# Linearization of the box-ball system: an elementary approach

@article{Kakei2017LinearizationOT,
title={Linearization of the box-ball system: an elementary approach},
author={Saburo Kakei and Jonathan J. C. Nimmo and Satoshi Tsujimoto and Ralph Willox},
journal={arXiv: Exactly Solvable and Integrable Systems},
year={2017}
}
• Saburo Kakei, +1 author R. Willox
• Published 28 September 2017
• Physics, Mathematics
• arXiv: Exactly Solvable and Integrable Systems
Kuniba, Okado, Takagi and Yamada have found that the time-evolution of the Takahashi-Satsuma box-ball system can be linearized by considering rigged configurations associated with states of the box-ball system. We introduce a simple way to understand the rigged configuration of $\mathfrak{sl}_2$-type, and give an elementary proof of the linearization property. Our approach can be applied to a box-ball system with finite carrier, which is related to a discrete modified KdV equation, and also to… Expand
4 Citations

#### Figures and Tables from this paper

Another generalization of the box-ball system with many kinds of balls
A cellular automaton that is a generalization of the box-ball system with either many kinds of balls or finite carrier capacity is proposed and studied through two discrete integrable systems:Expand
A Uniform Approach to Soliton Cellular Automata Using Rigged Configurations
• Physics, Mathematics
• Annales Henri Poincaré
• 2019
For soliton cellular automata, we give a uniform description and proofs of the solitons, the scattering rule of two solitons, and the phase shift using rigged configurations in a number of specialExpand
Darboux dressing and undressing for the ultradiscrete KdV equation
• Physics, Mathematics
• Journal of Physics A: Mathematical and Theoretical
• 2019
We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing boundExpand
Generalized Hydrodynamic Limit for the Box–Ball System
• Physics, Mathematics
• 2020
We deduce a generalized hydrodynamic limit for the box-ball system, which explains how the densities of solitons of different sizes evolve asymptotically under Euler space-time scaling. To describeExpand

#### References

SHOWING 1-10 OF 43 REFERENCES
Box-Ball Systems and Robinson-Schensted-Knuth Correspondence
We study a box-ball system from the viewpoint of combinatorics of words and tableaux. Each state of the box-ball system can be transformed into a pair of tableaux (P, Q) by theExpand
Integrable structure of box–ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
• Physics, Mathematics
• 2012
The box–ball system is an integrable cellular automaton on a one-dimensional lattice. It arises from either quantum or classical integrable systems by procedures called crystallization andExpand
The box?ball system and the N-soliton solution of the ultradiscrete KdV equation
• Mathematics
• 2008
Any state of the box–ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown thatExpand
Bethe ansatz and inverse scattering transform in a periodic box–ball system
• Physics, Mathematics
• 2006
Abstract We formulate the inverse scattering method for a periodic box–ball system and solve the initial value problem. It is done by a synthesis of the combinatorial Bethe ansatzes at q = 1 and q =Expand
LETTER TO THE EDITOR: Box and ball system with a carrier and ultradiscrete modified KdV equation
• Mathematics
• 1997
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 equationExpand
A bijection between Littlewood-Richardson tableaux and rigged configurations
• Mathematics
• 1999
Abstract. We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particleExpand
Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for $\mathfrak{sl}_{n}$ case
Abstract In proving the Fermionic formulae, a combinatorial bijection called the Kerov–Kirillov–Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest pathsExpand
Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Log-concavity
We will look at the Catalan numbers from the {\it Rigged Configurations} point of view originated \cite{Kir} from an combinatorial analysis of the Bethe Ansatz Equations associated with the higherExpand
Rigged Configurations and the Bethe Ansatz
These notes arose from three lectures presented at the Summer School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland, on September 11-18,Expand
The AM(1) automata related to crystals of symmetric tensors
• Mathematics, Physics
• 1999
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 theExpand