Lax Pairs for Discrete Integrable Equations via Darboux Transformations

doi:10.1088/0256-307X/29/5/050202

Chin. Phys. Lett.

2012, Vol. 29

Issue (5): 050202 DOI: 10.1088/0256-307X/29/5/050202

GENERAL

Lax Pairs for Discrete Integrable Equations via Darboux Transformations

CAO Ce-Wen^**,ZHANG Guang-Yao

Department of Mathematics, Zhengzhou University, Zhengzhou 450001

Cite this article:

CAO Ce-Wen**, ZHANG Guang-Yao 2012 Chin. Phys. Lett. 29 050202

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Abstract

A method is developed to construct discrete Lax pairs using Darboux transformations. More kinds of Lax pairs are found for some newly appeared discrete integrable equations, including the H1, the special H3 and the Q1 models in the Adler–Bobenko–Suris list and the closely related discrete and semi-discrete pKdV, pMKdV, SG and Liouville equations.

Keywords: 02.30.Ik 02.30.Jr 04.20.Jb

Received: 27 December 2011 Published: 30 April 2012

PACS:	02.30.Ik	(Integrable systems)
	02.30.Jr	(Partial differential equations)
	04.20.Jb	(Exact solutions)

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https://cpl.iphy.ac.cn/10.1088/0256-307X/29/5/050202 OR https://cpl.iphy.ac.cn/Y2012/V29/I5/050202

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	CAO Ce-Wen**
	ZHANG Guang-Yao

[1] Adler V E, Bobenko A I and Suris Yu B 2003 Commun. Math. Phys. 233 513

[2] Atkinson J, Hietarinta J and Nijhoff F 2008 J. Phys. A: Math. Theor. 41 142001

[3] Bobenko A and Pinkall U 1996 J. Diff. Geom. 43 527

[4] Grammaticos B, Kosmann-Schwarzbach Y and Tamizhmani T 2004 Discrete Integrable Systems (Berlin: Springer)

[5] Hietarinta J and Zhang D J 2009 J. Phys. A: Math. Theor. 42 404006

[6] Nijhoff F W 2006 Discrete Systems and Integrability MATH 3490/5491 (Leeds: University of Leeds)

[7] Nijhoff F, Atkinson J and Hietarinta J 2009 J. Phys. A: Math. Theor. 42 404005

[8] Nijhoff F W and Capel H W 1995 Acta Appl. Math. 39 133

[9] Suris Yu B2003 The Problem of Integrable Discretization: Hamiltonian Approach (Basel: Birkhäuser)

[10] Gu C, Hu H and Zhou Z 1999 Darboux Transformations in Soliton Theory and Its Geometric Applications (Shanghai: Shanghai Scientific and Technical)

[11] Levi D 1981 J. Phys. A: Math. Gen. 14 1083

[12] Gesztesy F, Holden H, Michor J and Teschl G2008 Soliton Equations and Their Algebro-geometric Solutions vol II (1+1)-dimensional Discrete Models (Cambridge: Cambridge University)

[13] Hirota R 1977 J. Phys. Soc. Jpn. 43 2079

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