From Nothing to Something: Discrete Integrable Systems
LOU Sen-Yue1,2**, LI Yu-Qi1,2, TANG Xiao-Yan3
1Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062 2Faculty of Science, Ningbo University, Ningbo 315211 3Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240
Abstract：Chinese ancient sage Laozi said that everything comes from 'nothing'. Einstein believes the principle of nature is simple. Quantum physics proves that the world is discrete. And computer science takes continuous systems as discrete ones. This report is devoted to deriving a number of discrete models, including well-known integrable systems such as the KdV, KP, Toda, BKP, CKP, and special Viallet equations, from 'nothing' via simple principles. It is conjectured that the discrete models generated from nothing may be integrable because they are identities of simple algebra, model-independent nonlinear superpositions of a trivial integrable system (Riccati equation), index homogeneous decompositions of the simplest geometric theorem (the angle bisector theorem), as well as the M?bious transformation invariants.
2013, Vol. 30 
