62 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.
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
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
Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation
A generalization of the max-plus transformation, which is known as a method
to derive cellular automata from integrable equations, is proposed for complex
numbers. Operation rules for this transformation is also studied for general
number of complex variables. As an application, the max-plus transformation is
applied to the discrete Fourier transformation. Stretched coordinates are
introduced to obtain the max-plus transformation whose imaginary part coinsides
with a phase of the discrete Fourier transformation
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.
Two-dimensional soliton cellular automaton of deautonomized Toda-type
A deautonomized version of the two-dimensional Toda lattice equation is
presented. Its ultra-discrete analogue and soliton solutions are also
discussed.Comment: 11 pages, LaTeX fil
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