# On one-soliton solutions of the Q2 equation in the ABS list

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## Abstract

In this paper, we derive seed and 1-soliton solutions of the Q2 equation in the Adler–Bobenko–Suris list. The seed solutions of Q2 are obtained using those of \(\mbox{Q}1(\delta)\) and an non-auto Bäcklund transformation connecting them. Then using an auto Bäcklund transformation, two types of Q2 one-soliton solutions are obtained based on its seed solutions. These obtained solutions are new and cannot be derived as degenerations from any known soliton solutions.

## Keywords

Soliton solutions ABS list The Q2 equation Bäcklund transformation## PACS Codes

02.30.Ik 02.30.Jr 04.20.Jb## 1 Introduction

Integrable systems involve the study of physically relevant nonlinear equations, which includes many families of well-known, highly important partial and ordinary differential equations. Over the past two decades, research in discrete integrable systems has undergone a truly remarkable development (see for instance the monograph [1]). One of the key achievements was the introduction of multi-dimensional consistency [2, 3, 4] as a defining criterion for discrete integrability, which can be seen as the discrete analog of the compatible hierarchies in continuous integrable theory. It turns out that quad-equations, two-dimensional lattice equations defined on square lattices, are said to be integrable, if they are three-dimensional consistent, geometrically meaning that the equations can be consistently embedded around a cube (CAC). This led, up to few other assumptions, to the classification of integrable affine linear quad-equations [5], known as the Adler–Bobenko–Suris (ABS) list. The list contains 9 equations, named Q4, \(\mbox{Q}3(\delta)\), Q2, \(\mbox{Q}1(\delta)\), A2, \(\mbox{A}1(\delta)\), \(\mbox{H}3(\delta)\), H2 and H1. Most of these equations were known or related to known equations. For example, Q4 is a fully discretized version of the famous Krichever–Novikov equation [6, 7], \(\mbox{Q}3(\delta)\) is related to the Nijhoff–Quispel–Capel equation [8, 9], \(\mbox{Q}1(0)\), \(\mbox{H}3(0)\) and H1 are, respectively, the lattice Schwarzian Korteweg–de Vries (KdV) equation, the lattice potential modified KdV equation and the lattice potential KdV equation [10].

*p*,

*q*are the lattice parameters associated with

*n*,

*m*respectively. See Fig. 1(a).

*k*(see Fig. 1(b)). The above equations also define an auto Bäcklund transformation (BT) of Q2 [5], namely, given

*u*as a solution,

*u̅*solves Q2 as well provided that

*u*and

*u̅*are connected through (1.2a)–(1.2b). In the auto BT approach,

*u*is commonly known as a

*seed*solution, while

*ū*as a new solution generated from

*u*.

One particular aim of this paper is to derive one-soliton solutions (1SSs) of Q2 by means of the auto BT (1.2a)–(1.2b). This requires knowledge of its seed solutions, which have not been well understood either in the literature. For instance the fixed-point method [15] only provides a solution of Q2 that is a special case of a more general solution (see Sect. 2.3). Our approach to obtaining Q2’s seed solutions is based on an non-auto BT between \(\mbox{Q}1(\delta)\) and Q2 [16]. There exist two different types of (seed) solutions of \(\mbox{Q}1(\delta)\) such as the exponential type cf. [15] and the rational type [15, 17]. These solutions allow one to derive exponential and rational types of seed solutions of Q2, and in turn, lead to different 1SSs of Q2. Note that although the idea of this paper is clear, the “integration” of the BT (1.2a)–(1.2b) to get 1SS is highly nontrivial. We also note that the solutions of Q2 we obtain here are essentially new, as they cannot be reduced as reductions of known results.

The paper is organized as follows. We will first make use of known solutions of \(\mbox{Q}1(\delta)\) and the non-auto BT between \(\mbox{Q}1(\delta)\) and Q2 to derive seed solutions for Q2. Fixed point idea will also be discussed. These will be done in Sect. 2. Then in Sect. 3 we derive three 1SSs for Q2 from different seed solutions. Section 4 serves for conclusions.

## 2 Seed solutions

### 2.1 Exponential case

*A*is an arbitrary non-zero constant, and

*α*,

*β*are connected to

*p*,

*q*through

*n*and

*m*, respectively, one gets

*r*is another constant.

### 2.2 Rational case

*u*and

*w*:

*δ*is a parameter and

*y*is to be determined. It follows from the BT (2.2a)–(2.2b) that

^{1}[17] \(w=\delta(x_{1}^{2}+ c_{0}) \) (\(c_{0}\) is a constant) into (2.10a)–(2.10b) yields

*y*satisfies

*z*reads

*γ*,

*δ*, \(c_{0}\) being constants. Using the ansatz (2.9), through (2.10a)–(2.10b), one has

*δ*. For (2.19), through (2.20) and

### 2.3 Fixed-point solution

*u̅*is considered to be

*u*, i.e.

*k*is not significant in generating solitons and then

*u*is solved usually as a background solution (seed solution) of solitons. By this idea the BT (1.2a)–(1.2b) of Q2 yields its fixed point version

## 3 One-soliton solutions of Q2

In this section, the solutions (2.6), (2.12) and (2.15) are used as seed solutions in the auto BT approach to generating 1SSs.

### 3.1 General procedure

Suppose *u* (denoted by \(u_{\theta}\) conventionally, cf. [15, 19, 20, 21, 22]) is a seed solution of Q2 and denote \(\overline{u}_{\theta}\) as a shifted *u* in the third direction in the light of the CAC property. In fact, the CAC property of Q2 indicates its solution \(u(n,m)\) can be consistently embedded into a 3-dimension cube (see Fig. 1(b)). Although there is no explicit independent variable *l* in \(u(n,m)\), one can introduce a bar shift (shift in *l*-direction) for it according to \(\widetilde{~~}\) or \(\widehat{~~~}\) shifts. For example, for *w* defined in (2.3), we have \(\widetilde{w}=\frac {1}{2}(A \alpha^{n} \beta^{m} \alpha+A^{-1}\alpha^{-n}\beta^{-m}\alpha^{-1})\) and *α* is related to *p* (the spacing parameter of *n*-direction) as in (2.4). Then *w̅* should be accordingly defined as \(\overline{w}=\frac{1}{2}(A \alpha^{n} \beta^{m} s +A^{-1}\alpha^{-n}\beta^{-m}s^{-1})\) and *s* is related to the spacing parameter *k* of *l*-direction by \(k=(1-s)^{2}/2s\), which is coincident with (2.4). For \(x_{i}\) defined in (2.7), we have \(\overline{x}_{i}=x_{i}+c^{i}\) where we suppose \(c^{2}=k\) to coincide with (2.8).

*u̅*as a solution of Q2 in the form

^{2}

*v*

*Λ*is some “balancing” factor to be determined to guarantee the compatibility \(\widehat{\widetilde{\varPhi}}=\widetilde{\widehat{\varPhi}}\). Then it remains to integrate the linear difference system (3.4) so that \(v=\frac{f}{g}\), hence

*ū*, can be derived. The so-obtained

*ū*gives a 1SS of Q2.

### 3.2 1SS from exponential seed solution (2.6)

*α*,

*β*and

*P*,

*Q*,

*s*and

*K*satisfy

*k*is understood as the lattice parameter in the \(\bar{~~}\) direction. Following the ansatz described above (3.1)–(3.4), direct but hard computations lead to the difference system

*v*in the form

*A*to \(\frac{A}{s}\) one obtains

### 3.3 1SSs from rational seed solutions (2.12) and (2.15)

We continue computing 1SSs of Q2 using the rational seed solutions (2.12) and (2.15). We will skip computational details and just put the main results.

*u*(2.12) and \(z_{n,m}\) given in (2.11). We depict its shape and motion, after removing the background \(\overline{u}_{\theta}\), in Fig. 3(a). One can find the wave asymptotically is governed by zero on one direction and by \(\frac{2 s z_{n,m}}{x_{1}^{2}-s^{2}}\) on the other direction.

*u*(2.15) and \(z_{n,m}\) given in (2.14). We depict its shape and motion, after removing the background \(\overline{u}_{\theta}\), in Fig. 3(b). This is a moving wave asymptotically governed by zero on one direction and by \(\frac{4 \delta s z_{n,m}}{(x_{1}-\delta)^{2}-s^{2}}\) on the other direction.

## 4 Conclusions

In this short paper, we manage to provide explicit formulas of solutions of the Q2 equation (1.1) in the ABS list, at the cost of a considerable computational effort (in particular, to determine the balancing factor *Λ*). We derive seed solutions of Q2 in Sect. 2 using solutions of \(\mbox{Q}1(\delta)\) and a non-auto BT connecting them. The results are then used in Sect. 3 to derive 1SSs of Q2 using an auto BT approach. Both the auto BT and the non-auto BT are realizations of the CAC property. The seed and the associated soliton solutions belong to either exponential type or rational type of solutions. They are essentially new solutions, i.e. (2.12), (2.15), (3.15) and (3.17), as they cannot be obtained as degenerated cases of known solutions.

*s*, which looks much neater than (3.10) derived from the exponential case. This is again different from the pure soliton form \(\frac{s\rho _{n,m}}{1+\rho_{n,m}}\) (cf. (3.15) in [15]). In addition, although rational solutions can usually be interpreted as some limits of solitons, so far in the literature we do not know any solutions that can yield the above form by imposing suitable limits.

In comparison with other equations in the ABS list, Q2 is rather special and needs further investigations. For instance, its bilinear form, continuous counterpart, geometric interpretations or physical significance have not been fully understood. In particular, a systematic approach to generating *N*-soliton solutions of given type is yet to be understood, into which we hope our results could provide insight.

## Footnotes

## Notes

### Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

### Funding

This project is supported by the NSFC (Nos. 11371241, 11631007, 11601312, 11875040, 11801289) and Shanghai Young Eastern Scholar scheme (2016–2019).

### Competing interests

The authors declare that there is no conflict of interest regarding the publication of this paper.

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