Analysis for integrable cellular automata by ultradiscretization method and its applications to non-integrable systems

我们提出了一种“含奇偶变量的超离散化”方法，它是通常超离散化方法的一种推广。利用这一过程，我们得到了关于可积细胞自动机的各种结果，例如，构造了q-Painleve II方程的超离散模拟及其一类特解。利用通常的超离散化方法研究了超离散系统的振荡解、基于代数几何方法的Box and Ball系统孤子解的推导等问题。本文还研究了一类不可积线性系统的最优速度模型和卡尔曼滤波器的超离散模拟。

We proposed a procedure `ultradiscretization with parityvariables’, which is an extended version of usual ultradiscretization method. By meansof this procedure, we obtained various results on integrable cellular automata, forexample, construction of an ultradiscrete analog of the q-Painleve II equation and a classof its special solutions. We studied oscillatory solutions of ultradiscrete systems,derivation of soliton solutions to the Box and Ball system based on an algebro-geometricmethod and so on by usual ultradiscretization method. We also investigated anultradiscrete analog of the optimal velocity model and Kalman filter for apiecewise-linear system, which are non-integrable systems.