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On integrability of a noncommutative q-difference two-dimensional Toda lattice equation
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On integrability of a noncommutative q-difference two-dimensional Toda lattice equation
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C.X. Li
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, J.J.C. Nimmo , Shoufeng Shen
d
a
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China b Department of Mathematics, College of Charleston, Charleston, SC 29401, USA c School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, UK d Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
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Article history: Received 11 May 2015 Received in revised form 18 September 2015 Accepted 15 October 2015 Available online xxxx Communicated by A.P. Fordy
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Keywords: The q-difference two-dimensional Toda lattice equation Bilinear Bäcklund transformation Lax pair Darboux transformations Quasideterminant solutions
In our previous work (C.X. Li and J.J.C. Nimmo, 2009 [18]), we presented a generalized type of Darboux transformations in terms of a twisted derivation in a unified form. The twisted derivation includes ordinary derivatives, forward difference operators, super derivatives and q-difference operators as its special cases. This result not only enables one to recover the known Darboux transformations and quasideterminant solutions to the noncommutative KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation, but also inspires us to investigate quasideterminant solutions to q-difference soliton equations. In this paper, we first construct the bilinear Bäcklund transformations for the known bilinear q-difference two-dimensional Toda lattice equation (q-2DTL) and then derive a Lax pair whose compatibility gives a formally different nonlinear q-2DTL equation and finally obtain its quasideterminant solutions by iterating its Darboux transformations. © 2015 Published by Elsevier B.V.
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1. Introduction
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Recently, noncommutative (nc) analogues of some well-known soliton equations and their integrability have been extensively studied. Examples include the KP equation, the modified KP equation, the KdV equation, the Hirota–Miwa equation and the two-dimensional Toda lattice equation [1–16]. Throughout the paper, we assume that the dependent variables do not commute in general. Results obtained under this assumption are applicable to most of nc integrable systems without regarding the reasons for noncommutativity. It has been shown that nc integrable systems often have quasideterminant solutions [17]. For instance, in [12], two families of quasideterminant solutions which were termed quasiwronskians and quasigrammians were presented in the nc KP equation. The origin of these solutions was explained by Darboux and binary Darboux transformations. The quasideterminant solutions were then verified directly using formulae for derivatives of quasideterminants (see also [9]). In [18], a twisted derivation was proposed following the terminology used in [19–21]. This twisted derivation includes ordinary derivations, difference operators, superderivatives and q-difference operators as some of its special cases. It has been shown that one can formulate Darboux transformations for such twisted derivations and the iteration formulae are expressed in terms of quasideterminants. As a remarkable example, we obtained quasideterminant solutions to the super KdV equation for the first time. The above results not only enable us to recover the Darboux transformations and therefore quasideterminant solutions to the existing nc KP equation, the non-Abelian (2 + 1)-dimensional Toda lattice equation and the non-Abelian Hirota–Miwa equation, but also inspire us to seek quasideterminant solutions to q-difference soliton equations. To a certain point, the proposal of the twisted derivation together with its Darboux transformations provides a systematic approach to study some supersymmetric equations, difference equations, differential equations and q-difference soliton equations and their quasideterminant solutions. The q-difference (also called q-analogue, q-deformed or q-discrete) integrable system is defined by means of q-difference operator (q-derivative or Jackson derivative) instead of the ordinary derivative ∂ with respect to independent variables x, y or t, etc. in a classical
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E-mail addresses: [email protected] (C.X. Li), [email protected] (J.J.C. Nimmo), [email protected] (S. Shen). http://dx.doi.org/10.1016/j.physleta.2015.10.027 0375-9601/© 2015 Published by Elsevier B.V.
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system [22]. It reduces to a classical integrable system as q → 1. The study of q-analogues of classical integrable systems in parallel with classical integrable systems has attracted much attention. Examples include q-difference Painéve equations, q-deformed KdV hierarchy and mKdV hierarchy, q-deformed KP hierarchy and constrained KP hierarchy, q-discrete two-dimensional Toda molecule (q-2DTM) equation and q-2DTL equation [23–28]. In [28], a q-2DTM equation and a q-2DTL equation as well as their determinant solutions were presented. The Bäcklund transformation and Lax pair were obtained for the q-2DTM equation. However, the Bäcklund transformation and Lax pair for the q-2DTL equation remains unknown. In this paper, we are aiming to derive the Bäcklund transformation and Lax pair for the q-2DTL equation and construct quasideterminant solutions to a nc q-2DTL equation by using the more general Darboux transformations for the twisted derivations. This work provides a concrete example for how to construct quasideterminant solutions to q-difference soliton equations under the general framework. The paper is organized as follows. In Section 2, we give a brief review of Darboux transformations for the twisted derivation and show how this much more general result could be applied to some typical nc integrable systems to recover their known quasideterminant solutions. In Section 3, we first construct the bilinear Bäcklund transformation for the known bilinear q-2DTL equation by using determinant identities and then derive the Lax pair for a formally different nonlinear q-2DTL equation. In Section 4, we construct quasideterminant solutions to the nc q-2DTL equation by its Darboux transformations. Conclusions are given in Section 5.
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Derivative. Here D = ∂/∂ x satisfies D (ab) = D (a)b + aD (b) and σ is the identity mapping. Forward difference. The homomorphism is the shift operator T , where T (a(x)) = a(x + h) and the twisted derivation is
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(a(x)) =
a(x + h) − a(x) h
a(qx) − a(x)
(q − 1)x
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,
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satisfying (ab) = (a)b + T (a)(b). Superderivative. For a, b ∈ A, a superalgebra, the superderivative D = ∂θ + θ∂x satisfies D (ab) = D (a)b + aD (b), where is the grade involution. Indeed, any odd derivation is a twisted derivation. Jackson derivative. The homomorphism is a q-shift operator defined by S q (a(x)) = a(qx) and the twisted derivation is
D q (a(x)) =
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.
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satisfying D q (ab) = D q (a)b + S q (a) D q (b).
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Remark 1. If we introduce a new q-deformed operator D q, f (a(x)) = with the difference interval given by an arbitrary function f (x), then we still have D q, f (ab) = D q, f (a)b + S q (a) D q, f (b). D q, f is again a twisted derivation. In this sense, we can generalize the results on q-2DTL equation by replacing the Jackson derivative with D q, f which can be done in the same way.
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2.2. Darboux transformations for twisted derivations
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Consider a general setting in which A is an associative, unital algebra over ring K , not necessarily graded. Suppose that there is a homomorphism σ : A → A (i.e. for all α ∈ K , a, b ∈ A, σ (αa) = ασ (a), σ (a + b) = σ (a) + σ (b) and σ (ab) = σ (a)σ (b)) and a twisted derivation or σ -derivation [19–21] D : A → A satisfying D ( K ) = 0 and D (ab) = D (a)b + σ (a) D (b). Here are some examples of the twisted derivation where a ∈ A and A consists of functions of x.
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2.1. A type of twisted derivations
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In [18], we proposed a type of twisted derivations and constructed Darboux transformations in terms of it in a unified way. Such twisted derivations include derivatives, difference operators, superderivatives and q-difference operators as special cases. Surprisingly, the formulae for the iteration of Darboux transformations were expressed in terms of quasideterminants. Here we avoid to talk about quasideterminants and their properties. But one can refer to [29] and [30] for details. In this section, to make the paper selfconsistent, we will first give a brief review of the twisted derivation and its corresponding Darboux transformation presented in [18] and then show how the results can be applied to recover quasideterminant solutions to some known nc soliton equations which involve ordinary derivatives, difference operators and superderivatives.
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2. A type of twisted derivations and Darboux transformations
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Let θ0 , θ1 , θ2 , . . . be a sequence in A. As was pointed out in [18], an elementary Darboux transformation expressed terms of the twisted derivation is determined by
G θ = σ (θ) D θ
−1
= D − D (θ)θ
−1
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, θ ∈A
(1)
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where θ is invertible, D and σ are the twisted derivation and homomorphism defined above. For any general twisted derivation D we have the following theorem
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Theorem 1. Let φ[0] = φ and for n ∈ N let
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φ[n] = D (φ[n − 1]) − D (θ[n − 1])θ[n − 1] where θ[n] = φ[n]|φ→θn . Then, for n ∈ N,
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φ[n − 1],
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θ0 D (θ0 ) .. . φ[n] = D n−1 (θ0 ) D n (θ0 )
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··· ··· ···
. D n (φ)
θn−1 D (θn−1 ) .. . D n−1 (θn−1 )
···
φ D (φ) .. . D n−1 (φ)
n
D (θn−1 )
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(2)
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Remark 2. In particular, if we have a Lax pair L , M with coefficients u , v , . . . , say, satisfying some given nonlinear system we choose θ0 , . . . , θn−1 as particular eigenfunctions of L , M. Then a generic eigenfunction φ of L , M is transformed to an eigenfunction of L [n], M [n] with coefficients u [n], v [n], . . . satisfying the same nonlinear system. Thereby we obtain a new solution u [n], v [n], . . . of the nonlinear system in terms of the ‘seed’ solution u , v , . . . and quasideterminants expressed in terms of the seed eigenfunctions. We illustrate this point in the examples in the next section.
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In the literature, Darboux transformations and quasideterminant solutions to the nc KP equation, the non-Abelian (2 + 1)-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation have been constructed, separately. In this section, we would like to illustrate how to derive Darboux transformations for the above-mentioned nc integrable systems from the more general Darboux transformation defined in terms of the twisted derivations.
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Lφ =
(∂x2
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+ v x − ∂ y )φ = 0,
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M φ = (4∂x3 + 6v x ∂x + 3v xx + 3v y + ∂t )φ = 0.
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Theorem 1, using the ordinary derivative D = ∂x , gives the nth transformation of the eigenfunction and the transformed potential is
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v [n] = v − 2Q x , where
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θ0 ∂x (θ0 ) .. . Q = ∂ n−1 (θ ) 0 x ∂ n (θ ) x 0
··· ···
0 0
···
θn−1 ∂x (θn−1 ) .. . ∂xn−1 (θn−1 )
···
∂xn (θn−1 )
0
.. . 1
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(3)
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which coincide with the results obtained in [12].
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Example 2 (The non-Abelian 2DTL equation [14,15]). The non-Abelian 2DTL equation is written as
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U m , x + U m V m +1 − V m U m = 0,
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V m,t + αm U m−1 − U m αm+1 = 0.
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It has the Lax pair
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It has the Lax pair
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( v t + v xxx + 3v x v x )x + 3v y y − 3[ v x , v y ] = 0.
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Example 1 (The nc KP equation [1–9,12]). This equation is
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2.3. Applications in some known noncommutative integrable systems
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φm,x = V m φm + αm φm−1 ,
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φm,t = U m φm+1 .
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In this case the Darboux transformation is expressed in terms of the twisted derivation D = T m , the translation operator in discrete variable m which leads to the results obtained in [15]. In particular, if the potential U m and V m are written in the form
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then the effect of the nth Darboux transformation is encapsulated as the transformation of X m :
··· ···
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−1 −1 U m = Xm Xm +1 , V m = X m , x X m ,
θ0 T m (θ0 ) .. n −1 . X m [n] = (−1) T n−1 (θ ) 0 m T n (θ ) m 0
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···
θn−1 T m (θn−1 ) .. . n −1 Tm (θn−1 )
1
···
n Tm (θn−1 )
0
.. .
Xm+n−1 .
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Example 3 (The super KdV equation [18,31–33]). The Manin–Radul super KdV equation is written as
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αt = (αxx + 3α D (α ) + 6α u )x ,
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(u xx + 3u 2 + 3α D (u ))x ,
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We take even/odd particular eigenfunction θ2i /θ2i +1 of the Lax pair, respectively, and use the twisted derivation (superderivative) D to define the Darboux transformation. The transformation of the odd potential is given by
α [n] = (−1) α − D ( Q ), where
θ0 D (θ0 ) .. . Q = D n−1 (θ ) 0 D n (θ ) 0
··· ···
···
0 0
.. . 1
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The formula for even potential u [n] is more complicated, but is again expressed in terms of quasideterminants, and is given in [18].
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3. The noncommutative q-2DTL equation
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In the previous section, we illustrated how to recover Darboux transformations and furthermore quasideterminant solutions for some known nc integrable systems which are defined in terms of ordinary derivatives, forward difference operators and super derivatives from the more general Darboux transformation for the twisted derivations. All these results inspire us to look for Darboux transformations and quasideterminant solutions to nc q-difference soliton equations. In this section, we will first give a brief review of some known facts about the q-2DTL equation, its bilinear form and determinant solutions. Next, we will construct a bilinear Bäcklund transformation for this bilinear form by using determinant identities and further derive a Lax pair for a formally different nonlinear q-2DTL equation.
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y ) = J n (x, y ) V n (qα x, y ) − J n+1 (x, y ) V n (x, y ),
δqβ , y J n (x, y ) = V n−1 (qα x, y ) − qβ V n (x, qβ y ),
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In [28], a q-discrete version of the two-dimensional Toda lattice equation was proposed
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δqα ,x V n (x,
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3.1. Some known results for the q-2DTL equation
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θn−1 D (θn−1 ) .. . D n−1 (θn−1 )
···
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α are even and odd dependent variables respectively. It has the Lax pair
φt = M φ = (∂x3 + ((α ∂x + ∂x α ) D + u ∂x + ∂x u ))φ.
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L φ = (∂x2 + α D + u )φ = λφ,
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where D = ∂θ + θ∂x with θ an odd Grassmann variable, u and
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(5) (6)
δqα ,x f (x, y ) =
(1 − q)x
, δqβ , y f (x, y ) =
(1 − q) y
.
(7)
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τn (qα x, y )τn−1 (x, y )) {1 + (1 − q) xy } J n (x, y ) = −1 , (1 − q)x τn (x, qβ y )τn−1 (qα x, y ) 2
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the q-2DTL equations (5) and (6) were transformed to the bilinear form
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{δqα ,x δqβ , y τn (x, y )}τn (x, y ) − {δqα ,x τn (x, y )}{δqβ , y τn (x, y )} = τn+1 (x, q y )τn−1 (qα x, y ) − τn (qα x, qβ y )τn (x, y ), β
(8)
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whose solutions were given by
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τn+1 (qα x, y )τn−1 (x, qβ y ) V n (x, y ) = , τn (x, qβ y )τn (x, y )
(1 ) f n (x, y ) (2 ) f (x, y ) τn (x, y ) = n . .. ( N ) f (x, y )
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f (x, y ) − f (x, qβ y )
Under the dependent variable transformations
1
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where δqα ,x and δqβ , y are q-difference operators defined by
f (x, y ) − f (qα x, y )
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(1 )
f n+1 (x, y ) · · · (2 ) f n+1 (x, y )
··· .. . ··· (N ) f n+1 (x, y ) · · ·
(1 ) f n+ N −1 (x, y ) (2 ) f n+ N −1 (x, y ) .. . (N ) f n+ N −1 (x, y )
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(k)
where f n , k = 1, . . . , N, satisfy the dispersion relations (k)
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(k)
(k)
1 2
(k)
δqα ,x f n (x, y ) = − f n+1 (x, y ), δqβ , y f n (x, y ) = f n−1 (x, y ).
(10)
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In the following, we will first present a bilinear Bäcklund transformation for (8) and then derive a Lax pair whose compatibility condition yields a formally different q-2DTL equation.
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3.2. Bilinear Bäcklund transformations for (8)
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By considering the determinantal identities between to the solutions given by (9), we obtain the following bilinear Bäcklund transformation for (8)
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δqα ,x τn (x, y ) · τn−1 (x, y ) − δqα ,x τn−1 (x, y ) · τn (x, y ) = τn−1
(11)
(qα x, y )τ (x, y ),
(12)
n
τn (x, y ) (N-soliton) given by (9) to another solution τn (x, y ) (( N + 1)-soliton) given by (1 ) (1 ) (1 ) f n (x, y ) f n+1 (x, y ) · · · f n+ N (x, y ) (2 ) (2 ) (2 ) f (x, y ) f n+1 (x, y ) · · · f n+ N (x, y ) . τn (x, y ) = n . .. .. .. . . · · · f ( N +1) (x, y ) f ( N +1) (x, y ) · · · f ( N +1) (x, y )
which transforms the solution
In what follows, we will prove that the bilinear Bäcklund transformation relates adopt the notations used in [28]. We denote
(1 )
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(15)
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(16)
−β
x, y ) = | − 1, 0, . . . , N − 2, N − 1q−α x |,
(17)
y ) = |0q−β y , 1, . . . , N − 1, t |,
τn (q
−α
x, y ) = |0, . . . , N − 2, N − 1q−α x , t |,
(1 − q)q−β y τn −1 (x, q−β y ) = |0q−β y , 0, . . . , N − 2, N − 1|, −α
(18)
0
·········
1 ··· N − 1
0
········· ∅
∅ N ········· ········· 1 ··· N − 1 N
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(23)
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(1 − q)q−β y τn −1 (x, q−β y )τn+1 (x, y ) − τn (x, q−β y )τn (x, y ) + τn (x, q−β y )τn (x, y ) = 0,
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(24)
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In the same way, we can prove that the Laplacian expansion of the following determinant identity
∅ N −1 ········· ········· 0 ··· N − 2 N −1
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which is nothing but (11) with y replaced by qβ y.
········· ∅
41
(20)
leads to the following equation
0 ··· N − 2
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(22)
t · · · · · · · · · ≡ 0 t
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(21)
x, y ) = |0, 1, . . . , N − 1, N − 1q−α x |.
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(19)
It is straightforward to prove that the Laplacian expansion of the determinant identity
−1 ········· −1
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τn (x, q−β y ) = |0q−β y , 1, . . . , N − 1, N |, τn −1 (x, q−β y ) = | − 1q−β y , 0, . . . , N − 2, N − 1|,
0q−β y ········· 0q−β y
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τn−1 (x, y ) = | − 1, 0, . . . , N − 2, N − 1|, τn (x, q
xτn (q
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− (1 − q)q
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τn+1 (x, y ) = |1, 2, . . . , N , t |, τn−1 (x, y ) = | − 1, 0, . . . , N − 2, t |,
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where t = (0, 0, . . . , 0, 1) T is an ( N + 1)-vector. Both τn (x, y ) given by (9) and τn (x, y ) given by (13) are expressed in term of Casoratian determinants. Concerning the applications of Casoratian determinants to integrable systems, we refer to [34,35]. Using the dispersion relations (10), we have
−α
14
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⎞
τn (x, y ) = |0, 1, . . . , N − 1, t |,
τn−1 (q
13
20
Similarly, we can denote
−α
12
18
(13)
(14)
⎜ (2 ) ⎟ ⎜ f (x, y ) ⎟ ⎜ n+ j ⎟ ⎜ ⎟ .. ⎟. “ j” = ⎜ ⎜ ⎟ . ⎜ (N ) ⎟ ⎜ f (x, y ) ⎟ ⎝ n+ j ⎠ ( N +1 ) f n+ j (x, y )
11
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where the number “ j” stands symbolically for a column vector given by
f n+ j (x, y )
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τn (x, y ) and τn (x, y ). For the sake of simplicity, we
τn (x, y ) = |0, 1, . . . , N − 1, N | ⎛
6
15
n+ N
n +1
n
5
8
δqβ , y τn (x, y ) · τn (x, y ) − δqβ , y τn (x, y ) · τn (x, y ) = τn −1 (x, y )τn+1 (x, qβ y ),
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N − 1q−α x
········· N − 1q−α x
· · · · · · · · · ≡ 0 t
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t
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(25)
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leads to
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−α
x, y ) − τn−1 (q
−α
x, y )τn (x, y ) − (1 − q)q
−α
xτn (q
−α
x, y )τn−1 (x, y ) = 0,
(26)
which is again nothing but (12) by replacing x with qα x.
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3.3. A new form of the q-2DTL equation and its Lax pair
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By introducing the eigenfunction φn+1 (x, y ) = τn (x, y )/τn (x, y ), we can derive a Lax pair from the bilinear Bäcklund transformation (11) and (12). We have
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δqβ , y φn+1 (x, y ) = W n (x, y )φn (x, y ),
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δqα ,x φn (x,
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y ) = −φn+1 (x, y ) + K n (x, y )φn (x, y ),
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48
11
(28)
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where
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τn−1 (x, y )τn+1 (x, qβ y ) W n (x, y ) = , τn (x, y )τn (x, qβ y )
1 τn−1 (x, y )τn (qα x, y ) . 1− K n (x, y ) = (1 − q)x τn (x, y )τn−1 (qα x, y )
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(29)
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δqβ , y K n (x, y ) = W n (x, y ) − W n−1 (qα x, y ),
(31)
δqα ,x W n (x, y ) = K n+1 (x, q y ) W n (x, y ) − W n (qα x, y ) K n (x, y ).
(32)
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Hereafter, for simplicity, we denote the q-difference operators D 1 = δqα ,x , D 2 = δqβ , y . σ1 and σ2 are homomorphism satisfying σ1 ( f (x, y )) = f (qα x, y ) and σ2 ( g (x, y )) = g (x, qβ y ) respectively. With the new notations, (31) and (32) can be rewritten as
D 2 ( K n ) = W n − σ1 ( W n−1 ),
(33)
D 1 ( W n ) = σ2 ( K n+1 ) W n − σ1 ( W n ) K n .
(34)
It is clear that D 1 and D 2 are twisted derivations with respect to σ1 and σ2 respectively. We will see later, in (38)–(40), that the effect of the Darboux transformation on the potentials W n and K n can be encapsulated in the effect on a new variable X n and by this we are led to define
W n = σ2 ( X n+1 ) X n−1 , K n = D 1 ( X n ) X n−1 .
(35)
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D 2 ( D 1 ( X n ) X n−1 ) = σ2 ( X n+1 ) X n−1 − σ1 (σ2 ( X n ) X n−−11 ).
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(36)
From now on, we refer to (36) as the nc q-2DTL equation. In this equation, the dependent variables are not assumed to commute in general. Through the change of variables (35) any solution X n of (36) gives a solution of the original q-2DTL system (31)–(32).
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Remark 4. In the commutative case, one can easily recover the bilinear q-2DTL equation (8) from the nc q-2DTL equation (36) by the substitution X n = τn /τn−1 .
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4. Quasicasoratian solutions for (36)
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In this section, we will first derive the Darboux transformations for the nc q-2DTL equation (36) and then construct its quasideterminant solutions by iterating Darboux transformations.
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4.1. Darboux transformations
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Then (33) and (34) can be surprisingly reduced to one single equation
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Remark 3. Although the q-2DTL equations (33)–(34) obtained in this paper formally looks different from the known one (5)–(6), the equations (33) and (34) can be transformed to (5) and (6) by letting β → −β and defining J n (x, y ) = − K n (x, q−β n y ) and V n (x, y ) = −q−β n W n (x, q−β n y ). We use the alternative form (31)–(32) from now on.
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The compatibility condition of the Lax pair (27) and (28) gives the following nonlinear q-2DTL equation
β
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(30)
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(27)
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As was pointed out in Section 2, a q-difference operator is an example of a twisted derivation. Therefore, from the general Darboux transformations defined in terms of twisted derivations, we can get the following Darboux transformation for the nc q-2DTL equation (36).
n = σ1 (θn ) D 1 θn−1 φn = ( D 1 − D 1 (θn )θn−1 )φn = −φn+1 + θn+1 θn−1 φn , φ K n = K n+1 + σ1 (θn+1 θn ) − θn+2 θn−+11 = { D 1 (θn+1 θn−1 ) + σ1 (θn+1 θn−1 ) K n }θn θn−+11 , W n = σ2 (θn+2 θn−+11 ) W n θn θn−+11 = W n+1 − D 2 (θn+2 θn−+11 ), −1
X n = −θn+1 θn X n , −1
where θn is a particular solution of the Lax pair (27) and (28).
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Let θn,i , i = 1, . . . , N, be a particular set of eigenfunctions of the Lax pair (27) and (28) and introduce the notation n = (θn,1 , . . . , θn, N ). The Darboux transformations for the nc q-2DTL equation may be iterated by defining
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φn [k + 1] = −φn+1 [k] + θn+1 [k]θn [k]φn [k],
(41)
K n [k + 1] = K n+1 [k] + σ1 (θn+1 [k]θn−1 [k]) − θn+2 [k]θn−+11 [k],
(42)
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W n [k + 1] = σ2 (θn+2 [k]θn−+11 [k]) W n [k]θn [k]θn−+11 [k],
(43)
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X n [k + 1] = −θn+1 [k]θn−1 [k] X n [k],
(44)
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where φn [1] = φn , X n [1] = X n and
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In what follows, we will show by induction that the results of N-repeated Darboux transformations φn [ N + 1] and X n [ N + 1] can be expressed in closed form as quasideterminants
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θn [k] = φn [k]|φn →θn,k .
n+ N N n+ N −1 φn [ N + 1] = (−1) .. . n
φn+ N φn+ N −1 .. . φn
n+ N . , Xn [ N + 1] = .. n+1 n
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0 .. . Xn . 0 1
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(46)
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The initial case N = 1 is obviously true for both φn [2] and X n [2]. In order to prove the inductive step, we shall use several quasideterminant identities. The source of these identities is [29,30] or they may be found in summary form in [15]. They include the noncommutative Sylvester/Jacobi identity (for expanding by row and column), homological relations (for moving the expansion point) and quasi-Plücker coordinate formula (identities amongst certain types of quasideterminants). In the following proof we indicate where these identities are used. From (41), we have
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φn [ N + 2] = −φn+1 [ N + 1] + θn+1 [ N + 1]θn [ N + 1]−1 φn [ N + 1] n+ N +1 n+ N +1 θn+ N +1, N +1 φ n + N + 1 φn+ N + (−1) N n+ N θn+ N , N +1 = −(−1) N n+ N .. .. . .. .. . . . n+1 n+1 φn+1 θn+1, N +1 ⎧ ⎪ ⎪ ⎪ ⎪ n+ N +1 ⎪ ⎪ ⎨ = (−1) N +1 n+ N ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ n+1
φn+ N +1 φn+ N .. . φn+1
n+ N +1 n+ N − .. . n+1
n+ N +1 θn+ N +1, N +1 n+ N θn+ N , N +1 = (−1) N +1 .. .. . . n+1 θn+1, N +1 n θn, N +1
φn+ N +1 φn+ N .. . φn+1 φn
θn+ N +1, N +1 θn+ N , N +1 .. . θn+1, N +1
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n+ N n+ N −1 .. . n
θn+ N , N +1 θn+ N −1, N +1 .. . θn, N +1
−1 n+ N n+ N −1 .. . n
−1 n+ N θn+ N , N +1 n+ N n+ N −1 θn+ N −1, N +1 n+ N −1 .. .. .. . . . n θn, N +1 n
φn+ N
φn+ N −1 .. . φn ⎫ φn+ N ⎪ ⎪ ⎪ ⎪ φn+ N −1 ⎪ ⎪ ⎬ .. . ⎪ ⎪ ⎪ ⎪ φn ⎪ ⎪ ⎭
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,
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where we use the homological relation and then the noncommutative Jacobi identity. Similarly, from the iteration (44), we have
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X n [ N + 1]
n+ N n+ N −1 .. . n
θn+ N , N +1 θn+ N −1, N +1 .. . θn, N +1
−1 n+ N .. . n+1 n
0 .. . Xn 0 1
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n+ N +1 θn+ N +1, N +1 n+ N θn+ N , N +1 = . .. .. . n+1 θn+1, N +1 n+ N +1 θn+ N +1, N +1 n+ N θn+ N , N +1 = .. .. . . n+1 θ n + 1 , N +1 n θn, N +1
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where we use the quasi-Plücker coordinate formula twice. These prove the inductive steps for both φn [ N + 1] and X n [ N + 1]. So far, we have proved quasideterminant solutions given by (46) to the nc q-2DTL equation (36).
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Remark 5. In the commutative case, we can express the quasideterminant as a ratio of determinants. So if we assume θn,i = f n vacuum solution X n = (−1) N +2 in (36), then we have X n [ N + 1] = τn+1 /τn from (46) with τn given by (9).
and the
5. Conclusions In our previous work [18], we proposed a twisted derivation which includes ordinary derivative, forward difference operators, q-difference operators and superderivatives as special cases and constructed its Darboux transformation which has an iteration formula written in terms of a quasideterminant. This result opens the opportunity for an unified approach to construct Darboux transformations and therefore quasideterminant solutions for noncommutative integrable systems defined in terms of twisted derivations. In this paper we show how this has been achieved for a noncommutative q-2DTL equation which involves q-difference operators besides the nc KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation which involve ordinary derivative, forward difference operators and superderivatives, respectively. In this paper, we first construct the bilinear Bäcklund transformation for the known bilinear q-2DTL equation by using determinant identities and then derive the Lax pair for a formally different nonlinear q-2DTL equation which can be surprisingly reduced to one single equation. Based on the Lax pair and the more general Darboux transformation defined in terms of the twisted derivation, we are able to get the Darboux transformation and therefore construct quasicasoratian solutions to the nc q-2DTL equation. However, how to construct the binary Darboux transformation and quasigrammian solutions for the nc q-2DTL equation still remains unsolved. In some sense, the results obtained in this paper extend the applications of quasideterminants in nc integrable systems and reveal the advantage of using quasideterminants in dealing with nc integrable systems, q-difference soliton equations and supersymmetric equations. Acknowledgements This work was supported by the National Natural Science Foundation of China under the grants 11271266 and 11371323, China Scholarship Council and Beijing Teachers Training Center for Higher Education. The authors would like to thank the reviewers particularly for their careful reading and valuable comments. References [1] B.A. Kupershmidt, KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems, Mathematical Surveys and Monographs, vol. 78, American Mathematical Society, New York, NY, 2000. [2] L.D. Paniak, Exact noncommutative KP and KdV multi-solitons, arXiv:hep-th/0105185, 2001. [3] M. Sakakibara, Factorization methods for noncommutative KP and Toda hierarchy, J. Phys. A 37 (2004) L599–L604. [4] N. Wang, M. Wadati, Noncommutative extension of ∂¯ -dressing method, J. Phys. Soc. Jpn. 72 (2003) 1366–1373. [5] N. Wang, M. Wadati, Exact multi-line soliton solutions of noncommutative KP equation, J. Phys. Soc. Jpn. 72 (2003) 1881–1888. [6] N. Wang, M. Wadati, Noncommutative KP hierarchy and Hirota triple-product relations, J. Phys. Soc. Jpn. 73 (2004) 1689–1698. [7] M. Hamanaka, Noncommutative solitons and D-branes, Ph.D. thesis, arXiv:hep-th/0303256, 2003. [8] M. Hamanaka, K. Toda, Towards noncommutative integrable systems, Phys. Lett. A 316 (2003) 77–83. [9] A. Dimakis, F. Müller-Hoissen, An algebraic scheme associated with the non-commutative KP hierarchy and some of its extensions, J. Phys. A 38 (2005) 5453–5505. [10] J.J.C. Nimmo, On a non-Abelian Hirota–Miwa equation, J. Phys. A 39 (2006) 5053–5065. [11] C.X. Li, J.J.C. Nimmo, K.M. Tamizhmani, On solutions to the non-Abelian Hirota–Miwa equation and its continuum limits, Proc. R. Soc. A 465 (2009) 1441–1451. [12] C.R. Gilson, J.J.C. Nimmo, On a direct approach to quasideterminant solutions of a noncommutative KP equation, J. Phys. A, Math. Theor. 40 (2007) 3839–3850. [13] C.R. Gilson, J.J.C. Nimmo, Y. Ohta, Quasideterminant solutions of a non-Abelian Hirota–Miwa equation, J. Phys. A, Math. Theor. 40 (2007) 12607–12617. [14] A.V. Mikhailov, Integrability of a two-dimensional generalization of the Toda chain, JETP Lett. 30 (1979) 443–448. [15] C.X. Li, J.J.C. Nimmo, Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation, Proc. R. Soc. A 464 (2008) 951–966. [16] C.X. Li, J.J.C. Nimmo, K.M. Tamizhmani, On solutions to the non-Abelian Hirota–Miwa equation and its continuum limits, Proc. R. Soc. A 465 (2009) 1441–1451. [17] I.M. Gelfand, V.S. Retakh, Determinants of matrices over noncommutative rings, Funct. Anal. Appl. 25 (1991) 13–25. [18] C.X. Li, J.J.C. Nimmo, Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation, Proc. R. Soc. A 466 (2009) 2471–2493. [19] J.T. Hartwig, D. Larsson, S.D. Silvestrov, Deformations of Lie algebras using σ -derivations, J. Algebra 295 (2006) 314–361. [20] S. De Concini, C. Procesi, Quantum Groups, Lecture Notes in Mathematics, vol. 1565, 1993, pp. 31–140. [21] A. Dimakis, F. Müller-Hoissen, Functional representations of integrable hierarchies, J. Phys. A 39 (2006) 9169–9186. [22] V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York, 2002. [23] V.G. Papageorgiou, F.W. Nijhoff, B. Grammaticos, A. Ramani, Isomonodromic deformation problems for discrete analogues of Painléve equations, Phys. Lett. A 164 (1992) 57–64.
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On integrability of a noncommutative q-difference two-dimensional Toda lattice equation
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C.X. Li
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, J.J.C. Nimmo , Shoufeng Shen
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School of Mathematical Sciences, Capital Normal University, Beijing 100048, China b Department of Mathematics, College of Charleston, Charleston, SC 29401, USA c School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QW, UK d Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China
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Article history: Received 11 May 2015 Received in revised form 18 September 2015 Accepted 15 October 2015 Available online xxxx Communicated by A.P. Fordy
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Keywords: The q-difference two-dimensional Toda lattice equation Bilinear Bäcklund transformation Lax pair Darboux transformations Quasideterminant solutions
In our previous work (C.X. Li and J.J.C. Nimmo, 2009 [18]), we presented a generalized type of Darboux transformations in terms of a twisted derivation in a unified form. The twisted derivation includes ordinary derivatives, forward difference operators, super derivatives and q-difference operators as its special cases. This result not only enables one to recover the known Darboux transformations and quasideterminant solutions to the noncommutative KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation, but also inspires us to investigate quasideterminant solutions to q-difference soliton equations. In this paper, we first construct the bilinear Bäcklund transformations for the known bilinear q-difference two-dimensional Toda lattice equation (q-2DTL) and then derive a Lax pair whose compatibility gives a formally different nonlinear q-2DTL equation and finally obtain its quasideterminant solutions by iterating its Darboux transformations. © 2015 Published by Elsevier B.V.
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Recently, noncommutative (nc) analogues of some well-known soliton equations and their integrability have been extensively studied. Examples include the KP equation, the modified KP equation, the KdV equation, the Hirota–Miwa equation and the two-dimensional Toda lattice equation [1–16]. Throughout the paper, we assume that the dependent variables do not commute in general. Results obtained under this assumption are applicable to most of nc integrable systems without regarding the reasons for noncommutativity. It has been shown that nc integrable systems often have quasideterminant solutions [17]. For instance, in [12], two families of quasideterminant solutions which were termed quasiwronskians and quasigrammians were presented in the nc KP equation. The origin of these solutions was explained by Darboux and binary Darboux transformations. The quasideterminant solutions were then verified directly using formulae for derivatives of quasideterminants (see also [9]). In [18], a twisted derivation was proposed following the terminology used in [19–21]. This twisted derivation includes ordinary derivations, difference operators, superderivatives and q-difference operators as some of its special cases. It has been shown that one can formulate Darboux transformations for such twisted derivations and the iteration formulae are expressed in terms of quasideterminants. As a remarkable example, we obtained quasideterminant solutions to the super KdV equation for the first time. The above results not only enable us to recover the Darboux transformations and therefore quasideterminant solutions to the existing nc KP equation, the non-Abelian (2 + 1)-dimensional Toda lattice equation and the non-Abelian Hirota–Miwa equation, but also inspire us to seek quasideterminant solutions to q-difference soliton equations. To a certain point, the proposal of the twisted derivation together with its Darboux transformations provides a systematic approach to study some supersymmetric equations, difference equations, differential equations and q-difference soliton equations and their quasideterminant solutions. The q-difference (also called q-analogue, q-deformed or q-discrete) integrable system is defined by means of q-difference operator (q-derivative or Jackson derivative) instead of the ordinary derivative ∂ with respect to independent variables x, y or t, etc. in a classical
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E-mail addresses: [email protected] (C.X. Li), [email protected] (J.J.C. Nimmo), [email protected] (S. Shen). http://dx.doi.org/10.1016/j.physleta.2015.10.027 0375-9601/© 2015 Published by Elsevier B.V.
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system [22]. It reduces to a classical integrable system as q → 1. The study of q-analogues of classical integrable systems in parallel with classical integrable systems has attracted much attention. Examples include q-difference Painéve equations, q-deformed KdV hierarchy and mKdV hierarchy, q-deformed KP hierarchy and constrained KP hierarchy, q-discrete two-dimensional Toda molecule (q-2DTM) equation and q-2DTL equation [23–28]. In [28], a q-2DTM equation and a q-2DTL equation as well as their determinant solutions were presented. The Bäcklund transformation and Lax pair were obtained for the q-2DTM equation. However, the Bäcklund transformation and Lax pair for the q-2DTL equation remains unknown. In this paper, we are aiming to derive the Bäcklund transformation and Lax pair for the q-2DTL equation and construct quasideterminant solutions to a nc q-2DTL equation by using the more general Darboux transformations for the twisted derivations. This work provides a concrete example for how to construct quasideterminant solutions to q-difference soliton equations under the general framework. The paper is organized as follows. In Section 2, we give a brief review of Darboux transformations for the twisted derivation and show how this much more general result could be applied to some typical nc integrable systems to recover their known quasideterminant solutions. In Section 3, we first construct the bilinear Bäcklund transformation for the known bilinear q-2DTL equation by using determinant identities and then derive the Lax pair for a formally different nonlinear q-2DTL equation. In Section 4, we construct quasideterminant solutions to the nc q-2DTL equation by its Darboux transformations. Conclusions are given in Section 5.
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Derivative. Here D = ∂/∂ x satisfies D (ab) = D (a)b + aD (b) and σ is the identity mapping. Forward difference. The homomorphism is the shift operator T , where T (a(x)) = a(x + h) and the twisted derivation is
33
(a(x)) =
a(x + h) − a(x) h
a(qx) − a(x)
(q − 1)x
30 31 32
35 36
,
37
satisfying (ab) = (a)b + T (a)(b). Superderivative. For a, b ∈ A, a superalgebra, the superderivative D = ∂θ + θ∂x satisfies D (ab) = D (a)b + aD (b), where is the grade involution. Indeed, any odd derivation is a twisted derivation. Jackson derivative. The homomorphism is a q-shift operator defined by S q (a(x)) = a(qx) and the twisted derivation is
D q (a(x)) =
29
34
38 39 40 41 42 43
.
44 45
satisfying D q (ab) = D q (a)b + S q (a) D q (b).
46 47 a(qx)−a(x) f (x)
Remark 1. If we introduce a new q-deformed operator D q, f (a(x)) = with the difference interval given by an arbitrary function f (x), then we still have D q, f (ab) = D q, f (a)b + S q (a) D q, f (b). D q, f is again a twisted derivation. In this sense, we can generalize the results on q-2DTL equation by replacing the Jackson derivative with D q, f which can be done in the same way.
48 49 50 51 52
2.2. Darboux transformations for twisted derivations
53
54 55
8
Consider a general setting in which A is an associative, unital algebra over ring K , not necessarily graded. Suppose that there is a homomorphism σ : A → A (i.e. for all α ∈ K , a, b ∈ A, σ (αa) = ασ (a), σ (a + b) = σ (a) + σ (b) and σ (ab) = σ (a)σ (b)) and a twisted derivation or σ -derivation [19–21] D : A → A satisfying D ( K ) = 0 and D (ab) = D (a)b + σ (a) D (b). Here are some examples of the twisted derivation where a ∈ A and A consists of functions of x.
52 53
7
26
47 48
6
27
45 46
5
25
2.1. A type of twisted derivations
43 44
4
17
In [18], we proposed a type of twisted derivations and constructed Darboux transformations in terms of it in a unified way. Such twisted derivations include derivatives, difference operators, superderivatives and q-difference operators as special cases. Surprisingly, the formulae for the iteration of Darboux transformations were expressed in terms of quasideterminants. Here we avoid to talk about quasideterminants and their properties. But one can refer to [29] and [30] for details. In this section, to make the paper selfconsistent, we will first give a brief review of the twisted derivation and its corresponding Darboux transformation presented in [18] and then show how the results can be applied to recover quasideterminant solutions to some known nc soliton equations which involve ordinary derivatives, difference operators and superderivatives.
27 28
3
16
25 26
2
15
2. A type of twisted derivations and Darboux transformations
17 18
1
54
Let θ0 , θ1 , θ2 , . . . be a sequence in A. As was pointed out in [18], an elementary Darboux transformation expressed terms of the twisted derivation is determined by
G θ = σ (θ) D θ
−1
= D − D (θ)θ
−1
55 56 57
, θ ∈A
(1)
58
where θ is invertible, D and σ are the twisted derivation and homomorphism defined above. For any general twisted derivation D we have the following theorem
59
Theorem 1. Let φ[0] = φ and for n ∈ N let
62
61 63
−1
φ[n] = D (φ[n − 1]) − D (θ[n − 1])θ[n − 1] where θ[n] = φ[n]|φ→θn . Then, for n ∈ N,
60
φ[n − 1],
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θ0 D (θ0 ) .. . φ[n] = D n−1 (θ0 ) D n (θ0 )
1 2 3 4 5 6 7
··· ··· ···
. D n (φ)
θn−1 D (θn−1 ) .. . D n−1 (θn−1 )
···
φ D (φ) .. . D n−1 (φ)
n
D (θn−1 )
3
1 2 3
(2)
5 6 7
8 9 10 11 12 13
8
Remark 2. In particular, if we have a Lax pair L , M with coefficients u , v , . . . , say, satisfying some given nonlinear system we choose θ0 , . . . , θn−1 as particular eigenfunctions of L , M. Then a generic eigenfunction φ of L , M is transformed to an eigenfunction of L [n], M [n] with coefficients u [n], v [n], . . . satisfying the same nonlinear system. Thereby we obtain a new solution u [n], v [n], . . . of the nonlinear system in terms of the ‘seed’ solution u , v , . . . and quasideterminants expressed in terms of the seed eigenfunctions. We illustrate this point in the examples in the next section.
14 15
18 19 20
23
26
In the literature, Darboux transformations and quasideterminant solutions to the nc KP equation, the non-Abelian (2 + 1)-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation have been constructed, separately. In this section, we would like to illustrate how to derive Darboux transformations for the above-mentioned nc integrable systems from the more general Darboux transformation defined in terms of the twisted derivations.
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
19 20
25
Lφ =
(∂x2
26
+ v x − ∂ y )φ = 0,
27 28
M φ = (4∂x3 + 6v x ∂x + 3v xx + 3v y + ∂t )φ = 0.
29 30
Theorem 1, using the ordinary derivative D = ∂x , gives the nth transformation of the eigenfunction and the transformed potential is
31
v [n] = v − 2Q x , where
32 33
θ0 ∂x (θ0 ) .. . Q = ∂ n−1 (θ ) 0 x ∂ n (θ ) x 0
··· ···
0 0
···
θn−1 ∂x (θn−1 ) .. . ∂xn−1 (θn−1 )
···
∂xn (θn−1 )
0
.. . 1
34 35 36 37
(3)
38 39 40 41 42
which coincide with the results obtained in [12].
43
Example 2 (The non-Abelian 2DTL equation [14,15]). The non-Abelian 2DTL equation is written as
44 45
U m , x + U m V m +1 − V m U m = 0,
46
V m,t + αm U m−1 − U m αm+1 = 0.
48
47 49
It has the Lax pair
50 51
18
24
It has the Lax pair
32 34
17
23
( v t + v xxx + 3v x v x )x + 3v y y − 3[ v x , v y ] = 0.
29
33
13
22
Example 1 (The nc KP equation [1–9,12]). This equation is
28
31
12
21
27
30
11
16
24 25
10
15
21 22
9
14
2.3. Applications in some known noncommutative integrable systems
16 17
4
50
φm,x = V m φm + αm φm−1 ,
51 52
φm,t = U m φm+1 .
53
In this case the Darboux transformation is expressed in terms of the twisted derivation D = T m , the translation operator in discrete variable m which leads to the results obtained in [15]. In particular, if the potential U m and V m are written in the form
57
then the effect of the nth Darboux transformation is encapsulated as the transformation of X m :
··· ···
55 56
−1 −1 U m = Xm Xm +1 , V m = X m , x X m ,
θ0 T m (θ0 ) .. n −1 . X m [n] = (−1) T n−1 (θ ) 0 m T n (θ ) m 0
54
0 0
···
θn−1 T m (θn−1 ) .. . n −1 Tm (θn−1 )
1
···
n Tm (θn−1 )
0
.. .
Xm+n−1 .
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4
1
Example 3 (The super KdV equation [18,31–33]). The Manin–Radul super KdV equation is written as
2 4
ut =
5 6 8
2
1
αt = (αxx + 3α D (α ) + 6α u )x ,
3
7
4 1 4
4
(u xx + 3u 2 + 3α D (u ))x ,
20 21 22 23 24 25 26
10 11 12
We take even/odd particular eigenfunction θ2i /θ2i +1 of the Lax pair, respectively, and use the twisted derivation (superderivative) D to define the Darboux transformation. The transformation of the odd potential is given by
α [n] = (−1) α − D ( Q ), where
θ0 D (θ0 ) .. . Q = D n−1 (θ ) 0 D n (θ ) 0
··· ···
···
0 0
.. . 1
n
D (θn−1 )
0
17
18 19 20
(4)
31 32 33 34 35 36
25
The formula for even potential u [n] is more complicated, but is again expressed in terms of quasideterminants, and is given in [18].
26 27 28
3. The noncommutative q-2DTL equation
29
In the previous section, we illustrated how to recover Darboux transformations and furthermore quasideterminant solutions for some known nc integrable systems which are defined in terms of ordinary derivatives, forward difference operators and super derivatives from the more general Darboux transformation for the twisted derivations. All these results inspire us to look for Darboux transformations and quasideterminant solutions to nc q-difference soliton equations. In this section, we will first give a brief review of some known facts about the q-2DTL equation, its bilinear form and determinant solutions. Next, we will construct a bilinear Bäcklund transformation for this bilinear form by using determinant identities and further derive a Lax pair for a formally different nonlinear q-2DTL equation.
37 38
41
y ) = J n (x, y ) V n (qα x, y ) − J n+1 (x, y ) V n (x, y ),
δqβ , y J n (x, y ) = V n−1 (qα x, y ) − qβ V n (x, qβ y ),
47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
33 34 35 36
40
In [28], a q-discrete version of the two-dimensional Toda lattice equation was proposed
43
46
32
39
δqα ,x V n (x,
45
31
38
3.1. Some known results for the q-2DTL equation
42 44
30
37
39 40
22 24
29 30
21 23
27 28
14 16
θn−1 D (θn−1 ) .. . D n−1 (θn−1 )
···
13 15
n
16
19
9
4
15
7 8
3
12
18
6
α are even and odd dependent variables respectively. It has the Lax pair
φt = M φ = (∂x3 + ((α ∂x + ∂x α ) D + u ∂x + ∂x u ))φ.
11
17
5
L φ = (∂x2 + α D + u )φ = λφ,
10
14
3
where D = ∂θ + θ∂x with θ an odd Grassmann variable, u and
9
13
1
41
(5) (6)
δqα ,x f (x, y ) =
(1 − q)x
, δqβ , y f (x, y ) =
(1 − q) y
.
(7)
48 49 51 52
53
τn (qα x, y )τn−1 (x, y )) {1 + (1 − q) xy } J n (x, y ) = −1 , (1 − q)x τn (x, qβ y )τn−1 (qα x, y ) 2
54 55 56
the q-2DTL equations (5) and (6) were transformed to the bilinear form
57
{δqα ,x δqβ , y τn (x, y )}τn (x, y ) − {δqα ,x τn (x, y )}{δqβ , y τn (x, y )} = τn+1 (x, q y )τn−1 (qα x, y ) − τn (qα x, qβ y )τn (x, y ), β
(8)
58 59
whose solutions were given by
n
47
50
τn+1 (qα x, y )τn−1 (x, qβ y ) V n (x, y ) = , τn (x, qβ y )τn (x, y )
(1 ) f n (x, y ) (2 ) f (x, y ) τn (x, y ) = n . .. ( N ) f (x, y )
44 46
f (x, y ) − f (x, qβ y )
Under the dependent variable transformations
1
43 45
where δqα ,x and δqβ , y are q-difference operators defined by
f (x, y ) − f (qα x, y )
42
60
(1 )
f n+1 (x, y ) · · · (2 ) f n+1 (x, y )
··· .. . ··· (N ) f n+1 (x, y ) · · ·
(1 ) f n+ N −1 (x, y ) (2 ) f n+ N −1 (x, y ) .. . (N ) f n+ N −1 (x, y )
61 62 63
(9)
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
(k)
where f n , k = 1, . . . , N, satisfy the dispersion relations (k)
30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
(k)
(k)
1 2
(k)
δqα ,x f n (x, y ) = − f n+1 (x, y ), δqβ , y f n (x, y ) = f n−1 (x, y ).
(10)
3
In the following, we will first present a bilinear Bäcklund transformation for (8) and then derive a Lax pair whose compatibility condition yields a formally different q-2DTL equation.
4
3.2. Bilinear Bäcklund transformations for (8)
7
By considering the determinantal identities between to the solutions given by (9), we obtain the following bilinear Bäcklund transformation for (8)
9
δqα ,x τn (x, y ) · τn−1 (x, y ) − δqα ,x τn−1 (x, y ) · τn (x, y ) = τn−1
(11)
(qα x, y )τ (x, y ),
(12)
n
τn (x, y ) (N-soliton) given by (9) to another solution τn (x, y ) (( N + 1)-soliton) given by (1 ) (1 ) (1 ) f n (x, y ) f n+1 (x, y ) · · · f n+ N (x, y ) (2 ) (2 ) (2 ) f (x, y ) f n+1 (x, y ) · · · f n+ N (x, y ) . τn (x, y ) = n . .. .. .. . . · · · f ( N +1) (x, y ) f ( N +1) (x, y ) · · · f ( N +1) (x, y )
which transforms the solution
In what follows, we will prove that the bilinear Bäcklund transformation relates adopt the notations used in [28]. We denote
(1 )
24
30 31
(15)
34 35 36 37
(16)
−β
x, y ) = | − 1, 0, . . . , N − 2, N − 1q−α x |,
(17)
y ) = |0q−β y , 1, . . . , N − 1, t |,
τn (q
−α
x, y ) = |0, . . . , N − 2, N − 1q−α x , t |,
(1 − q)q−β y τn −1 (x, q−β y ) = |0q−β y , 0, . . . , N − 2, N − 1|, −α
(18)
0
·········
1 ··· N − 1
0
········· ∅
∅ N ········· ········· 1 ··· N − 1 N
42 43 44 45
48 49 50 51 52 53 54
(23)
55 56 57 58
(1 − q)q−β y τn −1 (x, q−β y )τn+1 (x, y ) − τn (x, q−β y )τn (x, y ) + τn (x, q−β y )τn (x, y ) = 0,
59
(24)
62
In the same way, we can prove that the Laplacian expansion of the following determinant identity
∅ N −1 ········· ········· 0 ··· N − 2 N −1
60 61
which is nothing but (11) with y replaced by qβ y.
········· ∅
41
(20)
leads to the following equation
0 ··· N − 2
40
47
(22)
t · · · · · · · · · ≡ 0 t
39
46
(21)
x, y ) = |0, 1, . . . , N − 1, N − 1q−α x |.
38
(19)
It is straightforward to prove that the Laplacian expansion of the determinant identity
−1 ········· −1
32 33
τn (x, q−β y ) = |0q−β y , 1, . . . , N − 1, N |, τn −1 (x, q−β y ) = | − 1q−β y , 0, . . . , N − 2, N − 1|,
0q−β y ········· 0q−β y
26
29
τn−1 (x, y ) = | − 1, 0, . . . , N − 2, N − 1|, τn (x, q
xτn (q
25
28
− (1 − q)q
23
27
τn+1 (x, y ) = |1, 2, . . . , N , t |, τn−1 (x, y ) = | − 1, 0, . . . , N − 2, t |,
19
22
where t = (0, 0, . . . , 0, 1) T is an ( N + 1)-vector. Both τn (x, y ) given by (9) and τn (x, y ) given by (13) are expressed in term of Casoratian determinants. Concerning the applications of Casoratian determinants to integrable systems, we refer to [34,35]. Using the dispersion relations (10), we have
−α
14
21
⎞
τn (x, y ) = |0, 1, . . . , N − 1, t |,
τn−1 (q
13
20
Similarly, we can denote
−α
12
18
(13)
(14)
⎜ (2 ) ⎟ ⎜ f (x, y ) ⎟ ⎜ n+ j ⎟ ⎜ ⎟ .. ⎟. “ j” = ⎜ ⎜ ⎟ . ⎜ (N ) ⎟ ⎜ f (x, y ) ⎟ ⎝ n+ j ⎠ ( N +1 ) f n+ j (x, y )
11
17
where the number “ j” stands symbolically for a column vector given by
f n+ j (x, y )
10
16
τn (x, y ) and τn (x, y ). For the sake of simplicity, we
τn (x, y ) = |0, 1, . . . , N − 1, N | ⎛
6
15
n+ N
n +1
n
5
8
δqβ , y τn (x, y ) · τn (x, y ) − δqβ , y τn (x, y ) · τn (x, y ) = τn −1 (x, y )τn+1 (x, qβ y ),
28 29
5
N − 1q−α x
········· N − 1q−α x
· · · · · · · · · ≡ 0 t
63
t
64
(25)
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1
leads to
2
1
τn−1 (x, y )τn (q
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−α
x, y ) − τn−1 (q
−α
x, y )τn (x, y ) − (1 − q)q
−α
xτn (q
−α
x, y )τn−1 (x, y ) = 0,
(26)
which is again nothing but (12) by replacing x with qα x.
5 6
3.3. A new form of the q-2DTL equation and its Lax pair
7 8 9
7
By introducing the eigenfunction φn+1 (x, y ) = τn (x, y )/τn (x, y ), we can derive a Lax pair from the bilinear Bäcklund transformation (11) and (12). We have
10
δqβ , y φn+1 (x, y ) = W n (x, y )φn (x, y ),
11 12
δqα ,x φn (x,
13 14 15 16 17 18 19 20 21 22 23 24
y ) = −φn+1 (x, y ) + K n (x, y )φn (x, y ),
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
48
11
(28)
12 13
where
14
τn−1 (x, y )τn+1 (x, qβ y ) W n (x, y ) = , τn (x, y )τn (x, qβ y )
1 τn−1 (x, y )τn (qα x, y ) . 1− K n (x, y ) = (1 − q)x τn (x, y )τn−1 (qα x, y )
15
(29)
20 21
δqβ , y K n (x, y ) = W n (x, y ) − W n−1 (qα x, y ),
(31)
δqα ,x W n (x, y ) = K n+1 (x, q y ) W n (x, y ) − W n (qα x, y ) K n (x, y ).
(32)
53
Hereafter, for simplicity, we denote the q-difference operators D 1 = δqα ,x , D 2 = δqβ , y . σ1 and σ2 are homomorphism satisfying σ1 ( f (x, y )) = f (qα x, y ) and σ2 ( g (x, y )) = g (x, qβ y ) respectively. With the new notations, (31) and (32) can be rewritten as
D 2 ( K n ) = W n − σ1 ( W n−1 ),
(33)
D 1 ( W n ) = σ2 ( K n+1 ) W n − σ1 ( W n ) K n .
(34)
It is clear that D 1 and D 2 are twisted derivations with respect to σ1 and σ2 respectively. We will see later, in (38)–(40), that the effect of the Darboux transformation on the potentials W n and K n can be encapsulated in the effect on a new variable X n and by this we are led to define
W n = σ2 ( X n+1 ) X n−1 , K n = D 1 ( X n ) X n−1 .
(35)
58 59 60 61 62 63 64 65 66
24
D 2 ( D 1 ( X n ) X n−1 ) = σ2 ( X n+1 ) X n−1 − σ1 (σ2 ( X n ) X n−−11 ).
26 27 28 29 30 31 32 33 34 35 36 37 38 39 41
(36)
From now on, we refer to (36) as the nc q-2DTL equation. In this equation, the dependent variables are not assumed to commute in general. Through the change of variables (35) any solution X n of (36) gives a solution of the original q-2DTL system (31)–(32).
42 43 44 45 46
Remark 4. In the commutative case, one can easily recover the bilinear q-2DTL equation (8) from the nc q-2DTL equation (36) by the substitution X n = τn /τn−1 .
47 48 49 50
4. Quasicasoratian solutions for (36)
51
In this section, we will first derive the Darboux transformations for the nc q-2DTL equation (36) and then construct its quasideterminant solutions by iterating Darboux transformations.
52 53 54 55
4.1. Darboux transformations
56 57
23
40
Then (33) and (34) can be surprisingly reduced to one single equation
54 55
22
25
Remark 3. Although the q-2DTL equations (33)–(34) obtained in this paper formally looks different from the known one (5)–(6), the equations (33) and (34) can be transformed to (5) and (6) by letting β → −β and defining J n (x, y ) = − K n (x, q−β n y ) and V n (x, y ) = −q−β n W n (x, q−β n y ). We use the alternative form (31)–(32) from now on.
51 52
18 19
The compatibility condition of the Lax pair (27) and (28) gives the following nonlinear q-2DTL equation
β
16 17
(30)
49 50
9 10
46 47
8
(27)
25 26
3 4
5 6
2
56
As was pointed out in Section 2, a q-difference operator is an example of a twisted derivation. Therefore, from the general Darboux transformations defined in terms of twisted derivations, we can get the following Darboux transformation for the nc q-2DTL equation (36).
n = σ1 (θn ) D 1 θn−1 φn = ( D 1 − D 1 (θn )θn−1 )φn = −φn+1 + θn+1 θn−1 φn , φ K n = K n+1 + σ1 (θn+1 θn ) − θn+2 θn−+11 = { D 1 (θn+1 θn−1 ) + σ1 (θn+1 θn−1 ) K n }θn θn−+11 , W n = σ2 (θn+2 θn−+11 ) W n θn θn−+11 = W n+1 − D 2 (θn+2 θn−+11 ), −1
X n = −θn+1 θn X n , −1
where θn is a particular solution of the Lax pair (27) and (28).
57 58 59
(37)
60
(38)
61
(39)
63
(40)
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7
4.2. Quasicasoratian solutions
1
2 3 4
2
Let θn,i , i = 1, . . . , N, be a particular set of eigenfunctions of the Lax pair (27) and (28) and introduce the notation n = (θn,1 , . . . , θn, N ). The Darboux transformations for the nc q-2DTL equation may be iterated by defining
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
φn [k + 1] = −φn+1 [k] + θn+1 [k]θn [k]φn [k],
(41)
K n [k + 1] = K n+1 [k] + σ1 (θn+1 [k]θn−1 [k]) − θn+2 [k]θn−+11 [k],
(42)
60 61 62 63 64 65 66
6 7 8 9
W n [k + 1] = σ2 (θn+2 [k]θn−+11 [k]) W n [k]θn [k]θn−+11 [k],
(43)
10 11
X n [k + 1] = −θn+1 [k]θn−1 [k] X n [k],
(44)
12 13
where φn [1] = φn , X n [1] = X n and
14
(45)
15
In what follows, we will show by induction that the results of N-repeated Darboux transformations φn [ N + 1] and X n [ N + 1] can be expressed in closed form as quasideterminants
17
θn [k] = φn [k]|φn →θn,k .
n+ N N n+ N −1 φn [ N + 1] = (−1) .. . n
φn+ N φn+ N −1 .. . φn
n+ N . , Xn [ N + 1] = .. n+1 n
16 18 19
0 .. . Xn . 0 1
20 21 22
(46)
23 24 25 26
The initial case N = 1 is obviously true for both φn [2] and X n [2]. In order to prove the inductive step, we shall use several quasideterminant identities. The source of these identities is [29,30] or they may be found in summary form in [15]. They include the noncommutative Sylvester/Jacobi identity (for expanding by row and column), homological relations (for moving the expansion point) and quasi-Plücker coordinate formula (identities amongst certain types of quasideterminants). In the following proof we indicate where these identities are used. From (41), we have
27 28 29 30 31 32
φn [ N + 2] = −φn+1 [ N + 1] + θn+1 [ N + 1]θn [ N + 1]−1 φn [ N + 1] n+ N +1 n+ N +1 θn+ N +1, N +1 φ n + N + 1 φn+ N + (−1) N n+ N θn+ N , N +1 = −(−1) N n+ N .. .. . .. .. . . . n+1 n+1 φn+1 θn+1, N +1 ⎧ ⎪ ⎪ ⎪ ⎪ n+ N +1 ⎪ ⎪ ⎨ = (−1) N +1 n+ N ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ n+1
φn+ N +1 φn+ N .. . φn+1
n+ N +1 n+ N − .. . n+1
n+ N +1 θn+ N +1, N +1 n+ N θn+ N , N +1 = (−1) N +1 .. .. . . n+1 θn+1, N +1 n θn, N +1
φn+ N +1 φn+ N .. . φn+1 φn
θn+ N +1, N +1 θn+ N , N +1 .. . θn+1, N +1
33
n+ N n+ N −1 .. . n
θn+ N , N +1 θn+ N −1, N +1 .. . θn, N +1
−1 n+ N n+ N −1 .. . n
−1 n+ N θn+ N , N +1 n+ N n+ N −1 θn+ N −1, N +1 n+ N −1 .. .. .. . . . n θn, N +1 n
φn+ N
φn+ N −1 .. . φn ⎫ φn+ N ⎪ ⎪ ⎪ ⎪ φn+ N −1 ⎪ ⎪ ⎬ .. . ⎪ ⎪ ⎪ ⎪ φn ⎪ ⎪ ⎭
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
,
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where we use the homological relation and then the noncommutative Jacobi identity. Similarly, from the iteration (44), we have
58 59
4 5
−1
32 33
3
57 58
−1
X n [ N + 2] = θn+1 [ N + 1]θn [ N + 1]
n+ N +1 = n+ N .. . n+1
θn+ N +1, N +1 θn+ N , N +1 .. . θn+1, N +1
59
X n [ N + 1]
n+ N n+ N −1 .. . n
θn+ N , N +1 θn+ N −1, N +1 .. . θn, N +1
−1 n+ N .. . n+1 n
0 .. . Xn 0 1
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n+ N +1 θn+ N +1, N +1 n+ N θn+ N , N +1 = . .. .. . n+1 θn+1, N +1 n+ N +1 θn+ N +1, N +1 n+ N θn+ N , N +1 = .. .. . . n+1 θ n + 1 , N +1 n θn, N +1
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−1 n+ N θn+ N , N +1 n+ N −1 θn+ N −1, N +1 .. .. Xn . . n θn, N +1 0 0 .. Xn , . 0 1
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where we use the quasi-Plücker coordinate formula twice. These prove the inductive steps for both φn [ N + 1] and X n [ N + 1]. So far, we have proved quasideterminant solutions given by (46) to the nc q-2DTL equation (36).
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15 16 17
(i )
Remark 5. In the commutative case, we can express the quasideterminant as a ratio of determinants. So if we assume θn,i = f n vacuum solution X n = (−1) N +2 in (36), then we have X n [ N + 1] = τn+1 /τn from (46) with τn given by (9).
and the
5. Conclusions In our previous work [18], we proposed a twisted derivation which includes ordinary derivative, forward difference operators, q-difference operators and superderivatives as special cases and constructed its Darboux transformation which has an iteration formula written in terms of a quasideterminant. This result opens the opportunity for an unified approach to construct Darboux transformations and therefore quasideterminant solutions for noncommutative integrable systems defined in terms of twisted derivations. In this paper we show how this has been achieved for a noncommutative q-2DTL equation which involves q-difference operators besides the nc KP equation, the non-Abelian two-dimensional Toda lattice equation, the non-Abelian Hirota–Miwa equation and the super KdV equation which involve ordinary derivative, forward difference operators and superderivatives, respectively. In this paper, we first construct the bilinear Bäcklund transformation for the known bilinear q-2DTL equation by using determinant identities and then derive the Lax pair for a formally different nonlinear q-2DTL equation which can be surprisingly reduced to one single equation. Based on the Lax pair and the more general Darboux transformation defined in terms of the twisted derivation, we are able to get the Darboux transformation and therefore construct quasicasoratian solutions to the nc q-2DTL equation. However, how to construct the binary Darboux transformation and quasigrammian solutions for the nc q-2DTL equation still remains unsolved. In some sense, the results obtained in this paper extend the applications of quasideterminants in nc integrable systems and reveal the advantage of using quasideterminants in dealing with nc integrable systems, q-difference soliton equations and supersymmetric equations. Acknowledgements This work was supported by the National Natural Science Foundation of China under the grants 11271266 and 11371323, China Scholarship Council and Beijing Teachers Training Center for Higher Education. The authors would like to thank the reviewers particularly for their careful reading and valuable comments. References [1] B.A. Kupershmidt, KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems, Mathematical Surveys and Monographs, vol. 78, American Mathematical Society, New York, NY, 2000. [2] L.D. Paniak, Exact noncommutative KP and KdV multi-solitons, arXiv:hep-th/0105185, 2001. [3] M. Sakakibara, Factorization methods for noncommutative KP and Toda hierarchy, J. Phys. A 37 (2004) L599–L604. [4] N. Wang, M. Wadati, Noncommutative extension of ∂¯ -dressing method, J. Phys. Soc. Jpn. 72 (2003) 1366–1373. [5] N. Wang, M. Wadati, Exact multi-line soliton solutions of noncommutative KP equation, J. Phys. Soc. Jpn. 72 (2003) 1881–1888. 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