Rational Solutions to the ABS List: Transformation Approach

In the paper we derive rational solutions for the lattice potential modified Korteweg-de Vries equation, and Q2, Q1($\delta$), H3($\delta$), H2 and H1 in the Adler-Bobenko-Suris list. B\"acklund transformations between these lattice equations are used. All these rational solutions are related to a unified $\tau$ function in Casoratian form which obeys a bilinear superposition formula.

As for rational solutions, which are solutions expressed by fractions of polynomials, in general, such type of solutions can be derived from soliton solutions through a special limit procedure (or a Taylor expansion), which corresponds to a way to generate multiple zero eigenvalues for certain spectral problems (see [1,16] as examples). For the δ-dependent equations in the ABS list, for example, H3(δ) and Q1(δ), the existence of δ (i.e., δ = 0) plays a crucial role [21] in the procedure of obtaining rational solutions from their soliton solutions. For H1 which is independent of δ, its rational solutions were obtained recently by making use of the Hirota-Miwa equation and a continuous auxiliary variable [9]. Besides, as a generic (2+1)-D bilinear model, polynomial solutions of the Hirota-Miwa equation have been derived from several ways and presented via different forms [11,17].
In this paper we systematically construct rational solutions for the ABS list by means of Bäcklund transformations (BTs). A fundamental role playing in the paper is the lattice potential modified Korteweg-de Vries (lpmKdV) equation. There is a non-auto BT which connects the lpmKdV equation and Q1(0) (also known as the lattice Schwarzian Korteweg-de Vries equation and cross-ratio equation). The two equations and their BT constitute a consistent triplet, say, viewing the BT as a two-component system, then the compatibility of each component yields This paper is a contribution to the Special Issue on Symmetries and Integrability of Difference Equations. The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html a lattice equation of another component which is in the triplet. This means any pair of solutions of the BT provide solutions to the two equations that the BT connects. Details will be shown in Sections 3.1 and 3.2 on how such a consistent triplet works in generating rational solutions. We also make use of non-auto BTs between equations in the ABS list [4]. Starting from the lpmKdV equation and Q1(0), rational solutions of Q2, Q1(δ), A1(δ), H3(δ), H2 and H1 in the ABS list can be derived through the map: In the map the double-head arrow means the two equations it connects and their BT form a consistent triplet. Moreover, we find all the obtained rational solutions are related to a unified τ function in Casoratian form which obeys a bilinear superposition formula (see (5.22)). Compared with those rational solutions of H3(δ) and Q1(δ) derived in [21], here we obtain new solutions. In fact, we will see that rational solutions of Q1(δ) can explicitly be expressed through the rational solutions of Q1(0). Similar results hold for H3(δ) as well.
The paper is organized as follows. In Section 2 as preliminary we list quadrilateral equations that we consider in the paper and some notations. Then in Sections 3 and 4 we derive some rational solutions for the equations listed in Section 2. In Section 5 rational solutions in Casoratian form are proved. Finally in Section 6 we give conclusions.

Preliminary
We list quadrilateral equations that we consider in the paper: Here we use conventional notations u . = u n+1,m , u . = u n,m+1 . In the above equations, p and a are spacing parameters of n-direction and q and b are of m-direction; δ is an arbitrary constant.
Introduce a shift operator E ν to denote Then a N th-order Casoratian w.r.t. E l -shift is defined by and usually is compactly written as (cf. [10]) Another notation which is often used is | N − 2, N | = |0, 1, . . . , N − 2, N |. Besides, in Wronskian/Casoratian verification of solutions to a bilinear equation, the equation is usually reduced to a Laplace expansion of a zero-valued 2N × 2N determinant. The expansion is described as 10]). Suppose that B is an N × (N − 2) matrix and a, b, c, d are N th-order column vectors, then |B, a, b||B, c, d| − |B, a, c||B, b, d| + |B, a, d||B, b, c| = 0.
3 Rational solutions to lpmKdV, Q1, H3 and Q2 In this section we first investigate relation between the lpmKdV equation and Q1(0). Such a relation will be used to construct rational solutions to not only the two equations themselves but also to Q1(δ), H3(δ) and Q2.
Further than that, we have Proof . This lemma provides an approach to generate a sequence of solution pairs of the BT (3.1).

Theorem 3.3. For any solution pair
and it solves the lpmKdV equation (2.3). Next, the BT system (i.e., (3.1)) determines a function v N +1 that satisfies equation Q1(0) (2.5). v N and v N +1 obey the relation which is an auto BT of Q1(0).
Proof . The first part of the theorem holds due to Lemma 3.2. For the second part, since to map v to u from Q1(0) to Q1(δ). It holds as well for the degenerated case δ = 0, in which both v and u are solutions of Q1(0).

Rational solutions of Q1(0) and lpmKdV
We note that {V N } are different from the rational solutions of the lpmKdV equation obtained in [17] as a reduction of the Hirota-Miwa equation.
In the following we prove that if we start from (3.6), all the solutions generated from (3.3) are meaningful. First, let us look at non-zero property. Proof . Obviously, under assumption of the lemma, from (3.6a) we have v 1 > 0, V 1 > 0 and Next, we observe that in v 1 , v 2 and v 3 the order of leading terms (in terms of x 1 ) are respectively 1, 3 and 5. Now we prove all the v N defined through (3.3) with (3.6a) are distinct in the sense of having different leading orders in terms of x 1 .
Lemma 3.5. v N has a leading order 2N − 1 in terms of x 1 and V N has a leading order N − 1 in the same sense.
Proof . From (3.6) we can suppose the lemma is correct up to some integer N . Then one can find . Based on mathematical induction, the lemma holds.
We conclude the following. Note that not all solutions can be effectively iterated through (3.3). For example, Here the parameterizations for a and b are
Then the iteration relation (3.3) can be extended to N ∈ Z.
This lemma can be checked directly. Note that the extension does not lead to new solutions to Q1(0) and the lpmKdV equation because these two equations are invariant under transformations of type u → c/u. However, the extension does bring more rational solutions to Q1(δ).
gives a sequence of solutions to Q1(δ). u N and v N −1 are connected via which is the non-auto BT (3.5) between Q1(δ) and Q1(0). Also, (3.9) agrees with the chain Proof . We only need to prove u N defined by (3.9) satisfies (3.10). Since {v N } obey the BT (3.4), using which we can find Then, from (3.9) by direct calculation we immediately have which coincides with the first equation in (3.10). Similarly we can find u N − u N satisfies the second equation in (3.10) as well.
Formula (3.9) provides an explicit relation between solutions of Q1(δ) and Q1(0), where {v N } is a sequence generated from (3.3). For v N given as in (3.6), some rational solutions of Q1(δ) generated from (3.9) are where we have made use of relation (3.8) in order to get v −1 and v 0 .
Here we give two remarks.
Remark 3.9. There are some overlaps (dual forms) in the chain (3.9). Note that Q1(δ) equation is formally invariant under transformation first replacing u with εδ 2 u and then δ with 1/δ where ε = ±1, by which (3.9) is transformed into its dual form which gives a sequence of solutions to Q1(δ) as well. By the relation (3.8) given in Lemma 3.7, u N and u 3−N in (3.9) are dual forms of each other.
Remark 3.10. Not all the rational solutions of Q1(δ) are included in the chain (3.9). We give two exceptions. One is where γ is a constant and α, β are defined by parametrization with arbitrary constant c 0 , and the other is where x 1 is defined as (3.7), p, q are parameterized as in (3.2) and γ 0 is a constant.

Rational solutions to Q2
We make use of a non-auto BT between Q1(δ) (2.6) and Q2 (2.8) to derive rational solutions of Q2. The BT reads [4] δ When u is given by (3.13a) with parametrization (3.13b), from (3.15) we can find where γ 0 is a constant. This is not a pure rational solution.
For v N and V N constructed in Theorem 3.3 and Lemma 3.7, function solves A1(δ) (2.7), and  (3.16), which proves the present theorem.

Solutions of H1 and H2
In this section we first derive solutions of H1 using a relation between H1 and the lpmKdV equation. Then from H1 we derive solutions of H2.

H1
There is a non-auto BT [13] u to connect H1(u) (2.1) and the lpmKdV(V ) equation (2.3). We use it to derive solutions for H1 on the basis of the following fact.
Proof . We rewrite the lpmKdV equation Then, multiplying both equations in (4.1) and making use of (4.3) we immediately reach H1 (2.1) provided p, q are parameterized as (4.2).
From (4.1) we find of which we make use to derive u from known V . For V 1 = 1 and V 2 = x 1 given in (3.6), we derive same solution for H1, For V 3 and V 4 given in (3.6), we respectively find and One may wonder if the above relation can be used to generate more solutions for H1. However, making use of iterative relations (3.3) we find and a same formula for ( , b), which means V → 1 V does not lead to new solutions for H1. This can also explain the fact u 1 = u 2 due to V 1 = 1 V 1 = 1.

Rational solutions in determinant form
From the previous section it is understood that the sequence {V N } plays a crucial role in constructing solutions in the whole paper. With regard to rational solutions, it is hard to do "integration" from (3.3b) to get high order v N and consequently it is difficult to get high order V N . In this section we aim to construct Casoratian expressions for v N and V N , as well as rational solutions of other equations.

Bilinear relation of V N and v N
We express and it then follows from (3.3a) that and from (3.6) we find successively where P 4 is obtained from the relation P 4 P 3 = V 5 = v 4 V 4 . Viewing (5.1) as transformations, the BT (3.3b) yields P P − P P = aP P , P P − P P = bP P , This is a bilinear system for polynomials {P N }. Note that based on (3.8) the relations (5.1) and (5.2) can be extended to N ∈ Z by defining where [ · ] denotes the greatest integer function. In Section 5.3 we will give a Casoratian form of P . To achieve that, we make use of H1.

Casoratian form of rational solutions of H1
For H1 (2.1), using 3D consistency we have its BT where u stands for a new solution of H1, we adopt parametrization (4.2) and the arbitrary number k −2 = r acts as a "soliton number" which leads to a new soliton (cf. [14]). Now we remove the term k −2 from (5.4), i.e., taking r = 0, and consequently we have which can generate a rational part in the new solution u.
To find solutions from (5.5), first, we introduce which provide a factorization of (5.5). Such an assumption coincides with the previous results.
In fact, suppose V = f /f , then from (5.6) we can find u − u and u − u agree with (4.1) and u − u and u − u agree with (4.4). Then we introduce by which we bilinearize (5.6) as Next, we introduce Casoratian forms for f , f , g and g. Consider function where ± i and s i are nonzero constants 1 . This can be used to construct soliton solutions for H1 equation (cf. [14]) 2 . To derive the rational solutions obtained in the previous section, we take 1 If ± i are independent on si, in practice in (5.9) we replace (1 ± si) l with (1 ± si) l+l 0 and suppose l0 is either a large enough integer or a non-integer so that the derivative ∂ h s i (1 ± si) l+l 0 |s i =0 = 0. 2 One needs to use gauge property of bilinear H1 and make certain extension from (±si) l+l 0 to (1 ± si) l+l 0 .
Proof will be given in Appendix A.

Remark 5.2.
There is an alternative choice for the Casoratians (5.14), which are given by just replacing the basic column vector α given in (5.13) by where α + 2j are defined in (5.12), or equivalently,

Casoratian solutions to (5.2)
We can make use of the BT of H1 to obtain solutions to bilinear equation (5.2). By the compatibility of (5.6a) and (5.6b), i.e., (E n Similarly, This means Next we go to prove λ 1 (m, N ) = a and λ 2 (n, N ) = b. Again, from (5.6), we can derive Using (5.17) to eliminate f and f from the above equation, we find which means and it then follows that both λ 1 and λ 2 must be (n, m)-independent. We assume γ(N ) = λ 1 /a = λ 2 /b, and then (5.17) yields To determine the value of γ(N ), we investigate properties of f near the point (n, m) = (0, 0), which are presented through the following lemmas. Lemma 5.3. According to the definitions of u in (5.7), f and g in (5.14) and α ± h in (5.12), we find the value of α ± h | (n,m)=(0,0) is independent of (a, b), and so are f (0, 0), g(0, 0) and u(0, 0). Then, from (5.18) we find that γ(N ) must be independent of (a, b). Proof . First, noticing that relation With this lemma, for f = f N (α(n, m, l)) in (5.19), we have Then, since γ(N ) is independent of a, from (5.19) we arrive at We can sum up this subsection with the following theorem.

Casoratian rational solutions to H2 and a sum-up
We can derive Casoratian rational solutions for H2 through non-auto BT (4.6), in which we suppose Then BT (4.6) is bilinearized as Based on the bilinear form we have Theorem 5.7. The Casoratians solve the bilinear BT system (5.25), in which the basic Casoratian column vector α is given by (5.13). Consequently, (5.24) provides rational solutions to H1 and H2.
Proof will be given in Appendix B. Besides (5.15), some h of low orders are So far we have obtained Casoratian expressions for the rational solutions of Q1(0), lpmKdV, Q1(δ), H3(δ), H1 and H2. Noting that all these solutions are related to the rational solutions V N of the lpmKdV equation, it is necessary to express all these obtained solutions through the Casoratians with a unified N . We collect them in the following theorem.

Rational solutions to Q2
Now we come to the final equation, Q2. We start from the non-auto BT (3.15) in which we take parametrization (3.2) and u to be (5.28c) which is a solution of Q1(δ). Introduce auxiliary function by which the BT (3.15) yields Then, making use of the relation (5.22), from (5.28c) we can find On the basis of the above relations together with their (q, ) version, and introducing we then reduce (5.29) to To solve this system we expand It then follows from (5.30) that N +2 which is determined by (5.31c), the simplest two items are However, so far we do not find an explicit expression for θ N +2 in terms of f and other auxiliary functions.
As a conclusion of rational solutions of Q2, we give the following theorem.
Theorem 5.9. Suppose that f = f N is defined as in (5.27). Our construction provides rational solutions of Q2 in the following form where u N +2 is given by (5.28c) and θ (0) N +2 is determined by (5.31c). We note that it might be not sufficient to call (5.33) a rational solution for arbitrary N , because for this moment we do not have a general solution form (like (5.32)) for θ (0) N +2 .

Conclusions
In the paper we have derived rational solutions for the lpmKdV equation and some lattice equations in the ABS list. We make use of lpmKdV-Q1(0) consistent triplet to construct their rational solutions iteratively. This then becomes a starting point and through the route in Fig. 1 to generate solutions for other equations. All these rational solutions are related to a unified τ function in Casoratian form, f (α) = | N − 1|, which obeys the bilinear superposition formula (5.22).
There are several interesting points we would like to remark. First, formula (3.9) reveals an explicit relation between certain solutions of Q1(δ) and Q1(0). This formula holds not only for rational solutions but also for solitons. Once we obtain v N −2 and v N from (3.3), formula (3.9) gives a solution u N to Q1(δ), and these solutions provide a solution sequence for the chain (3.11) which is based on BT (3.5), i.e., (3.10). The second thing is about bilinear superposition formula (5.22) or (5.2). Casoratian f with ψ i (5.9) as a basic entry is also a solution of bilinear equation as well as its dual version by switching (a, ) and (b, ). (6.1) can be considered as a bilinear form of Hirota's discrete KdV equation (see [15] and [13,Section 8.4.1]). It was also derived from the Cauchy matrix approach as a bilinear form that is related to H1 (see [13,Section 9.4.3]). It is also well known that (6.1) can be derived as a reduction of the Hirota-Miwa equation, of which some rational solutions were derived from several different ways and reductions of few cases was already considered [11,17]. Here we can consider (5.22) as a bilinear superposition formula of (6.1) for rational solutions. Since between v N and f to formulate a mechanism for choosing γ j so that f is nonzero in the first quadrant. This will be done in Appendix C. At the end of the paper we would like to make a comparison for the rational solutions and their derivation between the present paper and [21]. In this paper the construction of rational solutions is based on iteration of a chain of transformations, and the unified τ function f (α) = | N − 1| is proved to satisfy the bilinear superposition formula (5.22). In [21], rational solutions (most of them with exponential background) for H3(δ) and Q1(δ) are obtained via a limiting procedure from soliton solutions in Casoratian expression. The method used in [21] can be extended to H1 and H2 (by selecting (5.9) as a basic Casoratian entry) and the results will be the same as the present paper. However, for H3(δ) and Q1(δ) it is obvious that our construction, which brings pure rational solutions, allows reduction δ = 0 and relies only on a unified τ function, has more advantage than the limiting procedure used in [21]. It is hard to say what is the reason of this difference, but a fact is all the BTs we used in our paper are only parametrically related to spacing parameters a, b without any extra parameters for solitons. These BTs are natural for generating rational solutions.

A Proof of Theorem 5.1 for H1
Here we prove Theorem 5.1 which gives Casoratian form of rational solutions of H1.
Besides, ω i obeys the same shift relation as (A.5), i.e., which leads to Then one can find the l.h.s. of (5.8b) yields which vanishes as (A.4). (5.8c) and (5.8d) can be proved similarly.

C Property of f
In the following we take a close look at Casoratian f N defined by (5.14a), i.e., where α is given by (5.13). Due to relation (5.23), we only consider the case N ∈ Z + . To investigate properties of f N , we introduce "degree" for a polynomial. For a monomial i≥1 x k i i where x i is defined in (3.7), we assign it a degree i≥1 ik i and denote this number by (iv) in construction, f N (0, 0) > 0 is guaranteed by successively choosing Proof . We prove the items of the theorem one by one.
(i) Consider Casoratian f N (C.1) in which α is given by (5.13). f N is written as |(f ij ) N ×N | where f ij = α + 2i−1 (n, m, j − 1). Noting that it is not α + h (n, m, l) but α + h (n, m, 0) that is homogeneous, in the following we make use of shift relation (5.21) to rewrite f N in terms of {α + h (n, m, 0)}. To do that, first from (5.21) we have α(l + 1) − α(l) = β(l), (C.2a) Here in (C.2) we have omitted n, m in α(n, m, l) and β(n, m, l) for convenience. Successively making use of (C.2a) we can rewrite (C.1) from the last to the first column, and we have Note that the degree of nonzero f ij is separable in terms of i and j. Meanwhile, f N is an algebraic summation in which each term is a nonzero product N i=1 f i,j i where j i runs over a permutation of the set {1, 2, . . . , N }. It is easy to get which means f N is homogeneous with degree D[f N ] = N (N +1) 2 . (ii) In the following we come to the statement that f N depends only on {x 1 , x 3 , . . . , x 2N −1 }, which has been shown correct for N = 1, 2, 3 in (5.15). Now we assume the statement is correct for f j where j ∈ {1, 2, . . . , N − 1}. For f N , first, it contains x 2N −1 . In fact, from the expression (C.3) we can see that x 2N −1 only appears in the element f N,1 , and doing Laplace expansion for (C.3) along the first two columns it is easy to find the only term involving x 2N −1 is Next, we note that for each monomial A = i≥1 x j i i , the commutating relation Taking derivative ∂ x 2i on both sides of double down bar shifted (5.22a) we have where the right hand side has vanished due to the assumption that f j is independent of x 2i for j ∈ {1, 2, . . . , N − 1}. The above relation indicates ∂x 2i f N f N −2 is independent of n. In a same manner, from (5.22b) we can find (iv) Finally, we formulate a mechanism to guarantee f N (0, 0) > 0 by choosing suitable γ j . From item (ii) we know that f N (0, 0) is only related to {γ 1 , γ 3 , . . . , γ 2N −1 }. Since in f N the term involving x 2N −1 is (C.4), we express f N as which, at point (0, 0), yields To guarantee f N (0, 0) > 0, we need to take Thus, as a starting step we take f 0 = 1 and f 1 = x 1 with γ 1 > 0, for N = 2, using the above formula we choose a value for γ 3 so that f 2 (0, 0) > 0. We can repeatedly use (C.6) and will successively choose γ 2N −1 and get f N (0, 0) > 0 for higher N .