Tetsuji Tokihiro

The AM(1) automata related to crystals of symmetric tensors

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) 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 Uq′(AM−1(1)). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete Kadomtsev–Petviashivili equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) 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 Uq′(AM−1(1)). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete Kadomtsev–Petviashivili equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.

Toda-type cellular automaton and its N-soliton solution

Abstract We show that the cellular automaton proposed by two of the authors (D.T. and J.M.) is obtained from the discrete Toda lattice equation through a special limiting procedure. Also by applying a similar kind of limiting procedure to the N -soliton solution of the discrete Toda lattice equation, we obtain the N -soliton solution for this cellular automaton.

Quantum affine symmetry in vertex models

We study the higher spin analogs of the six-vertex model on the basis of its symmetry under the quantum affine algebra . Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/ annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin 1/2, and that the n-particle space has an RSOS type structure rather than a simple tensor product of the one-particle space. This agrees with the picture proposed earlier by Reshetikhin.

Eigenstates in 2-Dimensional Penrose Tiling

We find a systematic procedure constructing the 2D and 3D “ periodic ” Penrose tilings, which tend to the infinite Penrose tilings. The electronic states in the 2D Penrose tiling are studied by using this sequence. We observe that the spectral measure is singular continuous in the limit of the infinite size. Most of eigenstates are critical, i.e. neither extended nor localized.

PROOF OF SOLITONICAL NATURE OF BOX AND BALL SYSTEMS BY MEANS OF INVERSE ULTRA-DISCRETIZATION

A soliton cellular automaton, which represents movement of a finite number of balls in an array of boxes, is investigated. Its dynamics is described by an ultra-discrete equation obtained from an extended Toda molecule equation. The rules for soliton interactions and factorization property of the scattering matrices (Yang-Baxter relation) are proved by means of inverse ultra-discretization. The conserved quantities are also presented and used for another proof of the solitonical nature.

Stochastic optimal velocity model and its long-lived metastability.

In this paper, we propose a stochastic cellular automaton model of traffic flow extending two exactly solvable stochastic models, i.e., the asymmetric simple exclusion process and the zero range process. Moreover, it is regarded as a stochastic extension of the optimal velocity model. In the fundamental diagram (flux-density diagram), our model exhibits several regions of density where more than one stable state coexists at the same density in spite of the stochastic nature of its dynamical rule. Moreover, we observe that two long-lived metastable states appear for a transitional period, and that the dynamical phase transition from a metastable state to another metastable/stable state occurs sharply and spontaneously.

Box and ball system as a realization of ultradiscrete nonautonomous KP equation

Cellular automata, which are realized by dynamics of several kinds of balls in an infinite array of boxes, are investigated. They show soliton patterns even in the case when each box has arbitrary capacity. The analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete KP equation.

Fundamental cycle of a periodic box?ball system

We investigate a box–ball system with periodic boundary conditions. Since the box–ball system is a deterministic dynamical system that takes only a finite number of states, it will exhibit periodic motion. We determine its fundamental cycle for a given initial state.

CONSTRUCTING SOLUTIONS TO THE ULTRADISCRETE PAINLEVE EQUATIONS

We investigate the nature of particular solutions to the ultradiscrete Painleve equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painleve equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painleve equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.

Darboux and binary Darboux transformations for the nonautonomous discrete KP equation

It is shown how Darboux and binary Darboux transformations for a nonautonomous discrete KP equation can be obtained from fermion analysis. This equation is obtained by considering a generalized Miwa transformation; it is also shown to be linked to the discrete KP equation by a special gauge transformation. The Darboux and binary Darboux transformations are used to discuss general classes of solutions in the form of Casorati- and Gramm-type determinants. N-soliton solutions are discussed as well.