### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 036, 18 pages      arXiv:1705.09896      https://doi.org/10.3842/SIGMA.2017.036

### Darboux and Binary Darboux Transformations for Discrete Integrable Systems. II. Discrete Potential mKdV Equation

Ying Shi a, Jonathan Nimmo b and Junxiao Zhao c
a) School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, P.R. China
b) Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, UK
c) School of Mathematics, University of Chinese Academy of Sciences, Beijing 100190, P.R. China

Received December 17, 2016, in final form May 16, 2017; Published online May 28, 2017

Abstract
The paper presents two results. First it is shown how the discrete potential modified KdV equation and its Lax pairs in matrix form arise from the Hirota-Miwa equation by a 2-periodic reduction. Then Darboux transformations and binary Darboux transformations are derived for the discrete potential modified KdV equation and it is shown how these may be used to construct exact solutions.

Key words: partial difference equations; integrability; reduction; Darboux transformation.

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