Shinsuke Iwao
We study a 2-dimensional Box-Ball system which is a ultradiscrete analog of the discrete KP equation. We construct an algorithm to calculate the fundamental cycle, which is an important conserved quantity of the 2-dim. Box-Ball system with periodic boundary condition, by using the tropical curve theory.
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Using the whurl relation of the first two authors, we define a new discrete solitonic system, which we call the box-basket-ball system, generalizing the box-ball system of Takahashi and Satsuma. In box-basket-ball systems balls may be put either into boxes or into baskets. While boxes stay fixed, both balls and baskets get moved during time evolution. Balls and baskets behave as fermionic and bosonic particles respectively. We classify the solitons of this system, and study their scattering.
A connection between the finite ultradiscrete Toda lattice and the box-ball system is extended to the case where each box has own capacity and a carrier has a capacity parameter depending on time. In order to consider this connection, new carrier rules "size limit for solitons" and "recovery of balls", and a concept "expansion map" are introduced. A particular solution to the extended system of a special case is also presented.
The box-ball system is studied 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 Robinson-Schensted-Knuth correspondence. In the language of tableaux, the $P$-symbol gives rise to a conserved quantity of the box-ball system, and the $Q$-symbol evolves independently of the $P$-symbol. The time evolution of the $Q$-symbol is described explicitly in terms of the box-labels.
Linearization of the box-ball system with box capacity L
本文通过构造盒容量为L的盒球系统状态与一对序列之间的双射，将其中一个序列的运动与容量为1的盒球系统规则相结合，并利用KKR双射实现线性化，从而推广到包含负值或超出盒容量值的状态。
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: nonautonomous discrete KP lattice and nonautonomous discrete two-dimensional Toda lattice. Applying reduction technique and ultradiscretization procedure to these discrete systems, we derive two types of time evolution equations of the proposed cellular automaton, and particular solutions to the ultradiscrete equations.
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 the combinatorial $R$-matrix of $A_1^{(1)}$-type. We also discuss combinatorial statistics and relate…
A nonautonomous version of the ultradiscrete hungry Toda lattice with a finite lattice boundary condition is derived by applying reduction and ultradiscretization to a nonautonomous two-dimensional discrete Toda lattice. It is shown that the derived ultradiscrete system has a direct connection to the box-ball system with many kinds of balls and finite carrier capacity. Particular solutions to the ultradiscrete system are constructed by using the theory of some sort of discrete biorthogonal polynomials.
RSK tableaux and box-ball systems
该研究聚焦盒球系统这一离散动力系统，证明排列的孤子分解在特定条件下与RS插入表相等，并借助行读词等工具研究系统性质。
Vertex models with quantum group symmetry give rise to integrable cellular automata at q=0. We study a prototype example known as the periodic box-ball system. The initial value problem is solved in terms of an ultradiscrete analogue of the Riemann theta function whose period matrix originates in the Bethe ansatz at q=0.
Duality between box-ball systems of finite box and/or carrier capacity
本文研究了无限初始配置下BBS(J,K)系统动力学，并证明其与BBS(K,J)系统对偶，通过Pitman型变换和详平衡方程分析了不变测度和标记粒子速度的对偶关系。
The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in comple…
We reformulate the Kerov-Kirillov-Reshetikhin (KKR) map in the combinatorial Bethe ansatz from paths to rigged configurations by introducing local energy distribution in crystal base theory. Combined with an earlier result on the inverse map, it completes the crystal interpretation of the KKR bijection for U_q(\hat{sl}_2). As an application, we solve an integrable cellular automaton, a higher spin generalization of the periodic box-ball system, by an inverse scattering method and obtain the solution of the initial value problem in terms of the ultradiscrete Riemann theta function.
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 ansa"tze at q=1 and q=0, which provides the ultradiscrete analogue of quasi-periodic solutions in soliton equations, e.g., action-angle variables, Jacobi varieties, period matrices and so forth. As an application we establish explicit formulas counting the states characterized by conserved quantities and the generic and fundamental period under the commuting family of time evolutions.
We analyse the concept of generalized Bose-Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in the Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the Off-Diagonal-Long-Range-Order (ODLRO), which are usually considered as some adequate way to describe the standard BEC on the ground state for the cubic boxes. First we show that these three criterions are equivalent in this latter case. Then, basing on a scaling approach, we revise formu- lation of these concepts to prove tha…
Box-ball system (BBS) is a prominent example of integrable cellular automata in one dimension connected to quantum groups, Bethe ansatz, ultradiscretization, tropical geometry and so forth. In this paper we study the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics. The results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions …
A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation
研究通过对可积的时滞离散 Lotka–Volterra 方程超离散化，构造箱球系统的时滞类比，并分类、数值发现及解析刻画其新型异常孤子。
Generalized hydrodynamic limit for the box-ball system
研究箱球系统在欧拉空间时间缩放下的广义水动力极限，推导不同大小孤子密度的渐进行为，提出连续状态空间模型并建立偏微分方程描述其演化。
A soliton cellular automaton on a one dimensional semi-infinite lattice with a reflecting end is presented. It extends a box-ball system on an infinite lattice associated with the crystal base of U_q(sl_n). A commuting family of time evolutions are obtained by adapting the K matrices and the double row construction of transfer matrices in solvable lattice models to a crystal setting. Factorized scattering among the left and the right moving solitons and the boundary reflection rule are determined.
A class of periodic soliton cellular automata is introduced associated with crystals of non-exceptional quantum affine algebras. Based on the Bethe ansatz at q=0, we propose explicit formulas for the dynamical period and the size of certain orbits under the time evolution in A^{(1)}_n case.
We introduce a probability distribution on the set of states in a generalized box-ball system associated with Kirillov-Reshetikhin (KR) crystals of type $A^{(1)}_n$. Their conserved quantities induce $n$-tuple of random Young diagrams in the rigged configurations. We determine their limit shape as the system gets large by analyzing the Fermionic formula by thermodynamic Bethe ansatz. The result is expressed as a logarithmic derivative of a deformed character of the KR modules and agrees with the stationary local energy of the associated Markov process of carriers.
Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures
本文综述了基于平稳马尔可夫链的盒球系统不变测度，并引入了新的周期配置不变测度，同时讨论了其与超离散Toda方程动力学的联系。
We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group $U_q(\widehat{sl}_n)$. Compared with the conventional $(n-1)$-color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall…
The Box-Ball System (BBS) is a cellular automaton introduced by Takahashi and Satsuma in the 1990s. The system is a discrete counterpart of the KdV equation and exhibits solitonic behavior. Recently, the BBS started from a random two-sided infinite particle configuration has been studied, by encoding the particle configuration to a certain discrete path on Z and defining the dynamic of the BBS in terms of the path. In this paper, we extend some results of the previous study in this direction to a generalization of the BBS with \k{appa}-color balls (particles), called the multicolor BBS. We fir…
We explore several aspects of the current fluctuations and correlations in the box-ball system (BBS), an integrable cellular automaton in one space dimension. The state we consider is an ensemble of microscopic configurations where the box occupancies are independent random variables (i.i.d. state), with a given mean ball density. We compute several quantities exactly in such homogeneous stationary state: the mean value and the variance of the number of balls $N_t$ crossing the origin during time $t$, and the scaled cumulants generating function associated to $N_t$. We also compute two spatial…
Scaling limit of soliton lengths in a multicolor box-ball system
本文研究多色盒球系统的两类随机初始构型，通过分析相关多维载流子马尔可夫链的性质，得到系统规模趋于无穷时孤子长度的精确标度极限，结果比单色情形更精细。
Skew column RSK dynamics and the box-ball system
本文提出斜列RSK动力学作为盒球系统的二维推广，通过构造显式双射线性化时间演化，并证明其孤子行为，同时给出与经典盒球系统的投影关系。
Soliton decomposition of the Box-Ball System
本文提出盒-球系统的槽分解，将孤子动力学线性化，构造不变测度族，并证明标记孤子速度收敛于有效速度，与广义吉布斯系综相关。
Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measures
本文研究了盒球系统中座位编号配置的扩展，引入了k跳映射，并探讨了其与10消除及不变测度的关系，同时计算了载具的期望值。
Slot decomposition of continuous Box-Ball Systems
将离散盒-球系统的槽分解推广到连续情形，通过将孤子映射为二维点，证明当权重函数在L^1中时，槽分解为泊松过程。
Relationships between two linearizations of the box-ball system : Kerov-Kirillov-Reshetikhin bijection and slot configuration
本文研究了Box-ball系统两种线性化方法之间的关系，通过引入新的描述方法揭示了Kerov-Kirillov-Reshetikhin双射与槽配置之间的联系，并展示了即使在有限载具容量情况下，槽配置也能实现线性化。
Dynamics of the box-ball system with random initial conditions via Pitman's transformation
本文研究随机初始条件下盒球系统动力学，利用Pitman变换分析配置集的可定义性和可逆性，探讨分布不变性、遍历性及概率性质，并引入新的连续盒球系统模型。
A box-ball system with more than one kind of balls is obtained by the generalized periodic discrete Toda equation (pd Toda eq.). We study the pd Toda equation in view of algebraic geometry. The time evolution of pd Toda eq. is linearized on an algebraic variety, and theta function solutions are obtained.
We introduce ultradiscrete tau functions associated with rigged configurations for A^{(1)}_n. They satisfy an ultradiscrete version of the Hirota bilinear equation and play a role analogous to a corner transfer matrix for the box-ball system. As an application, we establish a piecewise linear formula for the Kerov-Kirillov-Reshetikhin bijection in the combinatorial Bethe ansatz. They also lead to general N-soliton solutions of the box-ball system.
Several differential equations usually appearing in mathematical physics are solved through a power series expansion, which reduces in solving difference equations. In this paper a probability problem is presented whose solution follows a completely reversed but systematic approach. Hence, this work is about illustrating how complex probability problems could be tackled with the more powerful techniques of a better studied and well understood field, that of differential equations. The problem is defined as follows: Inside a box containing r red and w white balls random removals occur. The ball…
Rigged configurations are known to provide action-angle variables for remarkable discrete dynamical systems known as box-ball systems. We conjecture an explicit piecewise-linear formula to obtain the shapes of a rigged configuration from a tensor product of one-row crystals. We introduce cylindric loop Schur functions and show that they are invariants of the geometric R-matrix. Our piecewise-linear formula is obtained as the tropicalization of ratios of cylindric loop Schur functions. We prove our conjecture for the first shape of a rigged configuration, thus giving a piecewise-linear formula …
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