25 research outputs found
Euler-Lagrange correspondence of generalized Burgers cellular automaton
Recently, we have proposed a {\em Euler-Lagrange transformation} for cellular
automata(CA) by developing new transformation formulas. Applying this method to
the Burgers CA(BCA), we have succeeded in obtaining the Lagrange representation
of the BCA. In this paper, we apply this method to multi-value generalized
Burgers CA(GBCA) which include the Fukui-Ishibashi model and the quick-start
model associated with traffic flow. As a result, we have succeeded in
clarifying the Euler-Lagrange correspondence of these models. It turns out,
moreover that the GBCA can naturally be considered as a simple model of a
multi-lane traffic flow.Comment: 11 pages, 6 figures; accepted for publication in Int. J. Mod. Phys.
Casorati Determinant Solution for the Relativistic Toda Lattice Equation
The relativistic Toda lattice equation is decomposed into three Toda systems,
the Toda lattice itself, B\"acklund transformation of Toda lattice and discrete
time Toda lattice. It is shown that the solutions of the equation are given in
terms of the Casorati determinant. By using the Casoratian technique, the
bilinear equations of Toda systems are reduced to the Laplace expansion form
for determinants. The -soliton solution is explicitly constructed in the
form of the Casorati determinant.Comment: 19 pages in plain Te
Third-order integrable difference equations generated by a pair of second-order equations
We show that the third-order difference equations proposed by Hirota,
Kimura and Yahagi are generated by a pair of second-order difference
equations. In some cases, the pair of the second-order equations are equivalent
to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are
irrelevant to the QRT system. We also discuss an ultradiscretization of the
equations.Comment: 15 pages, 3 figures; Accepted for Publication in J. Phys.
Discrete mappings with an explicit discrete Lyapunov function related to integrable mappings
We propose discrete mappings of second order that have a discrete analogue of
Lyapunov function. The mappings are extensions of the integrable
Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the
mappings is identical to an explicit conserved quantity of the QRT mapping.
Moreover we can obtain a differential and an ultradiscrete limit of the
mappings preserving the existence of Lyapunov function. We also give
applications of a mapping with an adjusted parameter, a probabilistic mapping
and coupled mappings.Comment: submitted to Physica
Max-plus analysis on some binary particle systems
We concern with a special class of binary cellular automata, i.e., the
so-called particle cellular automata (PCA) in the present paper. We first
propose max-plus expressions to PCA of 4 neighbors. Then, by utilizing basic
operations of the max-plus algebra and appropriate transformations, PCA4-1, 4-2
and 4-3 are solved exactly and their general solutions are found in terms of
max-plus expressions. Finally, we analyze the asymptotic behaviors of general
solutions and prove the fundamental diagrams exactly.Comment: 24 pages, 5 figures, submitted to J. Phys.
周期写像を用いた高次保存量を持つ可積分方程式の生成
九州大学応用力学研究所研究集会報告 No.21ME-S7 「非線形波動研究の現状と将来 : 次の10 年への展望」RIAM Symposium No.21ME-S7 Current and Future Research on Nonlinear Waves : Perspectives for the Next Decade可積分な2階差分方程式としてQRT系が知られている。これまでに我々は、QRT系をもとに高次保存量を持つ差分方程式を生成してきた[1]。今回、2階可積分差分方程式の中でQRT系ではないと主張されている方程式から得た考察をもとに、新たな可積分な方程式を生成する