Frontiers of Mathematics in China >

2019 , Vol. 14 >Issue 1: 203 - 223

DOI: https://doi.org/10.1007/s11464-019-0753-0

RESEARCH ARTICLE

Addition formulae, Backlund transformations, periodic solutions, and quadrilateral equations

  • Danda ZHANG , 1 ,
  • Da-jun ZHANG , 2
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  • 1. School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
  • 2. Department of Mathematics, Shanghai University, Shanghai 200444, China

Received date: 21 Mar 2018

Accepted date: 29 Jan 2019

Published date: 22 Mar 2019

Copyright

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature
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Abstract

Addition formulae of trigonometric and elliptic functions are used to generate Backlund transformations together with their connecting quadrilateral equations. As a result, we obtain the periodic solutions for a number of multidimensionally consistent affine linear and multiquadratic quadrilateral equations.

Key words： Addition formulae; trigonometric functions; elliptic functions; Backlund transformation; quadrilateral equations

Cite this article

Danda ZHANG , Da-jun ZHANG . Addition formulae, Backlund transformations, periodic solutions, and quadrilateral equations[J]. Frontiers of Mathematics in China, 2019 , 14(1) : 203 -223 . DOI: 10.1007/s11464-019-0753-0

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Adler V E, Bobenko A I, Suris Yu B. Classification of integrable equations on quadgraphs. The consistency approach. Comm Math Phys, 2003, 233: 513–543

DOI

2
Atkinson J. Backlund transformations for integrable lattice equations. J Phys A, 2008, 41: 135202 (8pp)

3
Atkinson J, Nieszporski M. Multi-quadratic quad equations: integrable cases from a factorized-discriminant hypothesis. Int Math Res Not IMRN, 2014, 2014: 4215–4240

4
Bobenko A I, Suris Yu B. Integrable systems on quad-graphs. Int Math Res Not IMRN, 2002, 2002: 573–611

5
Hietarinta J, Joshi N, Nijhoff F W. Discrete Systems and Integrability. Cambridge: Cambridge Univ Press, 2016

DOI

6
Hietarinta J, Viallet C. Weak Lax pairs for lattice equations. Nonlinearity, 2011, 25: 1955–1966

DOI

7
Hietarinta J, Zhang D J. Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization. J Phys A, 2009, 42: 404006 (30pp)

8
Nijhoff F W. Lax pair for the Adler (lattice Krichever-Novikov) system. Phys Lett A, 2002, 297: 49–58

DOI

9
Nijhoff F W, Atkinson J. Elliptic N-soliton solutions of ABS lattice equations. Int Math Res Not IMRN, 2010, 2010: 3837–3895

10
Nijhoff F W, Walker A J. The discrete and continuous Painlevé VI hierarchy and the Garnier systems. Glasg Math J, 2001, 43A: 109–123

DOI

11
Xu X X, Cao C W. A new explicit solution to the lattice sine-Gordon equation. Modern Phys Lett B, 2016, 30: 1650166 (7pp)

12
Zhang D D, Zhang D J. Rational solutions to the ABS list: Transformation approach. SIGMA Symmetry Integrability Geom Methods Appl, 2017, 13: 078 (24pp)

13
Zhang D D, Zhang D J. On decomposition of the ABS lattice equations and related Backlund transformations. J Nonlinear Math Phys, 2018, 25: 34–53

DOI

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