A Revisit to the ABS H2 Equation

In this paper we revisit the Adler-Bobenko-Suris H2 equation. The H2 equation is linearly related to the $S^{(0,0)}$ and $S^{(1,0)}$ variables in the Cauchy matrix scheme. We elaborate the coupled quad-system of $S^{(0,0)}$ and $S^{(1,0)}$ in terms of their 3-dimensional consistency, Lax pair, bilinear form and continuum limits. It is shown that $S^{(1,0)}$ itself satisfies a 9-point lattice equation and in continuum limit $S^{(1,0)}$ is related to the eigenfunction in the Lax pair of the Korteweg-de Vries equation.


Introduction
In the past two decades the research of discrete integrable systems has undergone a true development (e.g., see [7] and the references therein). The property called multi-dimensional consistency (MDC) plays a central role in understanding integrability of discrete systems [1,3,13,16]. The MDC means a lattice equation can be consistently embedded into a higher dimensional system. For quadrilateral equations, this property is geometrically described as the consistency around a cube (CAC). Adler, Bobenko and Suris (ABS) classified all affine linear quadrilateral equations that are consistent-around-cube (CAC), D 4 symmetric and possess the so-called tetrahedron property [1]. In their full list there are amazingly 9 equations: namely, H1, H2, H3 δ , A1 δ , A2, Q1 δ , Q2, Q3 δ and Q4. Most of the equations in the ABS list are (or related to) equations known before the list was found. One of the new equation is H2, which is also special because only for H2 and H3 δ their continuous counterpart are not found in terms of Miwa's coordinates [18].
H2 equation reads [1] y − y y − y + p 2 − q 2 y + y + y + y − p 2 − q 2 = 0, (1.1) where p, q are spacing parameters in n-and m-direction, respectively, and tilde-hat notations stand for shifts in n-and m-direction (see equation (2.1) for definition). In this paper, we will revisit H2 equation with the help of H1 (i.e., the lattice potential Korteweg-de Vries equation (lpKdV) [15]) Note that H1 has a background solution z 0 = pn + qm. By removing the background from the equation, i.e., introducing z = z 0 − u so that u has a zero background, the resulting equation [15] p is ready to yield the potential KdV equation in its continuum limit. H2 has also a background solution y 0 = z 2 0 [8] but it is hard to remove it for H2 equation. This is why so far the continuous counterpart of H2 is unknown.
The main contents of the paper are the following. We will start from the following relations obtained from Cauchy matrix approach (see equation (2.6)), w + w = p u − pu + u u, (1.4a) w + w = q u − qu + u u, (1.4b) which provide a non-auto Bäcklund transformation (BT) between H1 and H2 with y = z 2 0 − 2z 0 u + 2w, (1.5a) (1.5b) From (1.4) we will derive a quadrilateral coupled system (in Section 3.1) which is CAC and therefore is integrable. As a consequence we will have a one-component 9-point equation of w, Considering the relation (1.5) we call (1.6) or (1.7) an alternative of H2. In this paper we will focus on the coupled system (1.6) to investigate its integrability, bilinear form, continuum limits, etc. It turns out that the continuous counterpart of (1.6) consists of the potential KdV equation and the time-part in the Lax pair of the (potential) KdV equation. Note that with some transformations the system (1.6) will be equivalent to a H1 × H2 two-component system which is first derived in [11] (see equation (3.9)). The paper is arranged as follows. In Section 2 we recall the Cauchy matrix approach and derive the BT (1.4). Then we derive the coupled quad-system (1.6) and investigate its 3D consistency and Lax pairs in Section 3, and present its bilinear form and Casoratian solutions in Section 4. After that, Section 5 consists of the investigation of the continuous counterpart of the quad-system (1.6), including their continuum limits and Cauchy matrix approach in continuous case, which leads to a connection between S (1,0) equation and the KdV equation. Finally, concluding remarks are given in Section 6. There is an appendix section to explore higher order equations in the continuum limits.
2 Cauchy matrix scheme

Preliminary
For a function U of (n, m) ∈ Z 2 , we introduce shorthand to express it and its shifts, such as Spacing parameters p and q serve as spacing parameters of n-direction and m-direction, respectively. These settings are depicted as in Figure 1(a). In 3-dimensional case, we introduce the third direction l together with its spacing parameter r, and denote the shift by U . = U (n, m, l+1).
We also introduce shift operators E n , E m and E l such that is CAC means the equation can be consistently embedded on six faces of the cube in Figure 1 In other words, given initial values U , U , U , U , the value of U is unique although it will be determined by the top-, front-and right-side, three equations.

Cauchy matrix approach to (1.4)
To derive the coupled system (1.4), we consider the Sylvester equation where M , K ∈ C N ×N , r and c are column vectors in C N defined as r = (r 1 , r 2 , . . . , r N ) T , c = (c 1 , c 2 , . . . , c N ) T .
Let us sketch some known results related to this equation.
Proposition 2.1. For given r, c and K, if K and −K do not share eigenvalues, then M will be uniquely determined by the equation (2.2) [17]. In addition, explicit formulae of M have been constructed in [22] with respect to canonical forms of K.

Proposition 2.2 ([22]). Introduce
where the elements r, c, K and M satisfy the Sylvester equation (2.2) and I is the N -th order unit matrix. For given r and c, the matrix K and any matrix similar to it give rise to the same S (i,j) , and the symmetric property holds.
The Cauchy matrix approach is a method to construct and study integrable equations by means of the Sylvester-type equations. In this approach integrable equations are presented as closed forms of S (i,j) and its shifts (or derivatives). It is first systematically used in [14] to investigate ABS equations and later developed in [20,22] to more general cases. To derive coupled system (1.4), we impose dispersion relation on r as (pI − K) r = (pI + K)r, (qI − K) r = (qI + K)r, and let c be a constant column vector, where K is a constant N × N matrix. Next, making use of the results obtained in [22], we have the following shift relations for M , from which one arrives at (see equations (27) in [22]) Now, taking (i, j) = (0, 0) in the above recurrence relation, defining and making use of the symmetric property (2.4), we have i.e., the coupled system (1.4).

Consistent triplet
Suppose that we have a coupled system The compatibility with respect to v i.e., v = v gives rise to a quadrilateral equation of u, Q u, u, u, u; p, q = 0, (2.8) while the compatibility for u yields a quadrilateral equation of v, When this happens we say that equations (2.7), (2.8) and (2.9) compose a consistent triplet. Note that such a triplet can be extended to the lattice systems defined on larger stencils, not necessary to be restricted to the quadrilateral case. In such a triplet, equation (2.7) acts as a BT to connect the other two equations (2.8) and (2.9), and due to the compatibility, any pair of solutions (u, v) of (2.7) will provide solutions to (2.8) and (2.9). Consistent triplets have played a useful role in constructing rational solutions for a number of quadrilateral equations [21]. Using the transformation (1.5), the coupled system (2.6) is written as Note that (1.5) has been implied from equation (5.26) in [14]. By checking compatibility of z and y, one immediately find that y and z have to satisfy H2 (1.1) and H1 (1.2), respectively. Thus we have the following. This means that solving (2.10) yields solutions to H1 and H2. Based on this fact in Section 4 we will bilinearize (2.10) and get a bilinear form for H2 which is simpler than the one given in [8]. Note that the fact of (2.10) being a BT for H1 and H2 was also found in [2] using the CAC property of H2 and the degeneration relation from H2 to H1.
3 Coupled system (1.6) The coupled system (2.6) is a set of 3-point equations. Initial values can be given on any line n+m = l, where l can be arbitrary integers. However, the system can only evolve towards downleft from the initial-value line. Initial values can also be given on a vertical line or on a horizontal line, but neither of them can yield a bidirectional evolution. Compared with (2.6), the quadsystem (1.6) is not only CAC, but also allows bidirectional evolution starting from initial values given on some staircase. Next, we derive (1.6) from (2.6) and investigate its integrability.

Derivation
First, in light of the compatibility E n E m w = E m E n w, from (2.6) we immediately get H1 (lp-KdV) equation (1.3). Next, eliminating w from (2.6) by (2.6b)-(2.6a) and by (2.6a)-(2.6b), respectively, we get Meanwhile, subtracting (2.6b) from (2.6a) and subtracting (2.6a) from (2.6b), respectively, yield Then, by multiplying u to (3.1) and u to (3.2) and eliminating their right-hand sides, we arrive at and similarly, from (3.3) and (3.4) we have It then directly follows that any two of them can be derived from the other two.
Other cases can be proved similarly.
In addition, eliminating u from (1.6) yields the nine-point single-component equation (1.7). Thus we have the following result parallel to Proposition 2.3. Besides, we also note that starting from the coupled system (1.6) and making use of (1.5) and (2.10), one can get another coupled system in terms of z and y, In fact, from (1.5) we have Substituting these into (1.6a), and then successfully using (2.10a) to eliminate y, using (2.10b) to eliminate y and using (1.2) to eliminate z, finally we arrive at (3.9a). Equation (3.9b) can be derived in a similar way. This coupled system was first derived in [11], known as the H1 × H2 2-component vertex equation. Using (1.5) and (2.10), the coupled system (1.6) can be recovered from (3.9) as well.

Integrability: CAC and Lax pair
In this part, let us focus on the integrability of the coupled system (1.6). In light of Proposition 3.1, in practice, we consider the equivalent equations (3.6b) and (3.6d), i.e., Let U = (u, w) T and put the above system on the six faces of the cube (see Figure 1(b)). Given initial values U , U , U , U , we have Then, with these in hand, we find that the value of U is same no matter we calculate it from the equation on the top, or the front, or the right face, where Thus, the system (3.10) (or (1.6)) is CAC. Note that the lpKdV equation (3.10b) itself is CAC as well. This also indicates that the 9-point equation (1.7) can be embedded into 3-dimensional space and is consistent around a large cube consisting of 4 elementary cubes.

Bilinear formulation
Now that both the coupled system (1.6) and H2 equation ( from which the coupled system (1.4) is bilinearized as In the following we present Casoratian solutions to the above bilinear equations. We introduce [8] where ± i and k i are constants, and define a vector  Proof . Let us first prove (4.2b). Introduce and rewrite (4.2b) as Next, we prove both A and B are zero when f , g, h, s are Casoratians defined above. Note that with ψ given by (4.3) as elementary entries, shifts of f , g, h satisfy (see formulae given in Appendix in [8])  In addition, since ± i in (4.3) are arbitrary parameters independent of (n, m, l), we introduce a dummy variable x and replace ± i by e ±k i x ± i , and then we have
Thus we have where the subscript [x] means the determinant is a Wronskian composed by ψ(x, l) and its derivatives w.r.t. x. Besides, for the Casoratians g and θ defined in (4.5), we have Thus we reach the following.

Corollary 4.3.
In terms of Wronskians composed by ψ(x, l) with entries (4.8), the bilinear system (4.2) is rewritten as where D is the Hirota bilinear operator defined as [10] 5 Continuous counterparts

Continuum limits
Let us start to investigate continuum limits of the coupled system (3.10) via a two-step scheme, which is based on the deformation of the H2 plane wave factor [8]: which are finite when n → ∞, p → ∞ and m → ∞, q → ∞.
In the first step, after Taylor expanding the coupled system (3.10) into power series of 1 p , the leading term in each equation w.r.t. 1 p gives rise to where for convenience we still employ u and w to denote functions of (ξ, m) in this step. Note that equation (5.1a) yields a form u − u = A(w), and substituting it into equation (5.1b) yields a form ∂ ξ (u + u) = B(w), where A(w) and B(w) are some functions of w, w and their derivatives w.r.t. ξ. Then, eliminating u from them we get a differential-difference equation for w, which is In the second step, first, expanding (5.1) in terms of 1 q , then, introducing and reorganizing the expansion w.r.t. (x, t) and the derivatives w.r.t. them, finally, from the leading terms we get which contains the potential KdV equation (5.3b) and its companion (5.3a). Again, one may eliminate u from the system and obtain an equation of w: where the shorthand w kx means ∂ k x w. Later we will have a closer look at this system in Section 5.3. We also note that higher order equations in the continuum limit of the coupled quad-system (3.10) will be explored in Appendix.

Cauchy matrix approach to (5.3)
A continuous version of the Cauchy matrix approach to the KdV equation has been developed in [20]. In the following we derive the coupled system (5.3) using this approach.

Derivation
We will start from the Sylvester equation (2.2) and the function S (i,j) defined as in (2.3), while in this case M , r and c T depends on (x, t). An auxiliary vector u (i) is introduced as Let us first recall some useful relations obtained in [20]. Proposition 2.2 holds as well in the continuous case. 1. S (i,j) satisfies the recursive relation for i, j ∈ Z, and in particular, when k = 1 we have 2. K and any matrix similar to it lead to same S (i,j) .
3. The symmetric property holds: In addition, a relation for u (i) is [20] ( Multiplied (I + M ) −1 from the left, one first has and then, in light of the definitions of u (i) and S (i,l) , it follows that Introducing dispersion relation (cf. [20]) one has (cf. [20]) and From the above general formulae, for u and w defined as in (2.5), one has and The function u obeys the potential KdV equation (cf. [20]) The right hand side vanishes by using the identity (5.5) for (i, j) = (2, 0) and (1, 0). Thus, we obtain equation (5.3a), i.e., Note that equation (5.8a) indicates a Miura-type transformation which connects the potential KdV equations (5.10) and (5.11). For this we have the following.
Proposition 5.2. In light of relation (5.12), the two equations in the system (5.3) are connected as

Lax Pair
A Lax pair for the coupled system (5.3) can be constructed as well from the Cauchy matrix approach (cf. [7]). Let us consider the relation (5.7) with i = 0 and i = 1, i.e., Then we do the following. Denote the first entries of u (0) and u (1) by φ 1 and φ 2 , respectively. Since in the complex field matrix K is always similar to a lower triangular matrix, in light of Proposition 5.1, without loss of generality, we assume in the following K is lower triangular and denote the entry K 1,1 in K by λ. Make use of the recursion relation (5.6) to express u (2) , u (3) and u (4) in terms of u (0) and u (1) . Then, for Φ = (φ 1 , φ 2 ) T , from (5.14) and (5.15), and after the above treatment we arrive at and we have made use of (5.9a) and (5.9c) to express S (1,1) , S (0,2) , S (1,2) and S (0,3) in terms of u, w and their derivatives. The compatibility Φ xt = Φ tx gives rise to i.e., in explicit form, gives the following equations This leads to three equations where (5.16a) is nothing but the Miura transformation (5.8a), i.e., (5.12), equation (5.16c) is a consequence of (5.16b) and (5.16a) in light of Proposition 5.2.   (3.13). This fact indicates the integrability of the 9-point equation (1.7). The coupled system (1.6) and the H1 × H2 system (3.9) derived in [11] are connected through (1.5) and (2.10). H2 is a consequence of the coupled system (1.4). Since in a consistent triplet the BT determines the other two equations in the triplet, in light of Propositions 2.3 and 3.2, one may consider the 9-point equation (1.7) is an alternative of H2 after removing the background z 2 0 and the term z 0 u from y. Again, due to the consistent triplets, the bilinear form (4.2) can serve as a bilinear form for H2 (1.1), the coupled quad-system (1.6) and the 9-point equation (1.7).

Connection to the KdV
As for the continuous counterparts, in continuum limits the coupled quad-system (3.10) gives rise to the semi-discrete system (5.1), which implies equation (5.2) for only w; and the full continuum limit leads (5.1) to (5. In addition to the Mirua-type connection with the potential KdV equation, equation (5.3a) appears as the t-part in the gauged Lax pair (5.21) of the potential KdV equation (5.3b), which indicates that w can be considered as an eigenfunction (in the Lax pair) of the (potential) KdV equation. Note that we derived the gauged Lax pair from a deformed bilinear BT (5.20), in which (5.20a) and the bilinear equation (4.9a) share the same form if we consider f as a shift f . We also note that higher order equations in the continuum limit of the coupled quad-system (3.10) will be investigated in Appendix A.

A Higher order equations in the continuum limits
In [19] Wiersma and Capel established a scheme to achieve higher order equations in continuum limits, and they obtained the (potential) KdV hierarchy from the continuum limit of the lpKdV equation (1.3). In their scheme, discrete equations are handled in the (skew) coordinates (N = n + m, m), and in the first round, take For the plane wave factor ρ given in Section 5.1, it turns out that The resulting semi-discrete equation usually has an evolution form where we adopt the notation e j∂ N V N = V N +j . In the full continuum limit, infinitely many independent variables (t 1 = x, t 3 , t 5 , . . . ) are introduced through It then follows that and (A.3b) Replacing ∂ τ 1 and ∂ N in the equation (A.2) with the above operators and expanding the equation in terms of 1/p, a hierarchy of equations will be obtained from the coefficients of 1/p j in the expansion.
With this scheme, the lpKdV equation (3.10b) is rewritten in terms of (N, m) as where V = V (N, m) = u(n, m). In light of (A.1), it gives rise to a semi-discrete pKdV equation In the full limit, redefining V = V (t 1 = x, t 3 , t 5 , . . . ) and using (A.3), the pKdV hierarchy is obtained by expanding (A.4) in terms of 1/p. The first two equations from the expansion are which are the 3rd-order potential KdV (pKdV) equation and 5th-order pKdV equation. For more details one may refer to [19].
where we have replaced ∂ τ 1 V N in the equation using (A.4). In the full limit of the above equation, the first two equations from the leading terms are and where W = W (t 1 = x, t 3 , t 5 , . . . ).
Noting that (A.5) and (5.16b) are connected by V = −2u, we introduce a gauge-transformed Schrödinger spectral problem (compared with (5.21a)) where W serves as the eigenfunction and γ for the spectral parameter. Then, it turns out that the pKdV equation (A.5) is a result of the compatibility of (A.9) and (A.7), i.e., (W xx ) t 3 = (W t 3 ) xx , and the compatibility of (A.9) and (A.8) yields the 5th-order pKdV equation (A.6). The same correspondence holds for the 7th-order equations obtained from the full continuum limits.