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Bäcklund transformations of $Z_{n}$ $Z_{n}$ -sine-Gordon systems

Xueping Yang

Department of Mathematics, Ningbo University, Ningbo 315211, China

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and

Chuanzhong Li

Department of Mathematics, Ningbo University, Ningbo 315211, China

E-mail Address: lichuanzhong@nbu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0217984917501895Cited by:13 (Source: Crossref)

Abstract

In this paper, from the algebraic reductions from the Lie algebra $g l (n, ℂ)$ to its commutative subalgebra $Z_{n}$ , we construct the general $Z_{n}$ -sine-Gordon and $Z_{n}$ -sinh-Gordon systems which contain many multi-component sine-Gordon type and sinh-Gordon type equations. Meanwhile, we give the Bäcklund transformations of the $Z_{n}$ -sine-Gordon and $Z_{n}$ -sinh-Gordon equations which can generate new solutions from seed solutions. To see the $Z_{n}$ -systems clearly, we consider the $Z_{2}$ -sine-Gordon and $Z_{3}$ -sine-Gordon equations explicitly including their Bäcklund transformations, the nonlinear superposition formula and Lax pairs.

Keywords:

PACS: 42.65.Tg, 42.65.Sf, 05.45.Yv, 02.30.Ik

References

1. E. Goldobin, A. Sterck, T. Gaber, D. Koelle and R. Kleiner, Phys. Rev. Lett. 92 (2004) 057005. Crossref, Web of Science, ADS, Google Scholar
2. V. G. Ivancevic and T. T. Ivancevic, J. Geom. Symmetry Phys. 31 (2013) 1. Google Scholar
3. G. A. Maugin, Nonlinear Waves in Elastic Crystals (Oxford University Press, UK, 1999). Crossref, Google Scholar
4. A. Kundu, Phys. Rev. Lett. 99 (2007) 154101. Crossref, Web of Science, ADS, Google Scholar
5. A. V. Mikhailov, G. Papamikos and J. P. Wang, Physica D: Nonlin. Phenom. 325 (2016) 53. Crossref, Web of Science, ADS, Google Scholar
6. I. A. B. Strachan and D. F. Zuo, J. Math. Phys. 56 (2015) 113509. Crossref, Web of Science, ADS, Google Scholar
7. A. P. Fordy, A. G. Reyman and M. A. Semenov-Tian-Shansky, Lett. Math. Phys. 17 (1989) 25. Crossref, Web of Science, ADS, Google Scholar
8. R. Hirota, X. B. Hu and X. Y. Tang, J. Math. Anal. Appl. 288 (2003) 326. Crossref, Web of Science, Google Scholar
9. W. X. Ma and B. Fuchssteiner, Phys. Lett. A 213 (1996) 49. Crossref, Web of Science, ADS, Google Scholar
10. C. Z. Li and J. S. He, Theor. Math. Phys. 185 (2015) 1614. Crossref, Web of Science, Google Scholar
11. C. Z. Li, J. Phys. A: Math. Theor. 49 (2016) 015203. Crossref, Web of Science, ADS, Google Scholar
12. R. K. Dodd and R. K. Bullough, Proc. R. Soc. A 351 (1976) 499. Crossref, ADS, Google Scholar
13. A. Grauel, Physica A 132 (1985) 557. Crossref, Web of Science, ADS, Google Scholar