Junta Matsukidaira

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.

Box and ball system with a carrier and ultradiscrete modified KdV equation

A new soliton cellular automaton is proposed. It is defined by an array of an infinite number of boxes, a finite number of balls and a carrier of balls. Moreover, it reduces to a discrete equation obtained from the discrete modified Korteweg - de Vries equation through a limit. An algebraic expression of soliton solutions is also proposed.

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.

On discrete soliton equations related to cellular automata

Abstract Discrete soliton equations are proposed. Features of the equations are: (1) Their variables including a dependent variable are all discrete. (2) They can be transformed into cellular automata with two-valued site values. (3) They have multi-soliton solutions.

Solutions of the Broer-Kaup System through Its Trilinear Form

Solutions of the Broer-Kaup system describing one-dimensional shallow water waves are discussed by using its trilinear form. It is shown that the soliton solutions have the interesting properties that (i) their speed is determined by a dispersion relation up to an overall constant, and (ii) there is a possibility of fusion and fission of solitons. These properties reflect the particular structure of the trilinear form.

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.

Two-Dimensional Burgers Cellular Automaton.

In this paper, 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 B...

Integrable Four-Dimensional Nonlinear Lattice Expressed by Trilinear Form

An exactly solvable four-dimensional lattice system is presented. The system is derived from a trilinear equation by applying dependent variable transformation, and its solutions can be expressed by a two-directional Casorati determinant. Higher-order equations are also discussed.

Bilinear form approach to the self-dual Yang-Mills equations and integrable systems in (2+1)-dimension

We apply Hirotas bilinear form approach to the analysis of the SU(2) self-dual Yang-Mills equations. Solutions of the bilinear forms are represented by using the Toda-molecule type determinants with new linear dispersion relations. We also discuss dimensional reductions of the bilinear forms and the (2+1)-dimensional integrable equations derived from them. These integrable equations are higher dimensional extension of the nonlinear Schroinger equation and the derivative nonlinear Schrodinger equation.

Max-plus analysis on some binary particle systems

We are concerned with a special class of binary cellular automata, i.e. the so-called particle cellular automata (PCA) in this paper. We first propose max-plus expressions to PCA of four 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.