On nonautonomous differential-difference AKP, BKP and CKP equations
aa r X i v : . [ n li n . S I] S e p ON NONAUTONOMOUS DIFFERENTIAL-DIFFERENCE AKP, BKP AND CKP EQUATIONS
WEI FU AND FRANK W. NIJHOFF
Abstract.
Three novel (2+2)-dimensional nonautonomous differential-difference equations of Toda-type are proposedwithin the so-called direct linearising framework for the discrete AKP, BKP and CKP equations established by theauthors [Proc. R. Soc. A, (2017) 20160915]. The direct linearising structure guarantees the integrability of thesenew equations in the sense of possessing nontrivial solutions expressed by integral with respect to continuous spectralvariables. Moreover, three novel (2+1)-dimensional integrable nonautonomous differential-difference equations are alsoestablished within the scheme, including two in the AKP class and the other one in the BKP class. Introduction
The study of discrete integrable systems has been becoming one of the most prominent branches in the theory ofintegrable systems in the past two decades, resulting in the establishment of many novel concepts and theories inmodern mathematics, see e.g. [11]. Compared with an integrable partial differential equation (PDE), its associatedintegrable partial difference equation (P∆E) often governs much richer structure, which mainly reflects on the factthat a discrete equation encodes the information of the whole hierarchy of its corresponding continuous equation inan implicit way. There are three typical equations (i.e. the discrete Kadomtsev–Petviashvili (KP)-type equations)which play the roles of master models in the theory of discrete integrable systems. The first one is the discrete AKPequation (often known as the Hirota equation or the Hirota–Miwa (HM) equation)( p i − p j )(T p h τ )(T p i T p j τ ) + ( p j − p h )(T p i τ )(T p j T p h τ ) + ( p h − p i )(T p j τ )(T p h T p i τ ) = 0 , (1.1)in which the dependent variable τ is a function of the discrete arguments n j (associated with their respect latticeparameter p j ) for j = 1 , , · · · , the notation T p j denotes the forward shift operation with respect to the correspondingdiscrete argument n j , and p h , p i and p j are distinct. Equation (1.1) was first introduced by Hirota [13] as a discreteanalogue of the generalised Toda equation, and was later reparametrised as the above form by Miwa [20] in the theoryof transformation groups, and the letter ‘A’ in its name is named after the infinite-dimensional algebra A ∞ . Thesecond one is the discrete BKP equation (also referred to as the Miwa equation)( p h − p i )( p i − p j )( p j − p h ) τ (T p h T p i T p j τ ) + ( p h + p i )( p h + p j )( p i − p j )(T p h τ )(T p i T p j τ )+ ( p i + p j )( p i + p h )( p j − p h )(T p i τ )(T p h T p j τ ) + ( p j + p h )( p j + p i )( p h − p i )(T p j τ )(T p h T p i τ ) = 0 , (1.2)which was introduced by Miwa in [20]. Although (1.2) has a fourth term in comparison with (1.1), its associatedalgebra is B ∞ , as a sub-algebra of A ∞ . The third model is the discrete CKP equation (also called the hyperdeterminantequation) ( A + A − A − A ) = 4 B B , (1.3)where the expressions A i and B i are given by A = ( p h − p i ) ( p i − p j ) ( p j − p h ) τ (T p h T p i T p j τ ) , A = ( p h + p i ) ( p h + p j ) ( p i − p j ) (T p h τ )(T p i T p j τ ) ,A = ( p i + p j ) ( p i + p h ) ( p j − p h ) (T p i τ )(T p h T p j τ ) , A = ( p j + p h ) ( p j + p i ) ( p h − p i ) (T p j τ )(T p h T p i τ ) ,B = ( p h − p i )( p j − p h ) (cid:2) ( p i + p j ) (T p i τ )(T p j τ ) − ( p i − p j ) τ (T p i T p j τ ) (cid:3) , and B = T p h B , respectively. This equation appeared in Kashaev’s paper [15] for the first time from the star-triangle transform in theIsing model, and was later identified by Schief [28] that such a model actually describes the superposition formula for thecontinuous KP equation of C ∞ -type. The form in (1.3) was parametrised by the authors for exact solutions in [10]. Ashigher-dimensional integrable models, the discrete AKP, BKP and CKP equations reduce to many lower-dimensionalP∆Es, such as the discrete Gel’fand–Dikii hierarchy including the famous discrete Korteweg–de Vries (KdV) andBoussinesq (BSQ) systems, see e.g. [4] and [23]. This observation implies that the discrete KP-type equations are onthe top among discrete integrable systems. The discrete KP-type equations are not only remarkable in their own rightin the integrable systems theory, but also play crucial roles in other subjects in modern mathematics, especially ingeometry. It was shown by Konopelchenko and Schief that the discrete AKP, BKP and CKP equations are connectedto fundamental theorems of plane geometry, i.e. Menelaus’ theorem [17], Reciprocal quadrangles [18] and Carnot’stheorem [28], respectively. Besides, Doliwa pointed out that the these models also arise in the multidimensionalquadrilateral lattice theory in discrete geometry [5–7]. Very recently, it was also revealed that the discrete KP-typeequations are closely related to discrete line complexes [2] and circle complexes [3].In addition to P∆Es, there are also the so-called generating PDEs, where the terminology ‘generating’ follows fromthe fact that it generates a whole continuous integrable hierarchy. From the perspective of integrable systems, thekey step of constructing generating PDEs is to take the lattice parameters in the associated P∆Es as independent Key words and phrases. differential-difference, nonautonomous, (2+2)-dimensional, tau function, KP, direct linearisation. variables. This technique is actually a reflection of introducing Miwa’s coordinates, see [16]. To be more precise,the derivative with respect to a lattice parameter is equivalent to the higher-order derivatives of the spatial variables,resulting in the higher-order symmetries in an integrable hierarchy. The first example was proposed in [22] for the KdVclass in the form of a nonautonomous fourth-order nonlinear equation, and it is integrable in its own right in sensethat it possesses soliton, Lax pair, Lagrangian structure, and Painlev´e reduction, etc. More importantly, it was shownin [30] that a proper generalisation of this equation incorporates the hyperbolic Ernst equation for a Weyl neutrinofield in general relativity. In spirit of this, a class of generating PDEs for the BSQ family, also as integrable nonlinearmodels, were constructed in [31], and they represent the hyperbolic Ernst equations for a source-free Maxwell fieldand a Weyl neutrino field, as a generalisation for the previous result of KdV.It turns out that it is very difficult to derive closed-form equations for a scalar field that would be the analoguesof the generating PDEs for the KP-type hierarchies. However, as we will show in this paper, it is possible to derivesemi-discrete analogues (in form of differential-difference equation (D∆E)) of these generating PDEs for the KP-typeequations, which simultaneously contain lattice variables and lattice parameters as independent variables. Such D∆Es,as nonautonomous semi-discrete equations, are significant in integrable systems theory in their own right. In fact,on the two-dimensional (2D) level, these semi-discrete equations appear in similarity reductions to discrete Painlev´eequations [26], and also play roles of master symmetries of 2D integrable difference equations, see e.g. [27, 33].We aim to construct the associated nonautonomous D∆Es for the KP-type equations, based on the direct lineari-sation (DL) framework for the KP-type equations given in [10]. The DL method was proposed by Fokas and Ablowitz(see e.g. [8, 9]) to solve the initial value problems for the KdV and KP equations, as a generalisation of the well-known Riemann–Hilebert problem (RHP) [1]. Subsequently, it was developed into a powerful tool to systematicallystudy integrable structures behind families of discrete and continuous nonlinear equations and their interrelations, seee.g. [21, 23–26]. The key idea in the DL is to associate a nonlinear equation with a linear integral equation, and byintroducing the infinite matrix structure, it allows us to observe the integrability and solution structures of a nonlinearsystem simultaneously. Recently, the authors further established the link between the linear integral equation andseveral affine Lie algebras. With the help of considering reductions on the measure of the linear integral equation, theDL scheme for the discrete AKP, BKP and CKP equations was proposed [10]. This makes it possible for us to furtherstudy the associated nonautonomous D∆Es for these KP-type equations from the general framework.In this paper, we establish the following novel (2+2)-dimensional nonautonomous D∆Es, which are parallel to theP∆Es (1.1), (1.2) and (1.3):12 ( p i − p j ) D p i D p j τ · τ = n i n j h (T p i T − p j τ )(T − p i T p j τ ) − τ i , (1.4)2 p i p j ( p i − p j )D p i D p j τ · τ = n i n j h ( p i + q j ) (T p i T − p j τ )(T − p i T p j τ ) − ( p i − q j ) (T p i T p j τ )(T − p i T − p j τ ) − p i p j ( p i + p j ) τ i , (1.5) p i p j ( p i − p j ) D p i D p j τ · τ = n i n j h ( p i − p j ) E F + ( p i + p j ) G H − p i p j ( p i + p j ) τ i , (1.6)where E , F , G and H are given by E = ( p i + p j ) (T − p i τ )(T − p j τ ) − ( p i − p j ) τ (T − p i T − p j τ ) , F = ( p i + p j ) (T p i τ )(T p j τ ) − ( p i − p j ) τ (T p i T p j τ ) ,G = ( p i − p j ) (T − p i τ )(T p j τ ) − ( p i + p j ) τ (T − p i T p j τ ) and H = ( p i − p j ) (T p i τ )(T − p j τ ) − ( p i + p j ) τ (T p i T − p j τ ) , respectively, as well as the novel (2+1)-dimensional nonautonomous D∆Es( p i − p j ) D p j [ τ (T p i T p j τ )] · [(T p i τ )(T p j τ )]= n j (T p j τ ) (T p i T p j τ )(T p i T − p j τ ) − ( n j + 1) τ (T p i τ ) (T p j τ ) + τ (T p i τ )(T p j τ )(T p i T p j τ ) . (1.7)and ( p i − p j )D p j (T p i τ ) · τ = n j h τ (T p i τ ) − (T p j τ )(T p i T − p j τ ) i . (1.8)in the AKP class, and2 p j ( p i − p j ) D p j (T p i τ ) · τ = n j h p i p j τ (T p i τ ) + ( p i − p j ) (T − p j τ )(T p i T p j τ ) − ( p i + p j ) (T p j τ )(T p i T − p j τ ) i , (1.9)for the BKP class, where the notation D · denotes Hirota’s bilinear derivative (see e.g. [14]) with respect to thecorresponding arguments p i and p j , which is defined byD x f · g = ( ∂ x − ∂ x ′ ) f ( x ) g ( x ′ ) | x ′ = x , for arbitrary differential functions f ( x ) and g ( x ).The tau functions for the above equations have their precise definitions in terms of infinite matrices and doubleintegral with regard to the spectral variables, which will be given in the sections of the derivations of these equations.We note that an autonomous version of (1.5) was given in [32] very recently, which plays the role of a higher-ordersemi-discrete BKP equation and was referred to as the (2+2)-dimensional Toda lattice. But here equation (1.5) is anonautonomous equation having the lattice parameters as the continuous independent variables.The paper is organised as follows. Section 2 is concerned with the formal structure of the direct linearisationapproach. In sections 3, 4 and 5, we provide the derivations of the six novel equations listed above. N NONAUTONOMOUS DIFFERENTIAL-DIFFERENCE AKP, BKP AND CKP EQUATIONS 3 Formal structure of the direct linearisation
In [10], the discrete AKP, BKP and CKP equations were studied within one coherent framework, given by thedirect linearising transform (DLT), which provides a dressing-type scheme for obtaining new solutions from given seedsolutions for those integrable equations. In contrast, what we mean by DL is a special case where the seed correspondsto a ‘free’ solution (namely when the initial solution is trivial) in the DLT. The latter restriction is useful if we wantto derive new equations from some basic assumptions about the initial solutions.We start with introducing some fundamental infinite matrices and vectors and their properties which are needed inthe DL framework, as later we need the infinite matrix representation in the DL. First we need a rank 1 projectionmatrix O = . . . 0 1 0 . . . , (2.1)where the ‘box’ denotes the location of the central element, namely the (0 , OU ) i,j = δ i, U ,j and ( U O ) i,j = U i, δ ,j , with δ i,j = (cid:26) , i = j, , i = j, ∀ i, j ∈ Z , for an arbitrary infinite matrix U = ... ... ... · · · U − , − U − , U − , · · ·· · · U , − U , U , · · ·· · · U , − U , U , · · · ... ... ... , where the operation ( · ) i,j denotes taking the ( i, j )-entry, namely only the entries in the central row and column arereserved after the action of the projection. Next, we introduce two infinite matrices Λ = . . . . . .0 10 10 . . .. . . and t Λ = . . .. . . 01 01 0. . . . . . . (2.2)The matrices Λ and t Λ are the transpose of each other, and have properties( Λ i ′ U ) i,j = U i + i ′ ,j and ( U t Λ j ′ ) i,j = U i,j + j ′ . This property indicates that the multiplications of Λ and t Λ raise the row index and the column index of U , respectively.For this reason, we refer to them as index-raising matrices. We also introduce the infinite-dimensional vectors c k = ( · · · , k − , , k, · · · ) T and t c k ′ = ( · · · , k ′− , , k ′ , · · · ) , (2.3)which obey the following identities: Λ c k = k c k , t c k ′ t Λ = k ′ t c k ′ . In the sections below, we also need the notions of trace and determinant for an infinite matrix. We give the formaldefinition of a trace of an arbitrary infinite matrix U as follows:tr U = X i ∈ Z U i,i . (2.4)Notice that this is a formal definition, since the infinite summation with respect to i over the integer ring may lead todivergence. In order to avoid this issue, in this paper we only deal with the trace of a matrix involving the projectionmatrix O , in which case the trace is always convergent. For example, we havetr( OU ) = tr( U O ) = U , = ( U ) , . In the convergent case, the trace also satisfies tr(
U V ) = tr
V U for arbitrary infinite matrices U and V . Thedeterminant of an infinite matrix is defined throughln(det U ) = tr(ln U ) . (2.5) WEI FU AND FRANK W. NIJHOFF
Again, this is a formal definition which could result in divergence problem. But if we restrict ourselves to the infinitematrix 1 + U , where 1 is the identity infinite matrix and U is an infinite matrix involving the projection matrix O ,the determinant is well-defined, since the right hand side of (2.5), namelytr [ln(1 + U )] = tr " ∞ X i =1 ( − i − i U i = ∞ X i =1 ( − i − i tr (cid:0) U i (cid:1) , has terms of convergent traces. We have the well-known Weinstein–Aronszajn formulae for the determinant. Forinstance, the following identities hold:det(1 + U OV ) = 1 + (
V U ) , , det(1 + U ( O Λ − t Λ O ) V ) = det (cid:18) Λ V U ) , − ( Λ V t Λ ) , ( V U ) , − ( V U t Λ ) , (cid:19) . which are the cases of rank 1 and rank 2.We provide the formal structure of the DL framework. The starting point is a linear integral equation u k + Z Z D d ζ ( l, l ′ ) ρ k Ω k,l ′ σ l ′ u l = ρ k c k , (2.6)in which the wave function u k is an infinite-dimensional column vector having its i -th component u ( i ) k (for i ∈ Z ) beinga function of the lattice variables n j (and also the associated lattice parameters p j ) for j = 1 , , · · · and the spectralvariable k (or l ), Ω k,l ′ is the kernel of the linear integral equation, depending on the spectral variables k and l ′ , d ζ and D are the measure and the domain for integration, and ρ k and σ l ′ are the so-called plane wave factors depending onthe discrete variables n j and lattice parameters p j as well as their respective spectral variables k and l ′ . Furthermore,to derive equations in given choice of independent variables (which could be either the discrete variables n j or thecontinuous variables p j ) we assume that the measure is independent of these chosen variables.We present the infinite matrix representation of (2.6). We first introduce the infinite matrix Ω defined throughΩ k,k ′ = t c k ′ Ω c k . (2.7)Following from the properties of the index-raising matrices and the projection matrix listed above, we can observethat Ω is actually an infinite matrix composed of Λ , t Λ and O , which relies on the precise expression of Ω k,k ′ , namelyit is an infinite matrix representation of the Cauchy kernel of the linear integral equation (2.6). If we replace Ω k,l ′ with the help of (2.7), the linear integral equation is reformulated as u k = (1 − U Ω ) c k ρ k , (2.8)where the infinite matrix U is given by U . = Z Z D d ζ ( k, k ′ ) u kt c k ′ σ k ′ . (2.9)Next, we consider the infinite matrix representation of the plane wave factors and introduce an infinite matrix C . = Z Z D d ζ ( k, k ′ ) ρ k c kt c k ′ σ k ′ . (2.10)The key characteristic of C is the product of the two plane wave factors, i.e. ρ k σ k ′ , which we normally refer to as theso-called effective plane wave factor. By acting the operation RR D d ζ ( k, k ′ ) · t c k ′ σ k ′ on equation (2.10), we obtain U = (1 − U Ω ) C , or equivalently U = C (1 + Ω C ) − . (2.11)The idea of the DL approach is to associate a nonlinear equation with a linear integral equation in the form of(2.6). Once the plane wave factors, the Cauchy kernel and the measure are given, the corresponding class of nonlinearintegrable systems is fully determined. To be more precise, for a certain class of nonlinear integrable equations, theinfinite matrix C describes the linear dispersion, the infinite matrix Ω together with the measure governs the nonlinearstructure of the corresponding integrable models, and u k and U are corresponding to the wave function in the Laxpair and the nonlinear potential, respectively. In the sections 3, 4 and 5, we follow the structures for the discreteAKP, BKP and CKP equations that we identified in [10], and establish the DL scheme for the nonautonomous D∆Es(1.4)–(1.9) given in section 1, respectively. 3. The AKP case
In the discrete AKP case, the discrete plane wave factors ρ k and σ k ′ are given by ρ k = ∞ Y i =1 ( p i + k ) n i and σ k ′ = ∞ Y i =1 ( p i − k ′ ) − n i , (3.1) N NONAUTONOMOUS DIFFERENTIAL-DIFFERENCE AKP, BKP AND CKP EQUATIONS 5 respectively. Substituting these into the infinite matrix C defined by (2.10) and considering the evolutions with regardto the discrete variables and lattice parameters, we obtain dynamical evolutions as follows:(T p j C )( p j − t Λ ) = ( p j + Λ ) C , (3.2a) ∂ p j C = n j (cid:18) p j + Λ C − C p j − t Λ (cid:19) , (3.2b)(T − a T b C ) b − t Λ a − t Λ = b + Λ a + Λ C . (3.2c)Below we provide the derivation of (3.2b). By differentiating ρ k σ k ′ with respect to p j , we have ∂ p j ρ k σ k ′ = ∂ p j ∞ X i =1 (cid:18) p i + kp i − k ′ (cid:19) n i = n j (cid:18) p j + k − p j − k ′ (cid:19) ∞ X i =1 (cid:18) p i + kp i − k ′ (cid:19) n i = n j (cid:18) p j + k − p j − k ′ (cid:19) ρ k σ k ′ , and thus, the same operation on C gives rise to ∂ p j C = Z Z D d ζ ( k, k ′ ) c k ∂ p j ( ρ k σ k ′ ) t c k ′ = Z Z D d ζ ( k, k ′ ) c k ∂ p j ( ρ k σ k ′ ) t c k ′ = n j (cid:18)Z Z D d ζ ( k, k ′ ) 1 p j + k c k ρ k σ k ′ t c k ′ − Z Z D d ζ ( k, k ′ ) c k ρ k σ k ′ t c k ′ p j − k ′ (cid:19) = n j (cid:18) p j + Λ C − C p j − t Λ (cid:19) , where in the last step we have made use of the property of the operation of the index-raising matrices on c k and t c k ′ given in section 2. The other two equations are derived similarly.The Cauchy kernel in the linear integral equation for the discrete AKP equation takes the form ofΩ k,k ′ = 1 k + k ′ , (3.3)and in this case we have no further requirement for the measure d ζ ( k, k ′ ) and the domain D , namely they can bearbitrary. According to (2.7), we can observe that in this case Ω = − P ∞ i =0 ( − t Λ ) − i − O Λ i ; in other words, it satisfies ΩΛ + t ΛΩ = O . (3.4)Equation (3.4) can also be written in other forms. Here we reformulate it in the following forms whose left hand sidesare compatible with (3.2): Ω ( p j + Λ ) + ( p j − t Λ ) Ω = O , (3.5a) Ω p j + Λ − p j − t Λ Ω = 1 − p j + t Λ O p j + Λ , (3.5b) Ω b + Λ a + Λ − b − t Λ a − t Λ Ω = − ( a − b ) 1 − a + t Λ O a + Λ . (3.5c)Equations (3.5) together with (3.2) will provide the dynamical evolutions of the infinite matrix U as follows:(T p j U )( p j − t Λ ) = ( p j + Λ ) U − (T p j U ) OU , (3.6a) ∂ p j U = n j (cid:18) p j + Λ U − U p j − t Λ − U − p j + t Λ O p j + Λ U (cid:19) , (3.6b) (cid:0) T a T − b U (cid:1) b − t Λ a − t Λ = a + Λ b + Λ U + ( a − b )(T − a T b U ) 1 − a + t Λ O a + Λ U . (3.6c)Again we only give the derivation of (3.6b) below and skip that of the other two, as the procedure is similar. Noticethat the infinite matrix U obeys (2.11) in the formal structure of the DL. Calculating the derivative of U with respectto p j , we obtain ∂ p j U = − ( ∂ p j U ) Ω C + (1 − U Ω )( ∂ p j C ) , which can equivalently be rewritten as( ∂ p j U )(1 + Ω C ) = n j (1 − U Ω ) (cid:18) p j + Λ C − C p j − t Λ (cid:19) = n j (cid:20) p j + Λ C − U p j − t Λ − U Ω p j + Λ C (cid:21) = n j (cid:20) p j + Λ C − U p j − t Λ (1 + Ω C ) − U − p j + t Λ O p j + Λ C (cid:21) , where in the first and third equalities we have used (3.2b) and (3.5b), respectively. Multiplying the above equationby (1 + Ω C ) − immediately gives rise to (3.6b).Next, we introduce the tau function in this class, which is defined as τ . = det(1 + Ω C ) , (3.7) WEI FU AND FRANK W. NIJHOFF where 1 denotes the infinite unit matrix. As we have mentioned in section 2, the determinant of an infinite matrixshould be understood as the formal expansion of exp { tr[ln(1 + Ω C )] } . Since Ω = − P ∞ i =0 ( − t Λ ) − i − O Λ i involves O in every term, the trace action is always convergent. For convenience, we introduce quantities u . = ( U ) , , V a . = 1 − (cid:18) U a + t Λ (cid:19) , , W a . = 1 − (cid:18) a + Λ U (cid:19) , and S a,b . = (cid:18) a + Λ U b + t Λ (cid:19) , . The tau function satisfies the dynamical evolutions with respect to the discrete variables and the lattices parametersas follows: T p j ττ = 1 − (cid:18) U − p j + t Λ (cid:19) , = V − p j , T − p j ττ = 1 − (cid:18) p j + Λ U (cid:19) , = W p j , (3.8a) ∂ p j ln τ = n j S p j , − p j , (3.8b)T − a T b ττ = 1 − ( a − b ) (cid:18) a + Λ U − b + t Λ (cid:19) , = 1 − ( a − b ) S a, − b , (3.8c)( p i − p j ) τ (T p i T p j τ )(T p i τ )(T p j τ ) = p i − p j + T p j u − T p i u. (3.8d)These equations are proven through direct computation in terms of infinite matrices. For instance, calculating thederivative of ln τ with respect to p j yields ∂ p j ln τ = ∂ p j ln[det(1 + Ω C )] = ∂ p j tr[ln(1 + Ω C )] = tr[ ∂ p j ln(1 + Ω C )] = tr[(1 + Ω C ) − Ω ( ∂ p j C )]= n j tr (cid:20) (1 + Ω C ) − Ω (cid:18) p j + Λ C − C p j − t Λ (cid:19)(cid:21) = n j tr (cid:20) C (1 + Ω C ) − (cid:18) Ω p j + Λ − p j − t Λ Ω (cid:19)(cid:21) . where in the last step the property of the cyclic permutation of the trace operation is used. With the help of (2.11)and (3.5b), this equation is reformulated as ∂ p j ln τ = n j tr (cid:18) U − p j + t Λ O p j + Λ (cid:19) = n j tr (cid:18) O p j + Λ U − p j + t Λ (cid:19) = n j (cid:18) p j + Λ U − p j + t Λ (cid:19) , , namely ∂ p j ln τ = n j S p j , − p j . To prove Equations (3.8a) and (3.8c), one needs to use (3.2a) and (3.5a) and (3.2c) and(3.5c), respectively. Since the idea of the proofs is similar to that of (3.8a), we skip them here. While equation (3.8d)is proven based on (3.8a) and (3.6), and it is a widely known relation which describes bilinear transformation for thediscrete KP equation (see e.g. [10]).Equations listed in (3.6) are the key formulae in the DL scheme to construct closed-form integrable equations, andthey together with (3.8) can produce the bilinear equations in the AKP class. In [10], the well-known HM equation,i.e. equation (1.1), is derived from (3.6a) and (3.8a). Here we start from the dynamical relations in terms of p j toconstruct the (2+2)-dimensional nonautonomous D∆E in this class. Considering ( a + Λ ) − (3.6b)( b + t Λ ) − and takingthe (0 , S a,b : ∂ p j S a,b = n j (cid:20)(cid:18) p j − a − p j + b (cid:19) S a,b − p j − a S p j ,b + 1 p j + b S a, − p j − S a, − p j S p j ,b (cid:21) . By setting a = p i and b = − p i , we reach to ∂ p j S p i , − p i = n j ( p i − p j ) (cid:2)(cid:0) − ( p j − p i ) S p j , − p i (cid:1) (cid:0) − ( p i − p j ) S p i , − p j (cid:1) − (cid:3) , which only involves the S -variable. If we replace all the S -variables in this equation by the tau function via (3.8) andnotice the identity ∂ p i ∂ p j ln τ = D p i D p j τ · τ τ for Hirota’s bilinear operator D · , the following bilinear equation arises:12 ( p i − p j ) D p i D p j τ · τ = n i n j h (T p i T − p j τ )(T − p i T p j τ ) − τ i , which is nothing but equation (1.4). This equation takes the form of a nonautonomous version of the 2D Toda equation,if we think of the discrete shift operations T p i T − p j and T − p i T p j as the forward and backward shifts along skew directionon the lattice. The difference is that here the lattice parameters p i and p j act the independent variables; while in thetwo-dimensional Toda lattice (2DTL), the bilinear derivatives are in terms of the continuous spacial variables.From the scheme, we can also construct two more nonautonomous semi-discrete equations of τ . Taking [(3.6b)( a + t Λ ) − ] , , one obtains ∂ p j V a = − n j (cid:20) V − p j S p j ,a + 1 a + p j ( V a − V − p j ) (cid:21) , and its a = − p i case gives rise to ∂ p j V − p i = n j p i − p j (cid:2) V − p i − V − p j (cid:0) − ( p j − p i ) S p j , − p i (cid:1)(cid:3) . N NONAUTONOMOUS DIFFERENTIAL-DIFFERENCE AKP, BKP AND CKP EQUATIONS 7
Substituting the V - and S -variables with the tau function with the help of (3.8), we are able to construct a newbilinear equation ( p i − p j )D p j (T p i τ ) · τ = n j h τ (T p i τ ) − (T p j τ )(T p i T − p j τ ) i , namely equation (1.8). Moreover, if we consider the equation for ∂ p j u by evaluating the (0 , ∂ p j u = n j (1 − V − p j W p j ) = n j " − (T p j τ )(T − p j τ ) τ , where formula (3.8) is used for the second equality, and therefore, we obtain from (3.8d) that ∂ p j (cid:20) ( p i − p j ) τ (T p i T p j τ )(T p i τ )(T p j τ ) (cid:21) = ∂ p j ( p i − p j + T p j u − T p i u ) = − p j ( ∂ p j u ) − T p i ( ∂ p j u ) . Finally, by substituting u with τ , a closed-form quartic equation of τ is constructed, which takes the form of( p i − p j ) D p j [ τ (T p i T p j τ )] · [(T p i τ )(T p j τ )]= n j (T p j τ ) (T p i T p j τ )(T p i T − p j τ ) − ( n j + 1) τ (T p i τ ) (T p j τ ) + τ (T p i τ )(T p j τ )(T p i T p j τ ) , i.e. equation (1.7). These equations are compatible with (1.4) in sense that the tau function defined by (3.7) is exactlythe same. 4. The BKP case
We select the following plane wave factors for BKP: ρ k = ∞ Y i =1 (cid:18) p i + kp i − k (cid:19) n i , σ k ′ = ρ k ′ = ∞ Y i =1 (cid:18) p i + k ′ p i − k ′ (cid:19) n i , (4.1)which should be understood as discrete odd flows, compared with (3.1) in the AKP case. In this case, it is observedthat the infinite matrix C defined in (2.10) evolves with regard to the discrete variables and the lattice parameters inthe following way: (T p j C ) p j − t Λ p j + t Λ = p j + Λ p j − Λ C , (4.2a) ∂ p j C = n j (cid:20)(cid:18) p j + Λ − p j − Λ (cid:19) C + C (cid:18) p j + t Λ − p j − t Λ (cid:19)(cid:21) . (4.2b)The derivation of these formulae is straightforward, as they follow from the formal definition of C , namely (2.10).The Cauchy kernel in the BKP case takes the form ofΩ k,k ′ = 12 k − k ′ k + k ′ , (4.3)which implies that the corresponding Ω satisfies ΩΛ + t ΛΩ = 12 ( ΩΛ − t Λ O ) . (4.4)For the purpose of looking for compatible relations of Ω with equations listed in (4.2), we reformulate (4.4) as Ω p j + Λ p j − Λ − p j − t Λ p j + t Λ Ω = p j p j + t Λ ( O Λ − t Λ O ) 1 p j − Λ (4.5a)and Ω (cid:18) p j + Λ − p j − Λ (cid:19) + (cid:18) p j + t Λ − p j − t Λ (cid:19) Ω = −
12 1 p j − t Λ ( O Λ − t Λ O ) 1 p j + Λ −
12 1 p j + t Λ ( O Λ − t Λ O ) 1 p j − Λ , (4.5b)respectively.Equations (4.2) and (4.5) are the fundamental relations to construct the dynamical evolutions of the infinite matrix U . By considering T p j U and ∂ p j U in (2.11), some straightforward calculation yields the following equations:(T p j U ) p j − t Λ p j + t Λ = p j + Λ p j − Λ U − p j (T p j U ) 1 p j + t Λ ( O Λ − t Λ O ) 1 p j − Λ U , (4.6a) ∂ p j U = n j (cid:20)(cid:18) p j + Λ − p j − Λ (cid:19) U + U (cid:18) p j + t Λ − p j − t Λ (cid:19) + 12 U p j − t Λ ( O Λ − t Λ O ) 1 p j + Λ U + 12 U p j + t Λ ( O Λ − t Λ O ) 1 p j − Λ U (cid:21) (4.6b)with the help of (4.2) and (4.5). WEI FU AND FRANK W. NIJHOFF
Besides the different kernel and plane wave factors, we also have to impose certain restrictions on the integrationmeasure and the integration domain for the BKP equation. We require that the integration domain D is symmetricin terms of the spectral variables k and k ′ , and then the measure d ζ ( k, k ′ ) is antisymmetric, i.e.d ζ ( k ′ , k ) = − d ζ ( k, k ′ ) . (4.7)These will together result in t C = Z Z D d ζ ( k, k ′ ) ρ k c k ′ t c k ρ k ′ = − ZZ D d ζ ( k ′ , k ) ρ k ′ c k ′ t c k ρ k = − C ;in other words, we have an antisymmetric infinite matrix C in the discrete BKP. Notice that the kernel given in (4.3)is also antisymmetric, or equivalently t Ω = − Ω . We can deduce from (2.11) that in this case the infinite matrix U obeys the antisymmetry property t U = − U . (4.8)Next, we present the dynamical evolutions of the tau function. For convenience we introduce quantities V a . = 1 − (cid:18) U aa − t Λ (cid:19) , , W a . = 1 + (cid:18) aa − Λ U (cid:19) , and S a,b . = (cid:18) aa − Λ U bb − t Λ (cid:19) , . Due to the antisymmetry property of the infinite matrix U , it is obvious to see that V a = W a and S a,b = − S b,a . We define the tau function in this class by τ = det(1 + Ω C ) , (4.9)since the antisymmetry of O and C will eventually make the above determinant a perfect square, namely the taufunction itself is corresponding to a Pfaffian. We now consider the evolution of the tau function with respect to n j ,and this gives us T p j τ = det (cid:2) Ω (T p j C ) (cid:3) = det (cid:20) Ω C + p j p j − t Λ ( O Λ − t Λ O ) 1 p j − Λ C (cid:21) = τ det " p ( Λ p − Λ U p − t Λ ) , − p ( Λ p − Λ U t Λ p − t Λ ) , p ( p − Λ U p − t Λ ) , − p ( p − Λ U t Λ p − t Λ ) , = τ V p j , where the rank 2 Weinstein–Aronszajn formula and equation (4.8) are used in the third and fourth equalities, re-spectively. Similarly, acting the backward shift on τ gives us T − p j τ /τ = V − p j . In order to simplify this formula, weevaluate [(4.6a)] , , which gives us V p j (T p j V − p j ) = 1. And thus, without loss of generality, we haveT p j ττ = V p j and T − p j ττ = V − p j . (4.10a)Next, we calculate ( p i p i − Λ (4.6a)) , . The following equation is obtained:1 + 2 V p j (T p j S − p i , − p j ) = p i − p j p i + p j ( V p j − V − p i ) + V p j (T p j V − p i ) . This equation provides a way to express the S -variable by the V -variables, and consequently, the S -variable can beexpressed by the tau function by making use of (4.10), and the formula is S − p i , − p j = 12 " T − p i τ − T − p j ττ + p i − p j p i + p j − T − p i T − p j ττ ! . (4.10b)The expressions S p i , − p j , S − p i ,p j and S p i ,p j in terms of the tau function are derived from the above equation, since inthe BKP case the identity T − p j = T − p j holds. Therefore, we have obtained the dynamical relations of τ in terms of S a,b for arbitrary a + b = 0. Furthermore, following the same idea of deriving (3.8b), we also have2 p j ∂ p j ln τ = n j ( V p j − V − p j − S p j , − p j ) (4.10c)for BKP. Equations listed in (4.10) establish the relations between the dynamics of the tau function and the V - and S -variables.To construct equation (1.5) in the DL framework, we compute [ aa − Λ (4.6b)] , and [ aa − Λ (4.6b) bb − t Λ ] , , which givesrise to p j ∂ p j V a = n j " V a ap j a − p j − V p j + 12 V − p j ! + V p j (cid:18) p j + ap j − a + S a, − p j (cid:19) − V − p j (cid:18) p j − ap j + a + S a,p j (cid:19) , (4.11a) N NONAUTONOMOUS DIFFERENTIAL-DIFFERENCE AKP, BKP AND CKP EQUATIONS 9 and p j ∂ p j S a,b = n j (cid:20) S p j ,b (cid:18) p j + ap j − a − W a (cid:19) + 12 S − p j ,b (cid:18) a − p j a + p j + W a (cid:19) + 12 S a,p j (cid:18) p j + bp j − b − V b (cid:19) + 12 S a, − p j (cid:18) b − p j b + p j + V b (cid:19) + S a,b ap j a − p j + 2 bp j b − p j ! − S a,p j S − p j ,b + S a, − p j S p j ,b , (4.11b)respectively. These two equations allow us to derive the closed-form equations in terms of the tau function. Noticethat the logarithm derivative of τ with respect to p j satisfies (4.10c), and hence we have p i p j ∂ p i ∂ p i ln τ = p j ∂ p j ( p i ∂ p i ln τ ) = p j ∂ p j [ n i ( V p i − V − p i − S p i ,p i )] = n i (cid:2) p j ( ∂ p j V p i ) − p j ( ∂ p j V − p i ) − p j ( ∂ p j S p i ,p i ) (cid:3) , in which the right hand side only involves the derivatives of the V - and S -variables with respect to p j , and they canbe further replaced by terms of V - and S -variables without derivatives with the help of (4.11a) and (4.11a) for special a and b . Finally, equations (4.10a) and (4.10b) help us to express every term on the right hand side of the aboveequation by the tau function. As a result, we end up with the following (2+2)-dimensional nonautonomous Toda-likeequation:2 p i p j ( p i − p j )D p i D p j τ · τ = n i n j h ( p i + q j ) (T p i T − p j τ )(T − p i T p j τ ) − ( p i − q j ) (T p i T p j τ )(T − p i T − p j τ ) − p i p j ( p i + p j ) τ i , which is exactly the same as equation (1.5).We can also derive an analogue of (1.8) in the BKP class, from the DL scheme in this section. Setting a = p i in(4.11a) yields p j ∂ p j V p i = n j " V p i p i p j p i − p j − V p j + 12 V − p j ! + V p j (cid:18) p j + p i p j − p i + S p i , − p j (cid:19) − V − p j (cid:18) p j − p i p j + p i + S p i ,p j (cid:19) . Substituting V and S according to (4.10), we obtain a nonautonomous differential-difference bilinear equation2 p j ( p i − p j ) D p j (T p i τ ) · τ = n j h p i p j τ (T p i τ ) + ( p i − p j ) (T − p j τ )(T p i T p j τ ) − ( p i + p j ) (T p j τ )(T p i T − p j τ ) i , which is nothing but equation (1.9).An interesting observation is that the right sides of (1.5) and (1.9) take the forms of Hirota’s discrete-time Todaequation (a 5-point equation) [12] and the bilinear discrete KdV equation (a 6-point equation), respectively (see alsochapter 8 of [11]), though the parametrisation here is entirely different.5. The CKP case
We choose the same plane wave factors as those of BKP in the CKP class, namely ρ k = ∞ Y i =1 (cid:18) p i + kp i − k (cid:19) n i and σ k ′ = ρ k ′ = ∞ Y i =1 (cid:18) p i + k ′ p i − k ′ (cid:19) n i , (5.1)and these provide the same dynamical relations for the infinite matrix C as follows:(T p j C ) p j − t Λ p j + t Λ = p j + Λ p j − Λ C , (5.2a) ∂ p j C = n j (cid:20)(cid:18) p j + Λ − p j − Λ (cid:19) C + C (cid:18) p j + t Λ − p j − t Λ (cid:19)(cid:21) , (5.2b)which describe the linear dispersion of the discrete CKP equation.We select the Cauchy kernel Ω k,k ′ = 1 k + k ′ (5.3)for the CKP class, which is the same as that in the AKP case, and therefore we still have the fundamental relationfor Ω given by ΩΛ + t ΛΩ = O . (5.4)Since the evolution of the infinite matrix C is governed by (5.2), we reformulate (5.4) and present the followingequations: Ω p j + Λ p j − Λ − p j − t Λ p j + t Λ Ω = 2 p j p j + t Λ O p j − Λ , (5.5a) Ω (cid:18) p j + Λ − p j − Λ (cid:19) + (cid:18) p j + t Λ − p j − t Λ (cid:19) Ω = − p j − t Λ O p j + Λ − p j + t Λ O p j − Λ , (5.5b)which makes these equations compatible with (5.2), in order to derive the dynamical relations of U below. Now we construct the dynamical evolutions of the infinite matrix U . Acting the shift operator T p j on equation(2.11) and taking (5.2a), we haveT p j U = (cid:2) − (T p j U ) Ω (cid:3) (T p j C ) = (cid:2) − (T p j U ) Ω (cid:3) p j + Λ p j − Λ C p j + t Λ p j − t Λ . Notice that the infinite matrix Ω obeys (5.5a). We can rewrite the above equation as(T p j U ) p j − t Λ p j + t Λ = p j + Λ p j − Λ C − (T p j U ) Ω p j + Λ p j − Λ C = p j + Λ p j − Λ C − (T p j U ) (cid:18) p j p j + t Λ O p j − Λ + p j − t Λ p j + t Λ Ω (cid:19) C , and this can be further simplified as(T p j U ) p j − t Λ p j + t Λ = p j + Λ p j − Λ U − p j (T p j U ) 1 p j + t Λ O p j − Λ U . (5.6a)Equation (5.6a) is the first dynamical relation we need and it describes how U evolves along the lattice directions n j .The other dynamical relation we need is the one evolving with regard to p j , and the derivation is very similar to thoseof (3.6b) and (4.6b), which reads ∂ p j U = n j (cid:20)(cid:18) p j + Λ − p j − Λ (cid:19) U + U (cid:18) p j + t Λ − p j − t Λ (cid:19) + U p j − t Λ O p j + Λ U + U p j + t Λ O p j − Λ U (cid:21) . (5.6b)Similar to the BKP case, we also need to impose a certain constraint on the spectral variables k and k ′ . This isrealised by setting the integration domain D symmetric and simultaneously requiring integration measure satisfyingd ζ ( k ′ , k ) = d ζ ( k, k ′ ) . (5.7)Such a reduction results in a symmetric infinite matrix C because t C = Z Z D d ζ ( k, k ′ ) ρ k c k ′ t c k ρ k ′ = Z Z D d ζ ( k ′ , k ) ρ k ′ c k ′ t c k ρ k = C . At the same time, we observe that the kernel (5.3) is symmetric in terms of k and k ′ , which implies that t Ω = Ω . Thesymmetry properties of C and Ω together guarantee that U from (2.11) satisfies t U = U , (5.8)i.e. it is an infinite symmetric matrix.During the derivation of (1.6) below, for convenience we introduce variables V a . = 1 − (cid:18) U a + t Λ (cid:19) , , W a . = 1 − (cid:18) a + Λ U (cid:19) , and S a,b . = (cid:18) a + Λ U b + t Λ (cid:19) , , (5.9)which satisfy V a = W a and S a,b = S b,a , (5.10)respectively, due to the symmetry condition (5.8). We define the tau function in this section as τ = det(1 + Ω C ) , (5.11)and after certain straightforward but relatively complex calculation based on (5.6a), we can deriveT p j ττ = 1 + 2 p j S − p j , − p j and T − p j ττ = 1 − p j S p j ,p j , (5.12a)as well as (cid:2) − ( p i + p j ) S p i ,p j (cid:3) = ( p i + p j ) (T − p i τ )(T − p j τ ) − ( p i − p j ) τ (T − p i T − p j τ )4 p i p j τ . (5.12b)Similar to the BKP case, the S − p i ,p j , S p i , − p j and S − p i , − p j analogues of (5.12b) follow from the identity T − p j = T − p j .Moreover, the dynamics of the tau function with respect to the lattice parameter p j is derived by carrying the samecalculation in the derivation of (3.8b), which establishes the link between τ and S and takes the form of ∂ p j ln τ = 2 n j S p j , − p j . (5.12c)Notice from (5.12) that the tau function is connected with the S -variable. We therefore multiply (5.6b) by ( a + Λ ) − from the left and ( b + t Λ ) − from the right simultaneously and take the (0 , S a,b arises in the form of ∂ p j S a,b = n j (cid:20) p j − a ( S a,b − S p j ,b ) + 1 p j + a ( S − p j ,b − S a,b )+ 1 b − p j ( S a,p j − S a,b ) + 1 p j + b ( S a, − p j − S a,b ) − S a, − p j S p j ,b − S a,p j S − p j ,b (cid:21) . N NONAUTONOMOUS DIFFERENTIAL-DIFFERENCE AKP, BKP AND CKP EQUATIONS 11
By setting a = p i and b = − p i , it is rewritten as ∂ p j S p i , − p i = n j ( p i + p j ) (cid:2) (1 − ( p i + p j ) S p i ,p j )(1 + ( p i + p j ) S − p i , − p j ) − (cid:3) + n j ( p i − p j ) (cid:2) (1 − ( p i − p j ) S p i , − p j )(1 − ( p j − p i ) S − p i ,p j ) − (cid:3) . It is then possible to replace the S -variable by τ through (5.12a) and (5.12b) in the above equation. As a result, itleads to a closed-form equation p i p j ( p i − p j ) D p i D p j τ · τ = n i n j h ( p i − p j ) E F + ( p i + p j ) G H − p i p j ( p i + p j ) τ i , where E , F , G and H are given by E = ( p i + p j ) (T − p i τ )(T − p j τ ) − ( p i − p j ) τ (T − p i T − p j τ ) , F = ( p i + p j ) (T p i τ )(T p j τ ) − ( p i − p j ) τ (T p i T p j τ ) ,G = ( p i − p j ) (T − p i τ )(T p j τ ) − ( p i + p j ) τ (T − p i T p j τ ) and H = ( p i − p j ) (T p i τ )(T − p j τ ) − ( p i + p j ) τ (T p i T − p j τ ) , respectively, which is exactly the (2+2)-dimensional nonautonomous differential-difference CKP equation (1.6). Wenote that this equation is still in the form of the (2+2)-dimensional Toda-type, but bilinearity is broken (though it isstill quadratic) compared with (1.4) and (1.5) since the square root operation is involved. This is not surprising as wehave seen the discrete CKP equation (1.3) is a quartic equation. Equation (1.6) can alternatively be written as " p i p j ( p i − p j ) n i n j D p i D p j τ · τ + 8 p i p j ( p i + p j ) τ − ( p i − p j ) EF − ( p i + p j ) GH = 4( p i − p j ) EF GH by taking square twice in order to eliminate the square root in the expression, namely an eighth-power equation.6.
Concluding remarks
Based on the DL framework for the discrete AKP, BKP and CKP equations, we introduce a new perspectiveto construct integrable nonlinear equations, by thinking of the lattice parameters as the independent variables andtreating them together with the discrete variables to be on the same footing. As a result, six novel nonautonomousdifferential-equations are proposed, including three (2+2)-dimensional nonautonomous D∆Es of Toda-type, namely(1.4), (1.5) and (1.6) from their respective class, as well as three (2+1)-dimensional nonautonomous semi-discreteequations, i.e. (1.7) and (1.8) from the AKP class and (1.9) from the BKP class.All the equations are integrable in sense that they are solved by their respective tau functions (which simultaneouslysolve the corresponding fully discrete equations) expressed by the determinant of an infinite matrix involving a doubleintegral over a continuous space of spectral parameters. Such a formulism also contains the well-known multi-solitonsolution as a very special case, reducing to solutions expressed by a finite determinant, cf. [10].It was already shown that nonautonomous D∆Es often play the role of master symmetries for lower-dimensionaldiscrete integrable systems [29], while the master symmetry theory for higher-dimensional lattice equations is not yetclear. These new equations potentially provide us with an insight into understanding master symmetries of three-dimensional (3D) integrable discrete models. Furthermore, the (2+2)-dimensional form (i.e. the 2D Toda-like form)for equations (1.4)–(1.6) is very rare to see in the literature. Their geometric interpretation still needs to be discovered.In addition to symmetries master symmetries, we also note that the integrability of discrete KP-type equationswas also proven from the perspective of conservations laws. In [19], conservation laws for the discrete AKP and BKPequations were constructed, while it remains a problem for the discrete CKP equations.The ultimate goal is to find closed-form generating PDEs for higher-dimensional integrable hierarchies and theirintegrability structures. This remains an open problem for future study.
Acknowledgments
WF was sponsored by the National Natural Science Foundation of China (grant nos. 11901198 and 11871396) andby Shanghai Pujiang Program (grant no. 19PJ1403200). This project was also partially supported by the Science andTechnology Commission of Shanghai Municipality (grant no. 18dz2271000).
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School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice,East China Normal University, 500 Dongchuan Road, Shanghai 200241, People’s Republic of China (FWN)(FWN)