Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAnother generalization of the box-ball system with many kinds of balls
126
0
0.0
(
0
)
Authors
Kazuki Maeda
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
We propose a new integrable generalization of the Toda lattice wherein the original Flaschka-Manakov variables are coupled to newly introduced dependent variables; the general case wherein the additional dependent variables are vector-valued is considered. This generalization admits a Lax pair based on an extension of the Jacobi operator, an infinite number of conservation laws and, in a special case, a simple Hamiltonian structure. In fact, the second flow of this generalized Toda hierarchy reduces to the usual Toda lattice when the additional dependent variables vanish; the first flow of the hierarchy reduces to a long wave-short wave interaction model, known as the Yajima-Oikawa system, in a suitable continuous limit. This integrable discretization of the Yajima-Oikawa system is essentially different from the discrete Yajima-Oikawa system proposed in arXiv:1509.06996 (also see https://link.aps.org/doi/10.1103/PhysRevE.91.062902) and studied in arXiv:1804.10224. Two integrable discretizations of the nonlinear Schrodinger hierarchy, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are contained in the generalized Toda hierarchy as special cases.
The Box-Ball System, shortly BBS, was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size $k$ solitons in each $k$-slot. The dynamics of the components is linear: the $k$-th component moves rigidly at speed $k$. Let $zeta$ be a translation invariant family of independent random vectors under a summability condition and $eta$ the ball configuration with components $zeta$. We show that the law of $eta$ is translation invariant and invariant for the BBS. This recipe allows us to construct a big family of invariant measures, including product measures and stationary Markov chains with ball density less than $frac12$. We also show that starting BBS with an ergodic measure, the position of a tagged $k$-soliton at time $t$, divided by $t$ converges as $ttoinfty$ to an effective speed $v_k$. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws.
The box-ball systems are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. In this paper, we consider multicolor box-ball system with two types of random initial configuration and obtain the scaling limit of the soliton lengths as the system size tends to infinity. Our analysis is based on modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.
In the present paper, we propose a two-component generalization of the reduced Ostrovsky equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and $N$-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Backlund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. Especially, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their $N$-soliton solutions in terms of pffafians are also provided.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
