Box and ball system with a carrier and ultradiscrete modified KdV equation

Abstract

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 obtained from discrete modified Korteweg-de Vries equation through a limit. Algebraic expression of soliton solutions is also proposed.

Integrable structure of box–ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

“….. The spins on the horizontal edges are "hidden variables" playing the role of carrier [73]. In the argument so far, one starts with q-dependent objects, e.g.…”

Box-ball systems related to the nonautonomous ultradiscrete Toda equation on the finite lattice

“…In this section, we discuss the well-known BBS with a carrier [8], which we call the BBS with speed limits in this letter. Fig.…”

“…In order to derive particular solutions, we introduce bilinear equations related to (5) and (8). The following theorem is proved by using the Plücker relation.…”

Yang–baxter Maps and the Discrete Kp Hierarchy

“…However, not only is the discrete system obtained from this reduction associated with a YB map, but it is also ultradiscretizable for an appropriate choice of the parameters (so as to ensure positivity). In fact, its ultradiscrete limit can be shown to be equivalent to the BBS with carrier introduced in [20], which can be interpreted as the soliton cellular automaton that corresponds to the crystal obtained from the symmetric tensor representation of U q (A (1) 1 ) in the limit q → 0 [11].…”

“…Recently a possible route emerged through which a firm connection between the classical and quantum settings might be established: the ultradiscretization of (classical) discrete integrable systems [8,22]. Crystal bases, arising in the zero-temperature limit of quantum enveloping algebras, have been shown to play an important rôle in the description of the dynamical properties of so-called box and ball systems (BBSs) [20], which in turn can be obtained from 1+1-dimensional discrete integrable systems through a special limiting procedure: the ultradiscrete limit [22]. On the other hand, geometric crystals [1] are classical analogues of such crystals.…”