Yang–baxter Maps and the Discrete Kp Hierarchy

Abstract

Abstract. We present a systematic construction of the discrete KP hierarchy in terms of Sato-Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang-Baxter maps is explained in two explicit examples.2000 Mathematics Subject Classification. 39A10, 39A12. Introduction.A problem widely recognized as one of the most fundamental open problems in the theory of integrable systems is that of the relation between classical and quantum integrable systems with infinitely many degrees of freedom. 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. These are such that (quantum) crystal bases can be obtained from them through a limiting procedure very much like the ultradiscrete limit. Geometric crystals have been studied extensively in connection with the combinatorial properties of BBSs [10, 11], but, most importantly, they also offer prime examples of Yang-Baxter (YB) maps [7], i.e. of set-theoretical solutions to the YB equation [6,26]. In particular, it can be shown that the R-matrix associated to such a YB map (or with the tropical version thereof, obtained from the geometric crystal) in the ultradiscrete limit turns into the combinatorial R-matrix that governs the time evolution of the BBS [8]. This R-matrix, in turn, is directly related to the crystal associated with that BBS. The question that remains however,

Linearization of the box–ball system: an elementary approach

“…One time step in the evolution corresponds to the carrier moving through all boxes, from left to right, which gives rise to the equations (3.1). The rule (3.1) can be obtained from the ultra-discrete limit of a modified discrete KdV equation [TM,KNW1]. It is equivalent to a combinatorial R-matrix of A (1) 1 -type [NY] as shown in [HHIKTT].…”

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