Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2007-08-06
J. Phys. Soc. Jpn. 78 (2009) 024003
Nonlinear Sciences
Exactly Solvable and Integrable Systems
31 pages. Section 1 changed and section 5 added
Scientific paper
10.1143/JPSJ.78.024003
Motivated by an attempt to develop a method for solving initial value problems in a class of one dimensional periodic cellular automata (CA) associated with crystal bases and soliton equations, we consider a generalization of a simple proposition in elementary mathematics. The original proposition says that any sequence of letters 1 and 2, having no less 1's than 2's, can be changed into a ballot sequence via cyclic shifts only. We generalize it to treat sequences of cells of common capacity s > 1, each of them containing consecutive 2's (left) and 1's (right), and show that these sequences can be changed into a ballot sequence via two manipulations, cyclic and "quasi-cyclic" shifts. The latter is a new CA rule and we find that various kink-like structures are traveling along the system like particles under the time evolution of this rule.
Takagi Taichiro
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