On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries Equation
aa r X i v : . [ n li n . S I] F e b ON THE THREE-DIMENSIONAL CONSISTENCY OF HIROTA’S DISCRETEKORTEWEG-DE VRIES EQUATION
NALINI JOSHI AND NOBUTAKA NAKAZONOA bstract . Hirota’s discrete Korteweg-de Vries equation (dKdV) is an integrable partialdi ff erence equation on Z , which approaches the Korteweg-de Vries equation in a con-tinuum limit. We find new transformations to other equations, including a second-degreesecond-order partial di ff erence equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that hasbeen widely used to study partial di ff erence equations on multidimensional lattices. Dedicated to Harvey Segur on the occasion of his 80th birthday.
1. I ntroduction
The main subject of this paper is the partial di ff erence equation u l + , m + − u l , m = u l , m + − u l + , m , u ∈ C , ( l , m ) ∈ Z , (1.1)and its non-autonomous version (1.2). This equation is interesting from at least two pointsof view: (i) it is an integrable discrete version of the Korteweg-de Vries equation andshares many of its famous properties; but, (ii) it does not satisfy a consistency property thathas been widely used to characterise and classify many other integrable partial di ff erenceequations. We describe new observations that overcome this disjunction.The nonlinear equation (1.1) is said to be integrable because it arises as a compatibilitycondition for a pair of linear partial di ff erence equations, called a Lax pair, dependingnon-trivially on a spectral parameter. On the discrete lattice Z , compatibility is equivalentto consistency, i.e., the result is independent of the order of in which we compose theiterations l l + m m +
1. There exist transformations of integrable equations thatgive rise to iterations in additional directions independent of ( l , m ) ∈ Z . A natural questionis to ask whether the resulting system of equations is consistent in such an extended multi-dimensional lattice.We answer this question by providing new transformations of Equation (1.1) and show-ing that the resulting system of nonlinear partial di ff erence equations (or, P ∆ Es) is con-sistent in a way that appears not to have been observed previously. It is notable that thesystem composed of Equation (1.1) and image P ∆ Es arising from transformations containsa second-degree P ∆ E.There is a non-autonomous generalization of Equation (1.1) with similar properties. Itis given by [5, 8, 10, 15] u l + , m + − u l , m = q m + − p l u l , m + − q m − p l + u l + , m , ( p l , q m ) ∈ C , (1.2) Mathematics Subject Classification.
Key words and phrases.
Korteweg-de Vries equation; integrable systems; partial di ff erence equations; latticeequations .Corresponding author: Nalini Joshi. NJ’s ORCID ID is 0000-0001-7504-4444. Her research was supportedby an Australian Research Council Discovery Projects where p l , q m are parameters that vary with one independent variable, either l or m respec-tively. Equation (1.1) is given by the reduction q m + − p l = q m − p l + =
1, which aresatisfied by p l = a , q m = a +
1, for an arbitrary constant a .Equation (1.1) was proposed by Hirota [7] as a di ff erence analogue of the Korteweg-deVries (KdV) equation w t + w w x + w xxx = , w ∈ C , ( x , t ) ∈ C , (1.3)in the sense that the continuum limit of Equation (1.1) leads to the KdV equation (see [7,§2]). Since Equation (1.1) is a special case of Equation (1.2), and they share similar prop-erties, we refer interchangebly to one or the other as the discrete KdV (dKdV) equation,where there is no confusion.Equation (1.1) shares distinctive properties with the KdV equation. It possesses N -soliton solutions, where N ∈ N \{ } . There exist transformations of the dependent variable u l , m that are discrete analogues of B¨acklund transformations of the KdV equation. The newtransformations we provide are new B¨acklund transformations. Equation (1.1) has a Laxpair, which gives rise to a discrete inverse scattering transform methodology. Our resultsprovide a new way of deducing a Lax pair.There is a graphical interpretation of consistency that provides a simple geometric wayof studying the evolution of the dKdV equation. Consider each iterate { u l , m , u l + , m , u l , m + , u l + , m + } as the value assigned to a vertex of a unit cell in a lattice isomorphic to Z ; seeFigure 1.1. (In the non-autonomous case, we assign the parameter p l to the edge connecting u l , m with u l + , m and q m to the edge connecting u l , m with u l , m + .) Then, the iteration l l + u l , m and parameters ( p l , q m ) assigned to a unit cell of Z .is equivalent to mapping the initial cell to an adjacent cell in the horizontal direction inFigure 1.1, while iterating m m + ∆ E, giving rise to a graphical description of theinitial value problem for the dKdV equation. (See [6, §1.3] for subtle issues that arise fordi ff erent initial configurations.) Under this interpretation, the dKdV equation is known asan integrable lattice equation [6].We now extend this two-dimensional lattice to a third dimension in the following way.Given a unit cell as shown in Figure 1.1, take a unit cell in an independent lattice andidentify two of its vertices, and the edge joining these two, with those in the original cell.We interpret the remaining edges in the new cell as lying along a third direction, indepen-dent of the two generating the original lattice. For the non-autonomous equation, we alsorequire a new parameter that evolves in the new direction.This generates a three-dimensional lattice given by Z , with unit cells given by cubes.If we are given a value at a vertex lying in the third direction, then the corresponding three-dimensional initial value problem for the P ∆ E starts with four initial values. If we assumethat the P ∆ E is given on each face of the cube, then we can find the values at every vertexof the resulting cube by solving the P ∆ E. D CONSISTENCY OF HIROTA’S DKDV EQUATION 3
While this construction is straightforward, there are possible inconsistencies that mayarise because a vertex value on the cube may be calculated in three di ff erent ways, cor-responding to the three di ff erent faces meeting at that vertex. Lattice equations that sat-isfy the resulting consistency conditions are said to be consistent around a cube (CAC).Equations satisfying the CAC property have been studied and classified under certain con-ditions [1, 14]. But, as mentioned above, the dKdV equation is known to not satisfy theseconsistency conditions [6, §3.8.1].The main new result of the current paper is to demonstrate a di ff erent construction bywhich the dKdV equation is consistently embedded in a higher dimensional lattice Z .Although the lattice structure we discover is di ff erent to the one conventionally used in theliterature, we show that it is not unique to dKdV equation − the same type of structure arisesin at least one other integrable P ∆ E, known as the lattice sine-Gordon equation [3, 16]: u l + , m + u l , m = ( γ u l , m + − γ − u l + , m )( γ − u l , m + )( γ u l + , m − , γ ∈ C . (1.4)1.1. Multi-quadratic equations.
Our results involve new transformations to multi-quad-ratic equations, involving a parameter λ ∈ C , which are listed below. The first group oftransformations given by Equations (3.1) maps the autonomous dKdV equation (1.1) to (cid:0) v l , m v l + , m − v l , m + v l + , m + (cid:1) + (cid:0) v l , m − v l + , m + (cid:1)(cid:0) v l + , m − v l , m + (cid:1)(cid:0) λ − v l , m v l + , m (cid:1)(cid:0) λ − v l , m + v l + , m + (cid:1) = . (1.5)The second group of transformations (see Equations (4.1)) maps the non-autonomousdKdV equation (1.2) to q m (cid:0) v l , m v l + , m − v l , m + v l + , m + (cid:1) + (cid:0) v l , m − v l + , m + (cid:1)(cid:0) v l + , m − v l , m + (cid:1)(cid:0) λ − v l , m v l + , m (cid:1)(cid:0) λ − v l , m + v l + , m + (cid:1) + (cid:0) p l + v l , m − p l v l + , m + (cid:1)(cid:0) p l v l + , m − p l + v l , m + (cid:1) − ( p l + p l + ) (cid:0) λ v l , m v l + , m + λ v l , m + v l + , m + − v l , m v l + , m v l , m + v l + , m + (cid:1) + (cid:0) p l + v l , m v l , m + + p l v l + , m v l + , m + (cid:1)(cid:0) λ − v l , m v l + , m − v l , m + v l + , m + (cid:1) = . (1.6)The third group of transformations (see Equations (4.7)) maps the lattice sine-Gordonequation (1.4) to γ (cid:0) v l , m − v l + , m + (cid:1) (cid:0) v l + , m − v l , m + (cid:1)(cid:0) λ + v l , m v l + , m (cid:1)(cid:0) λ + v l , m + v l + , m + (cid:1) − (1 − γ )( γ − λ ) (cid:0) v l , m v l + , m − v l , m + v l + , m + (cid:1) + γ (1 − γ ) (cid:0) v l , m v l + , m − v l , m + v l + , m + (cid:1) × (cid:16)(cid:0) λ + v l , m v l + , m (cid:1)(cid:0) v l , m + + v l + , m + (cid:1) − (cid:0) v l , m + v l + , m (cid:1)(cid:0) λ + v l , m + v l + , m + (cid:1)(cid:17) = . (1.7)1.2. Notation and Terminology.
For conciseness in the remainder of the paper, we adoptthe following notation: u = u l , m , u = u l + , m , e u = u l , m + , u = u l − , m , u e = u l , m − , e u = u l + , m + , (1.8)and extend the notation to p l , q m and other iterates of u as needed. To avoid overabundanceof decorations, we also use the notation u = u l , m , u = u l + , m , u = u l , m + , u = u l + , m + . (1.9)We write each lattice equation as the zero set of a polynomial of four variables, or itsequivalent rational form. For example, the dKdV equation is given by Q ( u , u , e u , e u ) = Q ( u , u , e u , e u ) = u e u ( e u − u ) − ( q m + − p l ) u + ( q m − p l + ) e u . NALINI JOSHI AND NOBUTAKA NAKAZONO
Note that, for conciseness, we omit the dependence of the polynomial Q on parameters.We assume that any parameters in the polynomial take generic values and that the corre-sponding polynomial is irreducible. Where the corresponding polynomial is linear in eachvariable, we describe it as an a ffi ne linear polynomial. Where the polynomial is quadraticin each variable, we refer to it as a multi-quadratic polynomial.Because of the association with a quadrilateral of Z , see Figure 1.1, a lattice equationrelating four vertex values is called a quad-equation . By a small abuse of terminology, wewill also refer to the corresponding function, whose zero set gives the lattice equation, as aquad-equation.1.3. Outline of the paper.
In Section 2, we describe a new way of embedding a latticeequation in Z , which di ff ers from the conventional one used for the CAC property. Thisprocess is applied to the autonomous dKdV equation (1.1) in Section 3, where we alsoshow how to deduce a Lax pair for this equation and how it is related to Equation (1.5).Finally, in Section 4, we show how to extend the construction to the non-autonomous dKdVequation (1.2) and to the lattice sine-Gordon equation (1.4).2. E mbedding into three - dimensions In this section, we describe a way to embed the dKdV equation in Z . To each elemen-tary cubic cell in Z , we associate 8 variables denoted by( u , u , u , u , v , v , v , v ) ∈ C , and assign each variable to a vertex of the cube.In contrast to the usual procedure assumed for proving the CAC property, we do notassign a quad-equation to each face of the cube. Instead, we describe a system of equationson the cube, which may (i) vary with each face; (ii) become a triangular equation, i.e.,those relating only three vertex values, on certain faces; and, (iii) involves vertices of aquadrilateral given by an interior diagonal slice of the cube.Three of the quad-equations occur on the bottom, front and back faces of the cube, whilethe fourth one occurs in the interior of the cube as a diagonal slice. Each triangular domainoccurs as a half of the left or right face of the cube. See Figure 2.1. We will refer to thisconfiguration as a broken cube .Figure 2.1. A cube with three quadrilateral faces labelled by A , C and C ′ , an interiordiagonal quadrilateral labelled by S and triangular domains labelled as B and B ′ . Notethat primes denote domains on parallel faces.Correspondingly, we define polynomials of 4 variables A , S , C , C ′ : C → C and thoseof 3 variables B , B ′ : C → C , such that B and B ′ written as functions of ( x , y , z ) satisfy(i) deg x B ≥
1, deg y B = deg z B = B = y and z , and each solution is a rational functionof the other two arguments. D CONSISTENCY OF HIROTA’S DKDV EQUATION 5
With the labelling of vertices given in Figure 2.1, we denote the system of six correspond-ing equations by A ( u , u , u , u ) = , S ( u , u , v , v ) = , (2.1a) B ( u , v , v ) = , B ′ ( u , v , v ) = , (2.1b) C ( u , u , v , v ) = , C ′ ( u , u , v , v ) = . (2.1c)The following definition describes how consistency holds for this system of equations. Definition 2.1 (CABC property) . Let { u , u , u , v } be given initial values. Using Equa-tions (2.1), we can express the variable v as a rational function in terms of the initialvalues in 3 ways. When the 3 results for v are equal, the system of Equations (2.1) issaid to be consistent around a broken cube or to have the consistency around a brokencube (CABC) property. In this case, we refer to the configuration of quadrilaterals andtriangular domains associated with the polynomials A , S , C , C ′ , B , B ′ as a CABC cube .Other equations arise from interrelationships between the above equations on the brokencube. For example, we show in the next section that Equation (1.5) arises on the top face,parallel to A . It is also useful to note equations that relate three vertices on a face to avertex on the opposite face. The following definition of such equations uses terminologyanalogous to existing ones in the literature on the CAC property. Definition 2.2 (Tetrahedron property) . A CABC cube is said to have a tetrahedron prop-erty , if there exist quad-equations K and K satisfying K ( u , u , v , v ) = , K ( u , u , v , v ) = . (2.2)In this case, each of the equations K = K = tetrahedronequation . Remark 2.3.
The above description of the broken cube (or the CABC property) and itsiteration in three-dimensional space cannot be replaced by a ffi ne linear transformationsof the standard cubic lattice. It is possible to reflect a broken cube around a horizontalplane through its centre and place a copy of it above the original one to create a verticalcolumn of alternating broken cubes. Stacking adjacent such vertical columns side-by-sideproduces a structure like that shown in Figure 2.2. However, it is not possible to rotate eachalternating column around its vertical axis of symmetry to create diagonal unit cubes fromthe triangular equations, because the iteration of the above polynomials does not have therequired symmetry. Figure 2.2. A stacking of vertically alternating broken cubes.
NALINI JOSHI AND NOBUTAKA NAKAZONO
By interpreting each vertex value as an iterate of a function in an appropriate way, wecan interpret the above equations as P ∆ Es. In particular, we use the terminology given inEquation (1.8) for u l , m and extend it to v l , m to give the following definition of P ∆ Es.
Definition 2.4 (CABC and tetrahedron properties for a system of P ∆ Es) . Define the P ∆ Es A (cid:0) u , u , e u , e u (cid:1) = , S (cid:0) u , u , e v , e v (cid:1) = , B (cid:0) u , v , e v (cid:1) = , C (cid:0) u , u , v , v (cid:1) = , (2.3)which give the following equations around each elementary cubic cell in Z : A ( u , u , u , u ) = A (cid:0) u , u , e u , e u (cid:1) = , S ( u , u , v , v ) = S (cid:0) u , u , e v , e v (cid:1) = , (2.4a) B ( u , v , v ) = B (cid:0) u , v , e v (cid:1) = , B ′ ( u , v , v ) = B (cid:0) u , v , e v (cid:1) = , (2.4b) C ( u , u , v , v ) = C (cid:0) u , u , v , v (cid:1) = , C ′ ( u , u , v , v ) = C (cid:0)e u , e u , e v , e v (cid:1) = . (2.4c)Then, the system (2.3) is said to have the CABC property if Definition 2.1 holds for theequations (2.4). We also transfer the definition of tetrahedron properties to P ∆ Es corre-sponding to K j , j = ,
2, in the obvious way. Moreover, the P ∆ E A (cid:0) u , u , e u , e u (cid:1) = Remark 2.5.
Note that Equations (2.3) are not necessarily autonomous. They may containparameters that evolve with ( l , m ) . In particular, this means that iteration from one face toa parallel face on the broken cube may give an equation with di ff erent coe ffi cients.
3. L attice structure of E quation (1.1)In this section, we show that Equation (1.1) has the CABC property, by deducing thecorresponding equations on a broken cube in a three-dimensional lattice. The main ad-vantage of this embedding is that it enables us to construct its Lax pair. We give thisconstruction below. We also show that the multi-quadratic quad-equation (1.5) arises fromthe CABC property.We start by defining equations on the broken cube: A (cid:0) u , u , e u , e u (cid:1) = e u − u − e u + u = , (3.1a) S (cid:0) u , u , e v , e v (cid:1) = u − e v − u + λ e v = , (3.1b) B (cid:0) u , v , e v (cid:1) = ( u − v ) (cid:16) u + e v (cid:17) − + λ = , (3.1c) C (cid:0) u , u , v , v (cid:1) = u − λ v − u + v = , (3.1d)where, as before, we have used the terminology given in Equation (1.8).It is straightforward to confirm that Definition 2.1 holds when A , B , B ′ , S , C and C ′ are defined by (2.4). Moreover, the tetrahedron properties hold. More explicitly, we have A = e u − u − e u + u = , S = u − e v − u + λ e v = , (3.2a) B = (cid:16) u − v (cid:17)(cid:16) u + e v (cid:17) − + λ = , B ′ = (cid:16) u − v (cid:17)(cid:16) u + e v (cid:17) − + λ = , (3.2b) C = u − λ v − u + v = , C ′ = e u − λ e v − e u + e v = , (3.2c)while the tetrahedron equations are given by K = (cid:16)e v + u (cid:17)(cid:16) u − λ v (cid:17) − + λ = , K = (cid:16) λ e v + u (cid:17)(cid:16) u − v (cid:17) − + λ = . (3.3)Hence, the following theorem holds. Theorem 3.1.
Equation (1.1) has the CABC property.
D CONSISTENCY OF HIROTA’S DKDV EQUATION 7
Now we show how to construct a Lax pair for the dKdV equation from the above systemof equations, through a method that parallels the well-known method for constructing theLax pair using the CAC property [4, 12, 17].To carry this out, we represent the auxiliary function v by v = FG , (3.4)where F = F l , m and G = G l , m . Substituting this into the equations (3.1c) and (3.1d),separating the numerators and denominators of the resulting equations, and using the 2-vector Ψ = Ψ l , m defined by Ψ = FG , (3.5)we obtain the following linear systems: Ψ = δ (1) l , m u − u λ Ψ , e Ψ = δ (2) l , m − u λ − u Ψ , (3.6)where δ (1) l , m and δ (2) l , m are arbitrary decoupling factors. The system of linear equations (3.6) isthe Lax pair of Equation (1.1). Indeed, we can easily verify that the compatibility condition e Ψ = e Ψ gives Equation (1.1) and δ (1) l , m δ (2) l + , m = δ (1) l , m + δ (2) l , m . (3.7)For simplicity, in what follows we take δ (1) l , m = δ (2) l , m = . (3.8)We next show how the multi-quadratic quad-equation (1.5) arises from the system (3.1)and investigate its properties. It arises immediately by eliminating the variable u from(3.1c) and (3.1d): A ′ (cid:16) v , v , e v , e v (cid:17) = (cid:16) vv − e v e v (cid:17) + (cid:16) v − e v (cid:17)(cid:16) v − e v (cid:17)(cid:16) λ − vv (cid:17)(cid:16) λ − e v e v (cid:17) = , (3.9)which is equivalent to Equation (1.5). The multivariate polynomial A ′ given by Equation(3.9) satisfies the Kleinian symmetry: A ′ (cid:16) v , v , e v , e v (cid:17) = A ′ (cid:16) v , v , e v , e v (cid:17) = A ′ (cid:16)e v , e v , v , v (cid:17) , (3.10)and the discriminants of A ′ = A ′ (cid:16) v , v , e v , e v (cid:17) are given by ∆ [ A ′ , v ] = G (cid:16) v , e v (cid:17) H (cid:16)e v , e v (cid:17) H (cid:16) v , e v (cid:17) , (3.11a) ∆ [ A ′ , v ] = G (cid:16) v , e v (cid:17) H (cid:16)e v , e v (cid:17) H (cid:16) v , e v (cid:17) , (3.11b) ∆ [ A ′ , e v ] = G (cid:16) v , e v (cid:17) H (cid:16) v , v (cid:17) H (cid:16) v , e v (cid:17) , (3.11c) ∆ [ A ′ , e v ] = G (cid:16) v , e v (cid:17) H (cid:16) v , v (cid:17) H (cid:16) v , e v (cid:17) , (3.11d)where the polynomials G ( x , y ), H ( x , y ) and H ( x , y ) are given by G ( x , y ) = ( x − y ) , H ( x , y ) = ( xy − λ ) , H ( x , y ) = ( xy − λ ) + xy . (3.12)Note that for a quadratic equation Q = ax + bx + c the discriminant ∆ [ Q , x ] is defined by ∆ [ Q , x ] = b − ac . (3.13)From the above information, we find that Equation (3.9) is dQ ∗ type in the sense givenin [2, 9]. NALINI JOSHI AND NOBUTAKA NAKAZONO
We finally show that Equations (1.1) and (1.5) can be solved by the lattice potentialmKdV equation (3.17). Rewrite the Lax pair (3.6) as follows F = − (cid:16) u − u (cid:17) F + λ G , G = F , e F = − u F + λ G , e G = F − uG . (3.14)Eliminating the variables u and F from Equations (3.14) leads to the following P ∆ E: (cid:16)e G − λ G (cid:17)(cid:16) G − e G (cid:17) + GG = . (3.15)Equation (3.15) is known as the lattice potential mKdV equation (lmKdV) [11, 13], alsolabelled H φ l , m = ( − λ ) − l / (1 − λ ) − m / G l , m , (3.16)Equation (3.15) can be rewritten in the following canonical form: e φφ = (1 − λ ) / φ − ( − λ ) / e φ (1 − λ ) / e φ − ( − λ ) / φ , (3.17)where φ = φ l , m . Moreover, from Equations (3.4), (3.14) and (3.16) we obtain u = F − e GG = G − e GG = ( − λ ) / φ − (1 − λ ) / e φφ , (3.18a) v = FG = GG = ( − λ ) / φφ . (3.18b)Therefore, we obtain the following lemma. Lemma 3.2.
Equations (1.1) and (1.5) are respectively solved byu = ( − λ ) / φ − (1 − λ ) / e φφ , v = ( − λ ) / φφ , (3.19) where φ = φ l , m is the solution of Equation (3.17) .Proof. The statement can be verified by direct calculation. (cid:3)
4. P roperties of E quations (1.2) and (1.4)In this section, we show that Equations (1.2) and (1.4) have the same properties asEquation (1.1). The process for demonstrating the result for each equation is exactly thesame as that for Equation (1.1) discussed in Section 3 and so, for consciseness, we omitdetailed arguments.4.1. Properties of Equation (1.2) . The system of P ∆ Es for Equation (1.2), which has theCABC and tetrahedron properties, is given by: A = e u − u − q m + − p l e u + q m − p l + u = , (4.1a) S = u − e v − q m − p l + u + λ − p l e v = , (4.1b) B = (cid:16) u − v (cid:17)(cid:16) q m − p l u + e v (cid:17) − q m + λ = , (4.1c) C = q m − p l u − λ − p l v − u + v = . (4.1d)Moreover, the tetrahedron equations are given by K = (cid:16)e v + q m − p l + u (cid:17)(cid:16) q m − p l u − λ − p l v (cid:17) − q m + λ = , (4.2a) K = (cid:16) λ − p l e v + u (cid:17)(cid:16) u − v (cid:17) − q m + λ = , (4.2b) D CONSISTENCY OF HIROTA’S DKDV EQUATION 9 and the Lax pair of Equation (1.2) is given by
Ψ = u − q m − p l u λ − p l Ψ , e Ψ = − q m − p l u λ − p l − u Ψ . (4.3)The corresponding multi-quadratic quad-equation associated with Equation (1.2) is givenby (1.6).As for Equation (1.1), there is a relation between the solutions of the non-autonomousdKdV equation (1.2) and that of Equation (1.6) with solutions of a non-autonomous lmKdVequation, as shown by the following lemma. Lemma 4.1.
Equations (1.2) and (1.6) are respectively solved byu = (cid:16) p l − λ (cid:17) / φ − (cid:16) q m − λ (cid:17) / e φφ , v = (cid:16) p l − λ (cid:17) / φφ , (4.4) where φ = φ l , m is the solution of the non-autonomous form of lmKdV [1, 11, 13]: e φφ = α l φ − β m e φα l e φ − β m φ , (4.5) where α l = (cid:16) p l − λ (cid:17) − / , β m = (cid:16) q m − λ (cid:17) − / . (4.6) Proof.
The statement can be verified by direct calculation. (cid:3)
Properties of Equation (1.4) . The system of P ∆ Es for the lattice sine-Gordon equa-tion (1.4), which has the CABC and tetrahedron properties, is given by: A = e uu − ( γ e u − γ − u )( γ − e u )( γ u − = , (4.7a) S = (cid:16) − γ u (cid:17)(cid:16) λ + γ e v (cid:17) − (cid:16) γ − u (cid:17)(cid:16) γ − e v (cid:17) u e v = , (4.7b) B = (cid:16) λ + γ e v − u e v (cid:17)(cid:16) − γ u + uv (cid:17) + (cid:16) − γ − λ (cid:17) uv = , (4.7c) C = (cid:16) − γ u (cid:17)(cid:16) λ + γ v (cid:17) − (cid:16) γ − u (cid:17)(cid:16) γ − v (cid:17) uv = . (4.7d)Moreover, the tetrahedron equations are given by K = (cid:16) λ − γλ u + (1 − γ ) uv (cid:17)(cid:16) − γ + γ e v − u e v (cid:17) + (cid:16) − γ − λ (cid:17)(cid:16) γ − u (cid:17)(cid:16) − γ u (cid:17) v = , (4.8a) K = (cid:16) λ + γ e v − u e v (cid:17)(cid:16) − γ u + u v (cid:17) + (cid:16) − γ − λ (cid:17) u e v = , (4.8b)and the Lax pair of Equation (1.4) is given by Ψ = γ + γ ( γ u − γ − u ) u λ ( γ u − γ − u ) u Ψ , e Ψ = ( γ − u γ − u λ ( γ u − γ − uu − γ u Ψ . (4.9)The corresponding multi-quadratic quad-equation associated with Equation (1.4) is givenby (1.7). Remark 4.2.
The multivariate polynomial A ′ for Equation (1.7): A ′ (cid:16) v , v , e v , e v (cid:17) = γ (cid:16) v − e v (cid:17)(cid:16) v − e v (cid:17)(cid:16) λ + vv (cid:17)(cid:16) λ + e v e v (cid:17) − (cid:16) − γ (cid:17)(cid:16) γ − λ (cid:17)(cid:16) vv − e v e v (cid:17) + γ (cid:16) − γ (cid:17)(cid:16) vv − e v e v (cid:17) (cid:16) λ + vv (cid:17)(cid:16)e v + e v (cid:17) − (cid:16) v + v (cid:17)(cid:16) λ + e v e v (cid:17) ! = satisfies the Kleinian symmetry (3.10) , and its discriminants are given by (3.11) withG ( x , y ) = γ ( x − y ) , H ( x , y ) = ( xy + λ ) , (4.11a) H ( x , y ) = (cid:16) γ xy + (1 − γ )( x + y ) − γλ (cid:17) − (cid:16) − γ − λ (cid:17) xy . (4.11b) From the information given above, we find that Equation (4.10) is an equation of dQ ∗ -typein the sense of [2, 9]. The solutions of Equations (1.4) and (1.7) are related to another P ∆ E as shown by thefollowing lemma.
Lemma 4.3.
Equations (1.4) and (1.7) are respectively solved byu = φ − e φγφ − φ , v = φφ , (4.12) where φ = φ l , m is the solution of the following P ∆ E: (cid:16) γ e φ + λφ (cid:17)(cid:16) φ − γ e φ (cid:17)(cid:16)e φ − φ (cid:17)(cid:16)e φ − φ (cid:17) = − γ . (4.13) Proof.
The statement can be verified by direct calculation. (cid:3)
5. C oncluding remarks
Much of what we know about the real world is modelled through continuous mathemat-ical equations that become discrete equations on the computer. Hirota’s dKdV equation isan important example in the study of such discretizations, because it shares the distinctiveproperties of the KdV equation.However, there are gaps in its study. The main open question studied in this paper con-cerns its embedding in a three-dimensional lattice and the question of its consistency. Byfinding previously unknown transformations to other P ∆ Es, we show that there is an un-usual embedding into a three-dimensional lattice along with a consistency property, whichwe call consistency around a broken cube . By using this property to construct a Lax pairfor the dKdV equation, we show that the embedding is related to its integrability. It isinteresting to note that a previously unknown transformation to a multi-quadratic latticeequation also arises from this construction.These observations lead to several open questions. One is how the construction mayextend to higher dimensional lattices, i.e., Z N , where N ≥
4. A second important questionis whether other integrable lattice equations, which do not satisfy the CAC property, turnout to satisfy the CABC property defined in Section 2. We anticipate that there may beother generalizations of consistency that remain to be found, particularly when we considernon-scalar, multi-component P ∆ Es.
Acknowledgment.
N. Nakazono would like to thank Dr. P. Kassotakis for inspiring andfruitful discussions about multi-quadratic equations and Dr. Y. Sun for those about thenon-autonomous form of the dKdV equation (1.2).R eferences [1] V. E. Adler, A. I. Bobenko, and Y. B. Suris. Classification of integrable equations on quad-graphs. Theconsistency approach.
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Ph.D. Thesis, University of Leeds , 2001.S chool of M athematics and S tatistics F07, T he U niversity of S ydney , NSW 2006, A ustralia . Email address : [email protected] I nstitute of E ngineering , T okyo U niversity of A griculture and T echnology , 2-24-16 N akacho K oganei ,T okyo apan . Email address ::