Xueli Wei¹, Peter H van der Kamp², Da-jun Zhang^,1,∗1Department of Mathematics, Shanghai University, Shanghai 200444, China
²Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

First author contact: ∗Author to whom any correspondence should be addressed
Received:2021-02-25Revised:2021-04-13Accepted:2021-04-29Online:2021-06-04


Abstract
An auto-Bäcklund transformation for the quad equation Q1₁ is considered as a discrete equation, called H2^a, which is a so called torqued version of H2. The equations H2^a and Q1₁ compose a consistent cube, from which an auto-Bäcklund transformation and a Lax pair for H2^a are obtained. More generally it is shown that auto-Bäcklund transformations admit auto-Bäcklund transformations. Using the auto-Bäcklund transformation for H2^a we derive a seed solution and a one-soliton solution. From this solution it is seen that H2^a is a semi-autonomous lattice equation, as the spacing parameter q depends on m but it disappears from the plane wave factor.
Keywords： auto-Bäcklund transformation;consistency;Lax pair;soliton solution;torqued ABS equation;semi-autonomous


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Xueli Wei, Peter H van der Kamp, Da-jun Zhang. Integrability of auto-Bäcklund transformations and solutions of a torqued ABS equation. Communications in Theoretical Physics, 2021, 73(7): 075005- doi:10.1088/1572-9494/abfcba

1. Introduction

The subtle concept of integrability touches on global existence and regularity of solutions, exact solvability, as well as compatibility and consistency (see [1]). In the past two decades, the study of discrete integrable systems has achieved a truly significant development, which mainly relies on the effective use of the property of multidimensional consistency (MDC). In the two-dimensional case, MDC means the equation is consistent around the cube (CAC) and this implies it can be embedded consistently into lattices of dimension 3 and higher [2–4]. In 2003, Adler, Bobenko and Suris (ABS) classified scalar quadrilateral equations that are CAC (with extra restrictions: affine linear, D4 symmetry and tetrahedron property) [5]. The complete list contains 9 equations.

In this paper, our discussion will focus on two of them, namely(1.1)$\begin{eqnarray}\begin{array}{l}{\rm{Q}}{1}_{\delta }(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)\\ \quad =\ p(u-\widehat{u})(\widetilde{u}-\widehat{\widetilde{u}})-q(u-\widetilde{u})(\widehat{u}-\widehat{\widetilde{u}})\\ \quad +\ \delta {pq}(p-q)=0\end{array}\end{eqnarray}$and(1.2)$\begin{eqnarray}\begin{array}{rcl}{\rm{H}}2(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q) & = & (u-\widehat{\widetilde{u}})(\widetilde{u}-\widehat{u})\\ & & +(q-p)(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}})\\ & & +{q}^{2}-{p}^{2}=0.\end{array}\end{eqnarray}$Here u = u(n, m) is a function on ${{\mathbb{Z}}}^{2}$, p and q are spacing parameters in the n and m direction respectively, δ is an arbitrary constant which we set equal to 1 in the sequel, and conventionally, tilde and hat denote shifts, i.e.(1.3)$\begin{eqnarray}\begin{array}{rcl}u & = & u(n,m),\,\widetilde{u}\,=\,u(n+1,m),\\ \widehat{u} & = & u(n,m+1),\,\widehat{\widetilde{u}}\,=\,u(n+1,m+1).\end{array}\end{eqnarray}$H2 is a new equation due to the ABS classification, while Q1_δ extends the well known cross-ratio equation, or lattice Schwarzian Korteweg–de Vries equation Q1_δ=0. Note that spacing parameters p and q can depend on n and m respectively, which leads to nonautonomous equations.

For a quadrilateral equation that is CAC the equation itself defines its own (natural) auto-Bäcklund transformation (auto-BT), see [5]. For example, the system$\begin{eqnarray*}{\rm{Q}}{1}_{\delta }(u,\widetilde{u},\overline{u},\widetilde{\overline{u}};p,r)=0,\,\,{\rm{Q}}{1}_{\delta }(u,\widehat{u},\overline{u},\widehat{\overline{u}};q,r)=0,\end{eqnarray*}$where r acts as a wave number, composes an auto-BT between ${\rm{Q}}{1}_{\delta }(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0$ and ${\rm{Q}}{1}_{\delta }(\overline{u},\widetilde{\overline{u}},\widehat{\overline{u}},\widehat{\widetilde{\overline{u}}};p,q)=0$. Such a property has been employed in solving CAC equations, see e.g. [6–10].

Some CAC equations allow auto-BTs of other forms. For example, in [11] it was shown that the coupled system(1.4a)$\begin{eqnarray}A:\ (u-\widetilde{u})(\widetilde{\overline{u}}-\overline{u})-p(u+\widetilde{u}+\overline{u}+\widetilde{\overline{u}}+p+2r)=0,\end{eqnarray}$(1.4b)$\begin{eqnarray}B:\ (u-\widehat{u})(\widehat{\overline{u}}-\overline{u})-q(u+\widehat{u}+\overline{u}+\widehat{\overline{u}}+q+2r)=0\end{eqnarray}$provides an auto-BT between(1.5)$\begin{eqnarray}Q:\ {\rm{Q}}{1}_{1}(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0\end{eqnarray}$and $\overline{Q}:\ {\rm{Q}}{1}_{1}(\overline{u},\widetilde{\overline{u}},\widehat{\overline{u}},\widehat{\widetilde{\overline{u}}};p,q)=0$, and, that H2 acts as a nonlinear superposition principle for the BT (1.4). One can think of the auto-BT as equations posed on the side faces of a consistent cube with Q and $\overline{Q}$ respectively on the bottom and the top face, as in figure 1. Here one interprets $\overline{u}=u(n,m,l+1)$, and r serves as a spacing parameter for the third direction l. The superposition principle can be understood as consistency of a 4D cube, see [12, 13].

Figure 1.

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Figure 1.Consistent cube with equations A, B and Q on its faces.


In [14] the auto-BT (1.4) and its superposition principle have been derived from the natural auto-BT for H2, employing a transformation of the variables and the parameters. The equation(1.6)$\begin{eqnarray}\begin{array}{l}{\rm{H}}{2}^{a}(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)={\rm{H}}2(u,\widehat{\widetilde{u}},\widehat{u},\widetilde{u};p+q,q)\\ =(u-\widetilde{u})(\widehat{\widetilde{u}}-\widehat{u})-p(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}}+p+2q)=0\end{array}\end{eqnarray}$was identified as a torqued version of the equation H2. The superscript ^a refers to the additive transformation of the spacing parameter. In [11], equation (1.6) appeared as part of an auto-BT for Q1₁. The corresponding consistent cube is a special case of [15, equation (3.9)]. In [14], equation (1.6) was shown to be an integrable equation in its own right, with an asymmetric auto-BT given by A = H2^a = 0 and B = H2 = 0. Here we provide an alternative auto-BT for equation (1.6) to the one that was provided in [14].

In section 2, we establish a simple but quite general result, namely that if a system of equations A = B = 0 comprises an auto-BT, then both equations A = 0 and B = 0 admit an auto-BT themselves. In particular, the equation H2^a given by (1.6) is CAC, with H2^a and Q1₁ providing its an auto-BT. We construct a Lax pair for H2^a, which is asymmetric. In section 3, we employ the auto-BT for H2^a to derive a seed-solution and the corresponding one-soliton solution. In the seed-solution the spacing parameter q depends explicitly on m, which makes H2^a inherent semi-autonomous. Some conclusions are presented in section 4.

2. Auto-BTs for auto-BTs and a Lax pair for H2^a

To have a consistent cube with H2^a and Q1₁ on the side faces, providing an auto-BT for H2^a, we assign equations to six faces as follows:(2.1a)$\begin{eqnarray}\begin{array}{l}Q:{\rm{H}}{2}^{a}(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0,\\ \quad \overline{Q}:{\rm{H}}{2}^{a}(\overline{u},\widetilde{\overline{u}},\widehat{\overline{u}},\widehat{\widetilde{\overline{u}}};p,q)=0,\end{array}\end{eqnarray}$(2.1b)$\begin{eqnarray}\begin{array}{l}A:{\rm{Q}}{1}_{1}(u,\widetilde{u},\overline{u},\widetilde{\overline{u}};p,r)=0,\\ \quad \widehat{A}:{\rm{Q}}{1}_{1}(\widehat{u},\widehat{\widetilde{u}},\widehat{\overline{u}},\widehat{\widetilde{\overline{u}}};p,r)=0,\end{array}\end{eqnarray}$(2.1c)$\begin{eqnarray}\begin{array}{l}B:{\rm{H}}{2}^{a}(u,\overline{u},\widehat{u},\widehat{\overline{u}};r,q)=0,\\ \quad \widetilde{B}:{\rm{H}}{2}^{a}(\widetilde{u},\widetilde{\overline{u}},\widehat{\widetilde{u}},\widehat{\widetilde{\overline{u}}};r,q)=0.\end{array}\end{eqnarray}$Then, given initial values $u,\widetilde{u},\widehat{u},\overline{u}$, by direct calculation, one can find that the value $\widehat{\widetilde{\overline{u}}}$ is uniquely determined. Thus, the cube in figure 1 with (2.1) is a consistent cube.

By means of such a consistency, the side equations A and B, i.e.(2.2a)$\begin{eqnarray}\begin{array}{l}A:\,p(u-\overline{u})(\widetilde{u}-\widetilde{\overline{u}})-r(u-\widetilde{u})(\overline{u}-\widetilde{\overline{u}})\\ \quad +{pr}(p-r)=0,\end{array}\end{eqnarray}$(2.2b)$\begin{eqnarray}B:\,(u-\overline{u})(\widehat{\overline{u}}-\widehat{u})-r(u+\overline{u}+\widehat{u}+\widehat{\overline{u}}+r+2q)=0,\end{eqnarray}$compose an auto-BT for the H2^a equation (1.6). Here r acts as the Bäcklund parameter.

We note that the order of the variables in the equations (2.1) is quite particular. Since equation (1.6) is not D4 symmetric, i.e. we have$\begin{eqnarray*}{\rm{H}}{2}^{a}(u,\overline{u},\widehat{u},\widehat{\overline{u}};r,q)\ne {\rm{H}}{2}^{a}(u,\widehat{u},\overline{u},\widehat{\overline{u}};q,r),\end{eqnarray*}$one has to be careful. The above result is explained by the following general result, see [16, section 2.1] where the same idea was used to reduce the number of triplets of equations to consider for the classification of consistent cubes.

Lemma 2.1.Let(2.3)$\begin{eqnarray}A(u,\widetilde{u},\overline{u},\widetilde{\overline{u}};p,r)=0,\,\,B(u,\widehat{u},\overline{u},\widehat{\overline{u}};q,r)=0\end{eqnarray}$be an auto-BT for(2.4)$\begin{eqnarray}Q(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0.\end{eqnarray}$Then we have (i)(2.5)$\begin{eqnarray}Q(u,\widetilde{u},\overline{u},\widetilde{\overline{u}};p,r)=0,\,\,B(u,\overline{u},\widehat{u},\widehat{\widetilde{u}};r,q)=0\end{eqnarray}$is an auto-BT for(2.6)$\begin{eqnarray}A(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0;\end{eqnarray}$and (ii)(2.7)$\begin{eqnarray}Q(u,\overline{u},\widetilde{u},\widetilde{\overline{u}};r,p)=0,\,\,A(u,\overline{u},\widehat{u},\widehat{\overline{u}};r,q)=0\end{eqnarray}$is an auto-BT for(2.8)$\begin{eqnarray}B(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0.\end{eqnarray}$

Proof.If $A=B=0$ is an auto-BT of Q = 0, then they compose a consistent cube as in figure 1. We prove the result by relabeling the fields at the vertices, see [13, lemma 2.1]. For (i) we interchange $\widehat{u}\leftrightarrow \overline{u}$ and $q\leftrightarrow r$, and for (ii) we perform the cyclic shifts $\widehat{u}\to \widetilde{u}\to \overline{u}\to \widehat{u}$ and $q\to p\to r\to q$.

Applying (i) to the consistent cube with (1.4a) and (1.5) we obtain (2.1a). Applying (ii) yields the same, as Q1₁ has D4 symmetry.

3D consistency can be used to construct Lax pairs for quadrilateral equations (see [3, 5, 17]). To achieve a Lax pair for H2^a, we rewrite (2.2a) as(2.9a)$\begin{eqnarray}\widetilde{\overline{u}}=\displaystyle \frac{u(p\widetilde{u}-r\overline{u})+(p-r)({pr}-\widetilde{u}\overline{u})}{(p-r)u+r\widetilde{u}-p\overline{u}},\end{eqnarray}$(2.9b)$\begin{eqnarray}\widehat{\overline{u}}=-r+\widehat{u}-\displaystyle \frac{2r(q+\widehat{u}+u)}{r-u+\overline{u}}.\end{eqnarray}$Then, introducing $\overline{u}=G/F$ and φ = (G, F)^T, from (2.9a) we have(2.10)$\begin{eqnarray}\widetilde{\varphi }=L\varphi ,\,\,\widehat{\varphi }=M\varphi ,\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{l}L=\gamma \left(\begin{array}{cc}-{ur}-(p-r)\widetilde{u} & {pu}\widetilde{u}+(p-r){pr}\\ -p & (p-r)u+r\widetilde{u}\end{array}\right),\\ \quad M=\gamma ^{\prime} \left(\begin{array}{cc}\widehat{u}-r & (-r+\widehat{u})(r-u)-2r(q+u+\widehat{u})\\ 1 & r-u\end{array}\right),\end{array}\end{eqnarray*}$with $\gamma =\tfrac{1}{\sqrt{{p}^{2}-{\left(u-\widetilde{u}\right)}^{2}}},\gamma ^{\prime} =\tfrac{1}{\sqrt{q+u+\widehat{u}}}$. The linear system (2.10) is compatible for solutions of (1.6) in the sense that H2^a is a divisor of ${\left(\widehat{L}M\right)}^{2}-{\left(\widehat{M}L\right)}^{2}$, where the square can be taken either as matrix multiplication, or as component-wise multiplication.

3. Seed and one-soliton solution

In this section, we use the auto-BT (2.2a) to construct solutions for (1.6). First, we need to have a simple solution as a ‘seed'. To find such a solution, we take $\overline{u}=u$ in the BT (2.2a), i.e.(3.1)$\begin{eqnarray}{\left(u-\widetilde{u}\right)}^{2}=p(p-r),\,\,u+\widehat{u}=-q-\displaystyle \frac{r}{2}.\end{eqnarray}$This so-called fixed point approach has proved to be effective in finding seed solutions [6, 8].

Proposition 3.1.Parametrizing(3.2)$\begin{eqnarray}p=\displaystyle \frac{\alpha }{a},\quad \alpha =-\displaystyle \frac{{ac}}{{a}^{2}-1},\quad q={\left(-1\right)}^{m}\beta -\displaystyle \frac{c}{2},\end{eqnarray}$and setting the seed BT parameter equal to r = c, the equations (3.1) allow the solution(3.3)$\begin{eqnarray}{u}_{0}={\left(-1\right)}^{m}(\alpha n+\beta m+{c}_{0}),\end{eqnarray}$where c₀ is a constant.

Proof.By direct calculation, with the given parameterizations the equations (3.1) read$\begin{eqnarray*}{\left(u-\widetilde{u}\right)}^{2}={\alpha }^{2},\,u+\widehat{u}={\left(-1\right)}^{m+1}\beta .\end{eqnarray*}$

It can be verified directly that (3.3) also provides a solution to (1.6). Next, we derive the one-soliton solution for (1.6), from the auto-BT (2.2a) with u = u₀ as a seed solution.Proposition 3.2.The equation (1.6), with lattice parameters (3.2) admits the one-soliton solution(3.4)$\begin{eqnarray}{u}_{1}={\left(-1\right)}^{m}\left(\alpha n+\beta m+{c}_{0}+\displaystyle \frac{{ck}}{1-{k}^{2}}\displaystyle \frac{1-{\rho }_{n,m}}{1+{\rho }_{n,m}}\right),\end{eqnarray}$where(3.5)$\begin{eqnarray}{\rho }_{n,m}={\rho }_{\mathrm{0,0}}{\left(\displaystyle \frac{a+k}{a-k}\right)}^{n}\,\prod _{i=0}^{m-1}\displaystyle \frac{{\left(-1\right)}^{i}-k}{{\left(-1\right)}^{i}+k}\end{eqnarray}$with constant ${\rho }_{\mathrm{0,0}}$, is the plane wave factor.


Proof.Let(3.6)$\begin{eqnarray}{u}_{1}={u}_{0}+{\left(-1\right)}^{m}(\kappa +\nu ),\end{eqnarray}$where $\kappa ={kr}$. With (3.2) and parametrizing the first BT parameter by(3.7)$\begin{eqnarray}r=\displaystyle \frac{c}{1-{k}^{2}},\end{eqnarray}$then substitution of $u={u}_{0}$ and $\overline{u}={u}_{1}$ into the auto-BT (2.2a) yields(3.8)$\begin{eqnarray}\widetilde{\nu }=\displaystyle \frac{\nu {E}_{+}}{\nu +{E}_{-}},\,\widehat{\nu }=\displaystyle \frac{\nu {F}_{+}(m)}{\nu +{F}_{-}(m)},\end{eqnarray}$where(3.9)$\begin{eqnarray}{E}_{\pm }=-r(a\pm k),\,{F}_{\pm }(m)=r({\left(-1\right)}^{m}\mp k).\end{eqnarray}$The difference system (3.8) can be linearized using $\nu =\tfrac{f}{g}$ and ${\rm{\Phi }}={\left(f,g\right)}^{{\rm{T}}}$, which leads to(3.10)$\begin{eqnarray}{\rm{\Phi }}(n+1,m)=M{\rm{\Phi }}(n,m),\,{\rm{\Phi }}(n,m+1)=N(m){\rm{\Phi }}(n,m),\end{eqnarray}$where(3.11)$\begin{eqnarray}\begin{array}{l}M=\left(\begin{array}{cc}{E}_{+} & 0\\ 1 & {E}_{-}\end{array}\right),\quad N(m)=\left(\begin{array}{cc}{F}_{+} & 0\\ 1 & {F}_{-}\end{array}\right).\end{array}\end{eqnarray}$By ‘integrating' (3.10) we have(3.12)$\begin{eqnarray}{\rm{\Phi }}(n,m)={ \mathcal M }(n){\rm{\Phi }}(0,m),\,\,{\rm{\Phi }}(n,m)={ \mathcal N }(m){\rm{\Phi }}(n,0),\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}{ \mathcal M }(n) & = & \left(\begin{array}{cc}{E}_{+}^{n} & 0\\ \displaystyle \frac{{E}_{-}^{n}-{E}_{+}^{n}}{2\kappa } & {E}_{-}^{n}\end{array}\right),\\ { \mathcal N }(m) & = & \left(\begin{array}{cc}\prod _{i=0}^{m-1}{F}_{+}(i) & 0\\ \displaystyle \frac{1-{\left(-1\right)}^{m}}{2}\prod _{i=0}^{m-2}{F}_{+}(i) & \prod _{i=0}^{m-1}{F}_{-}(i)\end{array}\right).\end{array}\end{eqnarray*}$Thus, we get a solution to (3.12):(3.13)$\begin{eqnarray}{\rm{\Phi }}(n,m)={ \mathcal M }(n){ \mathcal N }(m){\rm{\Phi }}(0,0),\end{eqnarray}$from which $\nu =f/g$ is obtained as(3.14)$\begin{eqnarray}\nu =\displaystyle \frac{{E}_{+}^{n}\prod _{i=0}^{m-1}{F}_{+}(i)\cdot {\nu }_{\mathrm{0,0}}}{{E}_{-}^{n}\prod _{i\,=\,0}^{m-1}{F}_{-}^{}(i)+\tfrac{\left({E}_{-}^{n}\prod _{i=0}^{m-1}{F}_{-}^{}(i)-{E}_{+}^{n}\prod _{i=0}^{m-1}{F}_{+}(i)\right){\nu }_{\mathrm{0,0}}}{2\kappa }},\end{eqnarray}$where ${\nu }_{\mathrm{0,0}}=\tfrac{{f}_{\mathrm{0,0}}}{{g}_{\mathrm{0,0}}}$. Introducing the plane wave factor(3.15)$\begin{eqnarray}\begin{array}{rcl}{\rho }_{n,m} & = & {\rho }_{\mathrm{0,0}}{\left(\displaystyle \frac{{E}_{+}}{{E}_{-}}\right)}^{n}\,\prod _{i\,=\,0}^{m-1}\displaystyle \frac{{F}_{+}(i)}{{F}_{-}(i)}\\ & = & {\rho }_{\mathrm{0,0}}{\left(\displaystyle \frac{a+k}{a-k}\right)}^{n}\,\prod _{i=0}^{m-1}\displaystyle \frac{{\left(-1\right)}^{i}-k}{{\left(-1\right)}^{i}+k}\end{array}\end{eqnarray}$with constant ${\rho }_{\mathrm{0,0}}$, the above ν is written as(3.16)$\begin{eqnarray}\nu =\displaystyle \frac{-2\kappa {\rho }_{n,m}}{1+{\rho }_{n,m}},\end{eqnarray}$where some constants are absorbed into ${\rho }_{\mathrm{0,0}}=\tfrac{-{\nu }_{\mathrm{0,0}}}{2\kappa +{\nu }_{\mathrm{0,0}}}$. Substituting (3.16) into (3.6) yields the one-soliton solution (3.4), which solves (1.6) with (3.2) and (3.7). Note that in the plane wave factor (3.15) $n,m\in {\mathbb{Z}}$, and when $m\leqslant 0$ the product ${\prod }_{i=0}^{m-1}(\cdot )$ is considered as ${\prod }_{i=m-1}^{0}(\cdot )$.

It is interesting that the solution has an oscillatory factor (− 1)^m in m-direction and in the plane wave factor ρ_n,m the spacing parameter q for m-direction does not appear. Considering the parameterization (3.2) where p is constant while q depends on m, we can say that the H2^a equation (1.6) is semi-autonomous.

4. Conclusions

In this paper, we have shown that equations which constitute an auto-BT for a quad equation admit auto-BTs themselves. We have focussed on one such equation, the torqued H2 equation denoted H2^a (1.6), which forms an auto-BT for Q1₁. This equation is not part of the ABS list of CAC quad equations, as it is not symmetric with respect to (n, p) ↔ (m, q). The integrability of this equation is guaranteed as it is part of a consistent cube, see [14]. The equations H2^a and Q1₁ comprise an auto-BT from which a Lax pair was obtained. Using this auto-BT we have derived a seed solution and a one-soliton solution. The parameterization of these solutions show that H2^a is a semi-autonomous equation. We hope to be able to construct higher order soliton solutions in a future paper.

Acknowledgments

This work was supported by a La Trobe University China studies seed-funding research grant, and by the NSF of China [Grant Numbers 11 875 040 and 11 631 007].

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