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Discrete integrable systems: Multidimensional consistency

## Discrete integrable systems: Multidimensional consistency

Zhang Da-Jun
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• #### Abstract

In contrast to the well-established theory of differential equations, the theory of difference equations has not quite developed so far. The most recent advances in the theory of discrete integrable systems have brought a true revolution to the study of difference equations. Multidimensional consistency is a new concept appearing in the research of discrete integrable systems. This property, as an explanation to a type of discrete integrability, plays an important role in constructing the Bäcklund transformations, Lax pairs and exact solutions for discrete integrable system. In the present paper, the multidimensional consistency and its applications in the research of discrete integrable systems are reviewed.

#### Cited By

• 图 1  $[x_0, x]$上的数值离散

Figure 1.  Numerical discretisation on $[x_0, x]$

图 2  Bäcklund变换解的交换性质

Figure 2.  Permutation property of Bäcklund transformation

图 3  变换与平面网格

Figure 3.  Map and lattice

图 4  相容立方体

Figure 4.  Consistent cube

图 5  离散sine-Gordon和势mKdV方程的相容立方体

Figure 5.  The consistent cube for the discrete sine-Gordon equation and potential mKdV equation

图 6  定义3维方程的6面体以及8面体

Figure 6.  Cube and octahedron for 3D equations

图 7  围绕超立方体的4D相容性

Figure 7.  4D consistency around the hyper cube

图 8  相容立方体

Figure 8.  The consistent cube

•  [1] Zhang Da-jun. Discrete integrable systems: Multidimensional consistency. Acta Physica Sinica, 2020, 69(1): 1-12. doi: 10.7498/aps.69.20191647 [2] Li Qi-Liang, Zhu Hai-Dong, Tang Xiang-Hong, Li Cheng-Jia, Wang Xiao-Jun, Lin Li-Bin. Integrability aspects of solitons’ coupled equation in multi-wavelength system. Acta Physica Sinica, 2004, 53(6): 1623-1628. doi: 10.7498/aps.53.1623 [3] Liu Ping, Xu Heng-Rui, Yang Jian-Rong. The Boussinesq equation: Lax pair, Bäcklund transformation, symmetry group transformation and consistent Riccati expansion solvability. Acta Physica Sinica, 2020, 69(1): 010203. doi: 10.7498/aps.69.20191316 [4] Tian Xiao-Dong, Yue Rui-Hong. Integrability of the generalized multi-component Fermi quantum derivative nonlin ear Schrdinger model. Acta Physica Sinica, 2005, 54(4): 1485-1489. doi: 10.7498/aps.54.1485 [5] Dong Wen-Jie, Feng Guo-Lin, He Wen-Ping, Li Jian-Ping. On the predictability of the Lorenz system. Acta Physica Sinica, 2006, 55(2): 969-977. doi: 10.7498/aps.55.969 [6] She Qing, Jiang Mei-Fu, Qian Nong, Pan Yue. Effects of preparation temperature of SiC intermediate layers on the hemocompatibility of SiC/F-DLC composite film. Acta Physica Sinica, 2014, 63(18): 185204. doi: 10.7498/aps.63.185204 [7] Li Bao-Sheng, Ding Rui-Qiang, Li Jian-Ping, Zhong Quan-Jia. Predictability of forced Lorenz system. Acta Physica Sinica, 2017, 66(6): 060503. doi: 10.7498/aps.66.060503 [8] DAI XIN-GANG, WANG AI-HUI, CHOU JI-FAN, FENG GUO-LIN. ON NUMERICAL PREDICTABILITY IN THE CHAOS SYSTEM. Acta Physica Sinica, 2001, 50(4): 606-611. doi: 10.7498/aps.50.606 [9] Chen Li-Qun, Fu Jing-Li, Shi Shen-Yang. Lie symmetries of discrete Lagrange systems. Acta Physica Sinica, 2007, 56(6): 3060-3063. doi: 10.7498/aps.56.3060 [10] Xiong Chuan-Hua. The Dressing symmetry and Affine integrability of type IIB superstring in AdS5○×S5 background. Acta Physica Sinica, 2005, 54(1): 47-52. doi: 10.7498/aps.54.47
•  Citation:
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• Abstract views:  140
• Cited By: 0
##### Publishing process
• Received Date:  28 October 2019
• Accepted Date:  14 November 2019
• Available Online:  05 December 2019
• Published Online:  01 January 2020

## Discrete integrable systems: Multidimensional consistency

###### Corresponding author: Zhang Da-Jun, djzhang@staff.shu.edu.cn
• Department of Mathematics, Shanghai University, Shanghai 200444, China

Abstract: In contrast to the well-established theory of differential equations, the theory of difference equations has not quite developed so far. The most recent advances in the theory of discrete integrable systems have brought a true revolution to the study of difference equations. Multidimensional consistency is a new concept appearing in the research of discrete integrable systems. This property, as an explanation to a type of discrete integrability, plays an important role in constructing the Bäcklund transformations, Lax pairs and exact solutions for discrete integrable system. In the present paper, the multidimensional consistency and its applications in the research of discrete integrable systems are reviewed.

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