraints on the operator B, namely,Ansatz for the operatorB:(2.4)B=∂qn∘Lm+B̃,
with ∂qn is the q-differential operator which acts on ℒm according to (1.10) and B̃ is another operator of same conformal weight than B. Then, with this ansatz, the problem reduces to find the operator B̃.To understand the situation, we will study two interesting examples to knowq-KdV and q-Burgers equations.


### 2.1.q-Deformed Burgers Equations

Theℒ-operator for the q-deformed Burgers equation is given by(2.5)Lq-burgers=∂q+u1
with(2.6)Lq-burgers∈qA1(0,1).
Let's consider the constraint n=1=m, for the q-deformed Burgers operator B can be written, from the ansatz (2.4), as follows:(2.7)B=∂q∘L+B̃=∂q2+ηq(u1)∂q+∂q(u1)+B̃.
Simply algebraic computations give(2.8)[L,B̃]=(ηq(u1)-u1)∂q2+[qηq(∂q(u1))+(ηq(u1))2+∂q(u1)-∂q(ηq(u1))-u1ηq(u1)]∂q+ηq(u1)∂q(u1)-u̇1,
where u̇1=∂u1/∂t.Now, our goal is to extract, from (2.1) and (2.8), the Lax equation called q-deformed Burgers or just q-Burgers equation. For this we will follow the following procedure:Ansatz for the operatorB̃:(2.9)B̃=α∂q+β,
where α and β are arbitrary functions on u and its derivatives. one finds(2.10)[L,B̃]=(ηq(α)-α)∂q2+[∂q(α)+ηq(β)+u1α-αηq(u1)-β]∂q+∂q(β)-α∂q(u1).
While identifying the two equations (2.8) and (2.10) we finds(2.11)a∂q2(u1)+(b-1)[ηq(u1)∂q(u1)+u1∂q(u1)]+u̇1=0,
with a and b are arbitrary real constant.Equation (2.11) is called q-deformed Burgers equation or q-Burgers equation.the characteristic of this equation is that it is linear for b=1 and that for q=1. (i.e.,ηq(u1)=u1) we recover the same equation gotten in works [4, 5, 9](2.12)au1′′+2(b-1)u1u1′+u̇1=0.


### 2.2.q-Deformed KdV Equations

In this second example, we go worked on anq-differentials operator of conformal weight 2, this operator is given by the KdV Lax operator(2.13)Lq-KdV=∂q2+u2.We are going to follow the same method of the previous example, therefore theAnsatz for the operatorB is(2.14)B=∂q∘L+B̃=∂q3+ηq(u2)∂q+∂q(u2)+B̃
and the associated Lax equation:(2.15)[∂t-B,L]q=0
after a calculation, one finds(2.16)[L,B]q=-u̇1,
by the same way of the case of Burgers, we finds the following q-KdV equation:(2.17)u̇2=[u2+ηq(u2)]∂q(u2)+∂q2[∂q(u2)+ηq(∂qu2)]
as for q=1, we finds the standard KdV(2.18)u̇2=u2u2′+u2′′′.


### 2.3.q-Deformed Burgers-KdV Mapping

In this section, we present an approach to define the correspondence between integrables systemsq-deformed-type Burgers and integrables systems q-deformed-type KdV. such correspondence named q-deformedBurgers-KdVmapping that is considered like a generalization of the Burgers-KdV mapping studied in works [7, 8, 11, 17].We illustrate this idea with the example of KdV and Burgers equation and then we are going to make a generalization for cameramanq-differentials-operators-type sln-KdV.Let's consider the Burgersq-differential operator (2.5):(2.19)Lq-burgers=∂q+u1∈qA1(0,1)
and the KdV q-differential operator (2.13):(2.20)Lq-KdV=∂q2+u2∈qA2(0,2)qA2(1,1).Proposition 2.1 (q-deformed Miura transformation).
If one considers the two previousq-differential operators, one can make the following decomposition:
(2.21)Lq-KdV(u2)=Lq-burgers(u1)∘Lq-burgers(v1),
with v1=-ηq(u1) and u2=∂q(-ηq(u1))-u1ηq(u1). This decomposition is called q-deformed Miura transformation. one can see this mapping under the following form:
(2.22)Lq-burgers(u1)↪Lq-KdV(u2)=Lq-burgers(u1)∘Lq-burgers(-ηq(u1))withu2=∂q(-ηq(u1))-u1ηq(u1).Proposition 2.2.
As basing on the conforms weights of the operators derivatives:[∂tq-KdV]=3 and [∂tq-Burgers]=2, one can make the following correspondence:
(2.23)∂tq-Burgers↪∂tq-KdV=α∂q∘∂tq-Burgers+β∂q3,
where α and β are arbitrary real constants.Proposition 2.3 (Généralisation).
Being given anq-deformed Burgers operator ℒq-Burgers and an q-deformed sln-KdV operator of type:
(2.24)Lq-sln-KdV=∂qn+u2∂qn-2+u3∂qn-3+⋯+un,
then we can make the following decomposition:
(2.25)Lq-sln-KdV=Lq-Burgers(v1)∘Lq-Burgers(v2)∘⋯∘Lq-Burgers(vn),
where vi,i=1,…,n are the fields of conformal weight 1 and which can be written in functions of the fields uj,j=2,…,n and their q-derivatives.


## 2.1.q-Deformed Burgers Equations

Theℒ-operator for the q-deformed Burgers equation is given by(2.5)Lq-burgers=∂q+u1
with(2.6)Lq-burgers∈qA1(0,1).
Let's consider the constraint n=1=m, for the q-deformed Burgers operator B can be written, from the ansatz (2.4), as follows:(2.7)B=∂q∘L+B̃=∂q2+ηq(u1)∂q+∂q(u1)+B̃.
Simply algebraic computations give(2.8)[L,B̃]=(ηq(u1)-u1)∂q2+[qηq(∂q(u1))+(ηq(u1))2+∂q(u1)-∂q(ηq(u1))-u1ηq(u1)]∂q+ηq(u1)∂q(u1)-u̇1,
where u̇1=∂u1/∂t.Now, our goal is to extract, from (2.1) and (2.8), the Lax equation called q-deformed Burgers or just q-Burgers equation. For this we will follow the following procedure:Ansatz for the operatorB̃:(2.9)B̃=α∂q+β,
where α and β are arbitrary functions on u and its derivatives. one finds(2.10)[L,B̃]=(ηq(α)-α)∂q2+[∂q(α)+ηq(β)+u1α-αηq(u1)-β]∂q+∂q(β)-α∂q(u1).
While identifying the two equations (2.8) and (2.10) we finds(2.11)a∂q2(u1)+(b-1)[ηq(u1)∂q(u1)+u1∂q(u1)]+u̇1=0,
with a and b are arbitrary real constant.Equation (2.11) is called q-deformed Burgers equation or q-Burgers equation.the characteristic of this equation is that it is linear for b=1 and that for q=1. (i.e.,ηq(u1)=u1) we recover the same equation gotten in works [4, 5, 9](2.12)au1′′+2(b-1)u1u1′+u̇1=0.


## 2.2.q-Deformed KdV Equations

In this second example, we go worked on anq-differentials operator of conformal weight 2, this operator is given by the KdV Lax operator(2.13)Lq-KdV=∂q2+u2.We are going to follow the same method of the previous example, therefore theAnsatz for the operatorB is(2.14)B=∂q∘L+B̃=∂q3+ηq(u2)∂q+∂q(u2)+B̃
and the associated Lax equation:(2.15)[∂t-B,L]q=0
after a calculation, one finds(2.16)[L,B]q=-u̇1,
by the same way of the case of Burgers, we finds the following q-KdV equation:(2.17)u̇2=[u2+ηq(u2)]∂q(u2)+∂q2[∂q(u2)+ηq(∂qu2)]
as for q=1, we finds the standard KdV(2.18)u̇2=u2u2′+u2′′′.


## 2.3.q-Deformed Burgers-KdV Mapping

In this section, we present an approach to define the correspondence between integrables systemsq-deformed-type Burgers and integrables systems q-deformed-type KdV. such correspondence named q-deformedBurgers-KdVmapping that is considered like a generalization of the Burgers-KdV mapping studied in works [7, 8, 11, 17].We illustrate this idea with the example of KdV and Burgers equation and then we are going to make a generalization for cameramanq-differentials-operators-type sln-KdV.Let's consider the Burgersq-differential operator (2.5):(2.19)Lq-burgers=∂q+u1∈qA1(0,1)
and the KdV q-differential operator (2.13):(2.20)Lq-KdV=∂q2+u2∈qA2(0,2)qA2(1,1).Proposition 2.1 (q-deformed Miura transformation).
If one considers the two previousq-differential operators, one can make the following decomposition:
(2.21)Lq-KdV(u2)=Lq-burgers(u1)∘Lq-burgers(v1),
with v1=-ηq(u1) and u2=∂q(-ηq(u1))-u1ηq(u1). This decomposition is called q-deformed Miura transformation. one can see this mapping under the following form:
(2.22)Lq-burgers(u1)↪Lq-KdV(u2)=Lq-burgers(u1)∘Lq-burgers(-ηq(u1))withu2=∂q(-ηq(u1))-u1ηq(u1).Proposition 2.2.
As basing on the conforms weights of the operators derivatives:[∂tq-KdV]=3 and [∂tq-Burgers]=2, one can make the following correspondence:
(2.23)∂tq-Burgers↪∂tq-KdV=α∂q∘∂tq-Burgers+β∂q3,
where α and β are arbitrary real constants.Proposition 2.3 (Généralisation).
Being given anq-deformed Burgers operator ℒq-Burgers and an q-deformed sln-KdV operator of type:
(2.24)Lq-sln-KdV=∂qn+u2∂qn-2+u3∂qn-3+⋯+un,
then we can make the following decomposition:
(2.25)Lq-sln-KdV=Lq-Burgers(v1)∘Lq-Burgers(v2)∘⋯∘Lq-Burgers(vn),
where vi,i=1,…,n are the fields of conformal weight 1 and which can be written in functions of the fields uj,j=2,…,n and their q-derivatives.


## 3. Conclusion

The importance of the theory of pseudodifferential operators in the study of nonlinear integrable systems is point out. Principally, the algebra of nonlinear (local and nonlocal) differential operators acte on the ring of analytic functionsus(x,t).In This paper, we have devoted to a brief account of the basic properties of the space ofq-pseudo differential Lax operators in the bosonic case. Presently, we know that any q-pseudodifferential operator is completely specified by a conformal spin s,s∈ℤ, two integers p, and q=p+n,n≥0 defining the lowest and the highest degrees, respectively, and finally (1+q-p)=n+1 analytic fields uj(z). We recall that the space 𝔮𝒜 of all local and nonlocal q-pseudodifferential operators admits a Lie algebra's structure with respect to the commutator buildout of the Leibnitz product. Moreover, we find that A splits into 3×2=6 subalgebr as 𝔮𝒜j+ and 𝔮𝒜j-,j=0,±1 related to each others by two types of conjugations, namely, the spin.Finally, we have focused in this work to present the basics steps towards constructing theq-deformed integrable systems and the associated Lax generating technique. Particular interest is devoted to the q-Burgers and the q-KdV systems and their underlying mapping.



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*Source: 503621-2012-03-22.xml*