Toda-type cellular automaton and its N-soliton solution

Matsukidaira, Junta; Satsuma, Junkichi; Takahashi, Daisuke; Tokihiro, Tetsuji; Torii, Makoto

doi:10.1016/s0375-9601(96)00899-7

Cited by 93 publications

(93 citation statements)

References 16 publications

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“…with Q l,m,n related to τ l,m,n by the transformation V l,m,n = ∆ l,m,n as [2] 6) where each of the functions {f…”

Section: The Fully Discrete 2d Toda Lattice Equationmentioning

confidence: 99%

“…A general method to obtain the SCA from discrete soliton equations was proposed in Refs. [6,18] and involves using an appropriate limiting procedure. Another issue which has received renewed interest in recent years is the phenomenon of soliton resonance, which was first discovered for the Kadomtsev-Petviashvili (KP) equation [8] (see also Refs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

Maruno

Biondini

2004

J. Phys. A: Math. Gen.

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“…with Q l,m,n related to τ l,m,n by the transformation V l,m,n = ∆ l,m,n as [2] 6) where each of the functions {f…”

Section: The Fully Discrete 2d Toda Lattice Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

Maruno

Biondini

2004

J. Phys. A: Math. Gen.

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“…As we have shown, GBCA can be expressed by (10). As (10) can be transformed into ultra-discrete diffusion type equation, asymptotic behavior of GBCA is similar to that of BCA.…”

Section: Fundamental Diagramsmentioning

confidence: 86%

Euler–lagrange Correspondence of Generalized Burgers Cellular Automaton

Matsukidaira

Nishinari

2004

Int. J. Mod. Phys. C

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Recently, we have proposed a 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.

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“…In recent years, many studies on ultradiscrete integrable systems, or integrable cellular automata have intensively been made by using various mathematical tools such as combinatorics [24,3,37,31], ultradiscretization procedure [35,27,34,14], crystal bases of quantum groups [9,7,8,11], and so forth. In particular, among many studies on geometric aspects of ultradiscrete integrable systems, several remarkable works have successively been made by Kimijima-Tokihiro [21], Inoue-Takenawa [12,13], and Iwao [17] in terms of tropical geometry and ultradiscretization procedure.…”

Section: Introductionmentioning

confidence: 99%

A geometric realization of the periodic discrete Toda lattice and its tropicalization

Nobe¹

2013

J. Phys. A: Math. Theor.

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An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the periodic discrete Toda lattice, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the periodic discrete Toda lattice. Finally, the formulation is lifted to the framework of tropical geometry, and a tropical geometric realization of the periodic box-ball system is constructed via tropical curve intersections.

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Toda-type cellular automaton and its N-soliton solution

Cited by 93 publications

References 16 publications

Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

Euler–lagrange Correspondence of Generalized Burgers Cellular Automaton

A geometric realization of the periodic discrete Toda lattice and its tropicalization

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