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On one-soliton solutions of the Q2 equation in the ABS list
- Qiang Cheng^{1},
- Cheng Zhang^{1}Email author,
- Danda Zhang^{2} and
- Da-jun Zhang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-019-1977-8
© The Author(s) 2019
- Received: 18 August 2018
- Accepted: 16 January 2019
- Published: 25 January 2019
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 [2–4] 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].
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
2.2 Rational case
2.3 Fixed-point solution
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, 19–22]) 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).
3.2 1SS from exponential seed solution (2.6)
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.
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.
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.
Declarations
Funding
This project is supported by the NSFC (Nos. 11371241, 11631007, 11601312, 11875040, 11801289) and Shanghai Young Eastern Scholar scheme (2016–2019).
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Hietarinta, J., Joshi, N., Nijhoff, F.W.: Discrete Systems and Integrablity. Cambridge University Press, Cambridge (2016) View ArticleGoogle Scholar
- Nijhoff, F.W., Walker, A.J.: The discrete and continuous Painlevé VI hierarchy and the Garnier systems. Glasg. Math. J. 43A, 109–123 (2001) MathSciNetView ArticleGoogle Scholar
- Nijhoff, F.W.: Lax pair for the Adler (lattice Krichever–Novikov) system. Phys. Lett. A 297, 49–58 (2002) MathSciNetView ArticleGoogle Scholar
- Bobenko, A.I., Suris, Y.B.: Integrable systems on quad-graphs. Int. Math. Res. Not. 2002(11), 573–611 (2002) MathSciNetView ArticleGoogle Scholar
- Adler, V.E., Bobenko, A.I., Suris, Y.B.: Classification of integrable equations on quad-graphs, the consistency approach. Commun. Math. Phys. 233, 513–543 (2002) MathSciNetView ArticleGoogle Scholar
- Krichever, I.M., Novikov, S.P.: Holomorphic fiberings and nonlinear equations: finite zone solutions of rank 2. Sov. Math. Dokl. 20, 650–654 (1979) MATHGoogle Scholar
- Adler, V.E.: Bäcklund transformation for the Krichever–Novikov equation. Int. Math. Res. Not. 1998, 1–4 (1998) View ArticleGoogle Scholar
- Nijhoff, F.W., Quispel, G.R.W., Capel, H.W.: Direct linearization of nonlinear difference–difference equations. Phys. Lett. A 97, 125–128 (1983) MathSciNetView ArticleGoogle Scholar
- Nijhoff, F.W., Atkinson, J., Hietarinta, J.: Soliton solutions for ABS lattice equations. I. Cauchy matrix approach. J. Phys. A, Math. Theor. 42, 404005 (2009) MathSciNetView ArticleGoogle Scholar
- Nijhoff, F.W., Capel, H.W.: The discrete Korteweg–de Vries equation. Acta Appl. Math. 39, 133–158 (1995) MathSciNetView ArticleGoogle Scholar
- Zhang, D.J., Zhao, S.L.: Solutions to ABS lattice equations via generalized Cauchy matrix approach. Stud. Appl. Math. 131, 72–103 (2013) MathSciNetView ArticleGoogle Scholar
- Nijhoff, F.W., Atkinson, J.: Elliptic N-soliton solutions of ABS lattice equations. Int. Math. Res. Not. 2010, 3837–3895 (2010) MathSciNetMATHGoogle Scholar
- Butler, S.: Multidimensional inverse scattering of integrable lattice equations. Nonlinearity 25, 1613–1634 (2012) MathSciNetView ArticleGoogle Scholar
- Zhang, D.D., Zhang, D.J.: Rational solutions to the ABS list: transformation approach. SIGMA 13, 078 (2017) MathSciNetMATHGoogle Scholar
- Hietarinta, J., Zhang, D.J.: Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization. J. Phys. A, Math. Theor. 42, 404006 (2009) MathSciNetView ArticleGoogle Scholar
- Atkinson, J.: Bäcklund transformations for integrable lattice equations. J. Phys. A, Math. Theor. 41, 135202 (2008) View ArticleGoogle Scholar
- Zhang, D.D., Zhang, D.J.: On decomposition of the ABS lattice equations and related Bäcklund transformations. J. Nonlinear Math. Phys. 25, 34–53 (2018) MathSciNetView ArticleGoogle Scholar
- Zhao, S.L., Zhang, D.J.: Rational solutions to \(\mbox{Q}3(\delta)\) in the Adler–Bobenko–Suris list and degenerations. J. Nonlinear Math. Phys. 26, 107–132 (2019) MathSciNetView ArticleGoogle Scholar
- Atkinson, J., Hietarinta, J., Nijhoff, F.: Seed and soliton solutions of Adler’s lattice equation. J. Phys. A, Math. Theor. 40, F1–F8 (2007) MathSciNetView ArticleGoogle Scholar
- Atkinson, J., Hietarinta, J., Nijhoff, F.W.: Soliton solutions for Q3. J. Phys. A, Math. Theor. 41, 142001 (2008) MathSciNetView ArticleGoogle Scholar
- Hietarinta, J., Zhang, D.J.: Multisoliton solutions to the lattice Boussinesq equation. J. Math. Phys. 51, 033505 (2010) MathSciNetView ArticleGoogle Scholar
- Hietarinta, J., Zhang, D.J.: Soliton taxonomy for a modification of the lattice Boussinesq equation. SIGMA 7, 061 (2011) MathSciNetMATHGoogle Scholar