# 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} }

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

#### Figures and Tables from this paper

#### 4 Citations

Another generalization of the box-ball system with many kinds of balls

- Mathematics, Physics
- 2018

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 special… Expand

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 bound… Expand

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 describe… Expand

#### References

SHOWING 1-10 OF 43 REFERENCES

Box-Ball Systems and Robinson-Schensted-Knuth Correspondence

- Mathematics, Physics
- 2001

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 the… Expand

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 and… Expand

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 that… Expand

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 equation… Expand

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-particle… Expand

Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for
$\mathfrak{sl}_{n}$
case

- Mathematics, Physics
- 2008

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 paths… Expand

Rigged Configurations and Catalan, Stretched Parabolic Kostka Numbers and Polynomials: Polynomiality, Unimodality and Log-concavity

- Mathematics
- 2015

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 higher… Expand

Rigged Configurations and the Bethe Ansatz

- Mathematics, Physics
- 2003

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 the… Expand