From nothing to something: discrete integrable systems

Details From nothing to something: discrete integrable systems From nothing to something: discrete integrable systems

2012-01-28

arxiv.org/abs/1201.5924v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

16 pages; 3 figures

Scientific paper

Chinese ancient sage Laozi said `\emph{Dao sheng yi, yi sheng er, er sheng san, san sheng wanwu}, ...' that means something even everything comes from \emph{\bf \em `nothing'} via `\bf \em Dao'. \rm In this paper, various discrete integrable models, including the known discrete Schwarzian KdV, KP, BKP, CKP, special Viallet equations and many other unknowns are derived from nothing. The ancient Greek geometric theorems, such as the angle bisector theorem, are also closely related to the discrete integrable models. It is conjectured that all the discrete models generated from nothing may be integrable, because they are identities of a same algebra, model independent nonlinear superpositions of the Riccati equations, index homogeneous decompositions of the angle bisector theorem and M\"obious transformation invariants.

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