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.
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
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
Multilinear Operators: The Natural Extension Of Hirota's Bilinear Formalism
We introduce multilinear operators, that generalize Hirota's bilinear
operator, based on the principle of gauge invariance of the functions.
We show that these operators can be constructed systematically using the
bilinear 's as building blocks. We concentrate in particular on the
trilinear case and study the possible integrability of equations with one
dependent variable. The 5th order equation of the Lax-hierarchy as well as
Satsuma's lowest-order gauge invariant equation are shown to have simple
trilinear expressions. The formalism can be extended to an arbitrary degree of
multilinearity.Comment: 9 pages in plain Te
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.
Two-dimensional Burgers Cellular Automaton
A two-dimensional cellular automaton(CA) associated with a two-dimensional
Burgers equation is presented. The 2D Burgers equation is an integrable
generalization of the well-known Burgers equation, and is transformed into a 2D
diffusion equation by the Cole-Hopf transformation. The CA is derived from the
2D Burgers equation by using the ultradiscrete method, which can transform
dependent variables into discrete ones. Some exact solutions of the CA, such as
shock wave solutions, are studied in detail.Comment: Latex2.09, 17 pages including 7 figure
Ablowitz-Ladik system with discrete potential. I. Extended resolvent
Ablowitz-Ladik linear system with range of potential equal to {0,1} is
considered. The extended resolvent operator of this system is constructed and
the singularities of this operator are analyzed in detail.Comment: To be published in Theor. Math. Phy
The A^{(1)}_M automata related to crystals of symmetric tensors
A soliton cellular automaton associated with crystals of symmetric tensor
representations of the quantum affine algebra U'_q(A^{(1)}_M) is introduced. It
is a crystal theoretic formulation of the generalized box-ball system in which
capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering
matrices of two solitons coincide with the combinatorial R matrices of
U'_q(A^{(1)}_{M-1}). A piecewise linear evolution equation of the automaton is
identified with an ultradiscrete limit of the nonautonomous discrete KP
equation. A class of N soliton solutions is obtained through the
ultradiscretization of soliton solutions of the latter.Comment: 45 pages, latex2e, 2 figure
Some properties of the k-dimensional Lyness' map
This paper is devoted to study some properties of the k-dimensional Lyness'
map. Our main result presentes a rational vector field that gives a Lie
symmetry for F. This vector field is used, for k less or equal to 5 to give
information about the nature of the invariant sets under F. When k is odd, we
also present a new (as far as we know) first integral for F^2 which allows to
deduce in a very simple way several properties of the dynamical system
generated by F. In particular for this case we prove that, except on a given
codimension one algebraic set, none of the positive initial conditions can be a
periodic point of odd period.Comment: 22 pages; 3 figure
Dynamical Reduction of Discrete Systems Based on the Renormalization Group Method
The renormalization group (RG) method is extended for global asymptotic
analysis of discrete systems. We show that the RG equation in the discretized
form leads to difference equations corresponding to the Stuart-Landau or
Ginzburg-Landau equations. We propose a discretization scheme which leads to a
faithful discretization of the reduced dynamics of the original differential
equations.Comment: LaTEX. 12pages. 1 figure include